\section{Introduction}
Group cohomology of a group $G$ with coefficients in a $G$-module $M$ is related to crosses extensions of $G$ by $M$. More precisely, equivalence classes of crossed $n$-fold extensions of $G$ by $M$ are classified by the $(n+1)$-th group cohomology $H^{n+1}(G,M)$ \cite{hueb}. A similar result for Hochschild cohomology was considered by Baues and Minian \cite{baues-minian}. Namely, they introduce crossed $n$-fold extensions of an associative algebra $A$ by an $A$-bimodule $M$ and prove that equivalence classes of such extensions are isomorphic to the $(n+1)$-th Hochschild cohomology $HH^{n+1}(A,M)$ as abelian groups. A further generalization of this result has been obtained in \cite{baues-minian-Richter}.

The cohomology of a Lie algebra $\mathfrak{g}$ with coefficients in a $\mathfrak{g}$-module $M$ is given by the Chevalley-Eilenberg cohomology. This cohomology theory controls the deformation of a given Lie algebra structure. The Chevalley-Eilenberg cohomology groups $H^\bullet_{CE} (\mathfrak{g}, M)$ are also related to special type of $L_\infty$-algebras \cite[Theorem 6.7]{baez-crans}. In this paper, we give another interpretation of Chevalley-Eilenberg cohomology in terms of crossed extensions. The idea is exactly same as Baues and Minian for associative algebra case. We introduce crossed $n$-fold extensions of a Lie algebra $\mathfrak{g}$ by a $\mathfrak{g}$-module $M$. The equivalence classes of such extensions are classified by $H^{n+1}_{CE} (\mathfrak{g}, M).$ We also find a similar result for Leibniz algebras.

 All vector spaces are over a field $\mathbb{K}$.

\section{Chevalley-Eilenberg cohomology}

Let $(\mathfrak{g},[~,~])$ a Lie algebra. A module over $\mathfrak{g}$ consists of a vector space $M$ together with a $\mathbb{K}$-bilinear map $[~,~]: \mathfrak{g} \times M \rightarrow M$ satisfying
\begin{align*}
[[x,y],m] = [x,[y,m]] - [y,[x,m]],
\end{align*}
for $x, y \in \mathfrak{g}$ and $m \in M$. It is clear that $\mathfrak{g}$ is a  module over $\mathfrak{g}$ with respect to the Lie bracket.

Given a Lie algebra $(\mathfrak{g},[~,~])$ and a module $M$, the corresponding Chevalley-Eilenberg cochain groups $\{ C^n_{CE} (\mathfrak{g}, M) \}_{n \geq 0}$ are given by $ C^0_{CE} (\mathfrak{g}, M) = M$ and $ C^n_{CE} (\mathfrak{g}, M) = Hom_{\mathbb{K}} (\wedge^n \mathfrak{g}, M)$, for $n \geq 1$. The coboundary map $\delta :  C^n_{CE} (\mathfrak{g}, M) \rightarrow  C^{n+1}_{CE} (\mathfrak{g}, M)$ is given by
\begin{align*}
(\delta m)(x) =~& [x, m],~~~ \text{ for } m \in M \text{ and } x \in \mathfrak{g},\\
(\delta f) (x_1, \ldots, x_{n+1}) =~& \sum_{i=1}^{n+1} (-1)^{i+1} [x_i, f (x_1, \ldots, \widehat{x_i}, \ldots, x_{n+1}) \\
~&+ \sum_{1 \leq i <  j \leq n+1} (-1)^{i+j} f ([x_i, x_j], x_1, \ldots, \widehat{x_i}, \ldots, \widehat{x_j}, \ldots, x_{n+1}),
\end{align*}
for $f \in C^n_{CE} (\mathfrak{g}, M)$ and $x_1, \ldots, x_{n+1} \in \mathfrak{g}$. The corresponding cohomology groups are denoted by $H^\bullet_{CE} (\mathfrak{g}, M)$ and called the Chevalley-Eilenberg cohomology of $\mathfrak{g}$ with coefficients in the module $M$.

It is easy to see that $H^0_{CE}(\mathfrak{g}, M)$ is the submodule of invariants of $M$:
\begin{align*}
H^0_{CE} (\mathfrak{g}, M) = \{ m \in M |~ [x,m] = 0, \forall x \in \mathfrak{g} \}.
\end{align*}
The first cohomology group $H^1_{CE} (\mathfrak{g}, M)$ can be seen as the space of outer derivations
\begin{align*}
H^1_{CE} (\mathfrak{g}, M) = \text{OutDer} (\mathfrak{g}, M) := \frac{\text{Der} (\mathfrak{g}, M)}{\text{InnDer} (\mathfrak{g}, M)},
\end{align*}
where a derivation is a map $f : \mathfrak{g} \rightarrow M$ satisfying $f [x,y] = [x, fy] - [y, fx]$, for all $x, y \in \mathfrak{g}$ and is called inner if $f(x) = [x, m]$, for some $m \in M$.

The second cohomology group $H^2_{CE} (\mathfrak{g}, M)$ can be described as the space of equivalence classes of Lie algebra extensions
of $\mathfrak{g}$ by the module $M$.

\medskip

\noindent \textbf{Pushout of $\mathfrak{g}$-modules.}
Pushout in a category is an important tool to study some nice properties of the category.
\begin{defn}
Let $\mathcal{C}$ be a category. Given two morphisms $f : A \rightarrow B$ and $g : A \rightarrow C$, a pushout (or fibered sum) is a triple $(D, i, j)$ where $D \in \text{Ob}(\mathcal{C})$ and $i : B \rightarrow D$ and $j : C \rightarrow D$ are morphisms in $\mathcal{C}$ such that $jg = if$ that satisfies the following universal property: for any triple $(D', i', j')$ with $j' g = i' f$, there is a unique morphism $\theta : D \rightarrow D'$ making the following diagram commute
\[
\xymatrix{
A \ar[r]^{g} \ar[d]_{f} & C & & A \ar[r]^{g} \ar[d]_{f} & C \ar[d]^{j} \ar@/^/[ddr]^{j'} &\\
B & & & B \ar[r]_{i} \ar@/_/[drr]_{i'} & D \ar@{.>}[rd]^{\theta} & \\
& & & & & D'.
}
\]

\end{defn}

Let $\mathfrak{g}$ be a Lie algebra. Consider the category $ \mathfrak{g}\textbf{-mod}$ of $\mathfrak{g}$-modules and $\mathfrak{g}$-module morphisms. Then the pushout of two maps $f : A \rightarrow B$ and $g : A \rightarrow C$ in $ \mathfrak{g}\textbf{-mod}$ exists.

Let $S = \{ (f(a), - g(a) ) |~ a \in A \}$. Then it is easy to see that $S$ is a $\mathfrak{g}$-submodule of $B \oplus C$. Take $D = \frac{B \oplus C}{S}$ and define $i : B \rightarrow D$ by $i (b) = (b,0)+ S$ and $j : C \rightarrow D$ by $j(c) = (0, c) + S$. It is easy to see that $jg = if$. Moreover, if $(D', i', j')$ is another triple with  $j'g = i' f$, we define $\theta : D \rightarrow D'$ by $\theta ((b,c)+ S) = i'(b) + j'(c)$. It is also easy to check that $\theta$ is unique. Hence the claim.

\section{Crossed modules of Lie algebras}
In this section we give an interpretation of $H^3_{CE} (\mathfrak{g}, M)$ in terms of crossed module over Lie algebras. However our notion of crossed module is different from the traditional one (see for example \cite{kassel-loday, wagemann}). Our definition is motivated from the one given by Baues and Minian for associative algebras \cite{baues-minian}.

\begin{defn}
A crossed module over a Lie algebra is a triple $(V, L, \partial)$ in which $L$ is a Lie algebra, $V$ is a $L$-module and $\partial : V \rightarrow L$ is a map of $L$-modules such that
\begin{align*}
[\partial v, w] = - [ \partial w, v ], ~~ \text{ for } v, w \in V.
\end{align*}
\end{defn}

\begin{defn}
Let $(V, L, \partial)$ and $(V', L', \partial')$ be two crossed modules. A map between them consists of a linear map $\alpha : V \rightarrow V'$ and a Lie algebra map $\beta : L \rightarrow L'$ such that the following diagram commute
\[
\xymatrix{
V \ar[r]^{\partial} \ar[d]_{\alpha} & L \ar[d]^{\beta} \\
V' \ar[r]_{\partial'} & L'
}
\]
and satisfying $\alpha [x,v] = [\beta(x), \alpha (v)],$ for $x \in L,~ v \in V.$
\end{defn}

Given a crossed module $(V , L, \partial)$, we consider $\mathfrak{g} = \text{coker}(\partial)$ and $M = \text{ker} (\partial)$. The Lie algebra structure on $L$ induces a Lie algebra structure on $\mathfrak{g}$ by $[\pi(x), \pi(y)] = \pi [x,y]$, where $\pi : L \rightarrow \mathfrak{g}$ is the projection map. Moreover, the action of $L$ on $V$ induces an action of $\mathfrak{g}$ on $M$ via $[\pi (x), m] = [x,m]$, for $m \in M$. Hence a crossed module yields an exact sequence
\begin{align*}
0 \rightarrow M \xrightarrow{i} V \xrightarrow{\partial} L \xrightarrow{\pi} \mathfrak{g} \rightarrow 0.
\end{align*}
We call $(V , L, \partial )$ a crossed module over the Lie algebra $\mathfrak{g}$ with kernel a $\mathfrak{g}$-module $M$. Two crossed modules
$(V, L, \partial)$ and $(V', L', \partial')$ over $\mathfrak{g}$ with kernel $M$ are said to be equivalent if there is a morphism of crossed modules $(V, L, \partial) \rightarrow (V', L', \partial')$ which induces identity maps on $\mathfrak{g}$ and $M$. Let $\text{Cross}(\mathfrak{g}, M)$ be the equivalence classes of such crossed modules.
In the next theorem we see that equivalence classes of crossed modules are in one-to-one correspondence with the third Chevalley-Eilenberg cohomology.

\begin{thm}\label{cross-3}
There is a bijection
\begin{align*}
\psi : \text{Cross}(\mathfrak{g}, M) \rightarrow H^3_{CE} (\mathfrak{g}, M).
\end{align*}
\end{thm}

\begin{proof}
Let
\begin{align*}
\mathcal{E} : 0 \rightarrow M \xrightarrow{i} V \xrightarrow{\partial} L \xrightarrow{\pi} \mathfrak{g} \rightarrow 0
\end{align*}
be a crossed module. Choose $\mathbb{K}$-linear sections $s : \mathfrak{g} \rightarrow L$ with $\pi s = \text{id}$ and $q : \text{Im}(\partial) \rightarrow V$ with $\partial q = \text{id}$. For any $x , y \in \mathfrak{g},$ we have
\begin{align*}
\pi ([s(x), s(y)] - s[x,y]) = 0.
\end{align*}
This shows that $[s(x), s(y)] - s[x,y]$ is in ker $(\pi) = \text{Im} (\partial)$. Take $g (x,y) = q ([s(x), s(y)] - s[x,y]) \in V$. Define a map $\theta_{\mathcal{E}, s, q} : \mathfrak{g}^{\otimes 3} \rightarrow M$ by
$$\theta_{\mathcal{E}, s, q} (x,y,z) = [s(x), g(y,z)] - [s(y), g(x,z)] + [s(z), g (x,y)]
 - g ([x,y], z) + g ([x,z], y) - g ([y,z],x).$$
Since $\partial$ is a map of $L$-modules, it follows that $\partial (\theta_{\mathcal{E}, s, q} (x,y,z)) = 0.$ Therefore, $\theta_{\mathcal{E}, s, q} (x,y,z) \in \text{ker} (\partial) = M$. The map $\theta_{\mathcal{E}, s, q} : \mathfrak{g}^{\otimes 3} \rightarrow M$ is skew-symmetric in $x, y, z$. Hence $\theta_{\mathcal{E}, s, q} : \wedge^3 \mathfrak{g} \rightarrow M$. The map $\theta_{\mathcal{E}, s, q}$ defines a $3$-cocycle in the Chevalley-Eilenberg cohomology of $\mathfrak{g}$ with coefficients in $M$.

We first show that the class of $\theta_{\mathcal{E}, s, q}$ in $H^3_{CE}(\mathfrak{g}, M)$ does not depend on the section $s$. Suppose $\overline{s} : \mathfrak{g} \rightarrow L$ is another section of $\pi$ and let $\theta_{\mathcal{E}, \overline{s}, q}$ be the corresponding $3$-cocycle defined by using $\overline{s}$ instead of $s$. Then there exists a linear map $h : \mathfrak{g} \rightarrow V$ with $s - \overline{s} = \partial h$. Observe that
\begin{align*}
[s(x), g(y,z)] - [\overline{s}(x) , \overline{g} (y,z)]
=~& [s(x) - \overline{s}(x), g (y,z)] + [ \overline{s}(x), (g - \overline{g})(y,z)] \\
=~& [\partial h (x) , q ([s(y), s(z)] - s[y,z]) ] + [ \overline{s}(x), (g - \overline{g})(y,z)] \\
=~& - [ ([s(y), s(z)] - s[y,z]), h(x) ] + [ \overline{s}(x), (g - \overline{g})(y,z)].
\end{align*}
Therefore,
\begin{align}\label{e-e'}
( \theta_{\mathcal{E}, s, q} - \theta_{\mathcal{E}, \overline{s}, q}) (x,y,z) = ~&
- [ ([s(y), s(z)] - s[y,z]), h(x) ] + [ ([s(x), s(z)] - s[x,z]), h(y) ] \\
~&- [ ([s(x), s(y)] - s[x,y]), h(z) ]  + [ \overline{s}(x), (g - \overline{g})(y,z)]  \nonumber \\
~& - [ \overline{s}(y), (g - \overline{g})(x,z)] + [ \overline{s}(z), (g - \overline{g})(x,y)] \nonumber \\
~~&- (g - \overline{g}) ([x,y],z) + (g - \overline{g}) ([x,z], y) - (g - \overline{g}) ([y,z], x). \nonumber
\end{align}

Define a map $b : \wedge^2 \mathfrak{g} \rightarrow V$ by
\begin{align*}
b (x,y) = [s(x), h(y)] - h ([x,y]) - [s(y), h(x)] - [ \partial h(x), h (y)].
\end{align*}
Then a easy calculation shows that  $\partial b = \partial (g - \overline{g})$. Hence $(g - \overline{g} - b) : \wedge^2 \mathfrak{g} \rightarrow M$. It follows from (\ref{e-e'}) that
\begin{align*}
( \theta_{\mathcal{E}, s, q} - \theta_{\mathcal{E}, \overline{s}, q}) (x,y,z) =~& - [ ([s(y), s(z)] - s[y,z]), h(x) ] + [ ([s(x), s(z)] - s[x,z]), h(y) ] \\
~& - [ ([s(x), s(y)] - s[x,y]), h(z) ] + (\delta (g - \overline{g} - b)) (x,y,z) \\
~&+ [ \overline{s}(x), b (y, z)] - [ \overline{s}(y), b (x,z)] + [\overline{s}(z) , b (x,y)] \\
~&- b ([x,y],z) + b  ([x,z], y) - b ([y,z], x).
\end{align*}
In the right hand side of the above equation, we substitute the definition of $b$ in last six terms. After many cancellations on the right hand side (one has to use the fact that $(V , L, \partial)$ is a crossed module for some cancellations), we are only left with the term $(\delta (g - \overline{g} - b)) (x,y,z)$. Hence the class of $\theta_{\mathcal{E}, s, q}$ does not depend on the section $s$.

\medskip

Next consider a map
\[
\xymatrix{
\mathcal{E}:= \quad  0 \ar[r] & M \ar[r]^{i} \ar@{=}[d] & V \ar[r]^{\partial} \ar[d]_{\alpha} & L \ar[r]^{\pi} \ar[d]^{\beta} & \mathfrak{g} \ar[r] \ar@{=}[d] & 0 \\
\mathcal{E}' := \quad  0 \ar[r] & M \ar[r]_{i'} & V' \ar[r]_{\partial'} & L' \ar[r]_{\pi'} & \mathfrak{g} \ar[r] & 0
}
\]
of crossed modules. Let $s' : \mathfrak{g} \rightarrow L'$ and $q' : \text{Im} (\partial') \rightarrow V'$ be sections of $\pi'$ and $\partial'$, respectively.
Note that $(\pi' \beta s)(x) = (\pi s) (x) = x$, for all $x \in \mathfrak{g}$. Therefore, $\beta s : \mathfrak{g} \rightarrow L'$ is another section of $\pi'$. Thus, we have
\begin{align*}
(\theta_{\mathcal{E}, s, q} &- \theta_{\mathcal{E}', \beta s, q'} ) (x,y,z) \\
= ~&[s(x), g(y,z)] - [s(y), g(x,z)] + [s(z), g (x,y)]  - g ([x,y], z) + g ([x,z], y) - g ([y,z],x) \nonumber \\
~-& [\beta s(x), g'(y,z)] + [\beta s(y), g'(x,z)] - [\beta s(z), g' (x,y)]  + g' ([x,y], z) - g' ([x,z], y) - g' ([y,z],x), \nonumber
\end{align*}
where $g'(x,y) = q' ([\beta s (x), \beta s (y)] - \beta s [x,y])$. Here we have used the same notation $[~,~]$ to denote the action of $L$ on $V$ and the action on $L'$ on $V'$. Hence, we have
\begin{align*}
(\theta_{\mathcal{E}, s, q} &- \theta_{\mathcal{E}', \beta s, q'} ) (x,y,z) \\
=~& [\beta s (x), (\alpha q - q' \beta) ([s(y), s(z)] - s[y,z]) ]
- [\beta s (y), (\alpha q - q' \beta) ([s(x), s(z)] - s[x,z]) ] \\
~&+ [\beta s (z), (\alpha q - q' \beta) ([s(x), s(y)] - s[x,y]) ]
- (\alpha q - q' \beta) \big(  [  s[x,y], s(z)] - s[[x,y],z]  \big) \\
~&+ (\alpha q - q' \beta) \big(  [  s[x,z], s(y)] - s[[x,z],y]  \big)
- (\alpha q - q' \beta) \big(  [  s[y,z], s(x)] - s[[y,z],x]  \big).
\end{align*}
It follows from the above expression that $(\theta_{\mathcal{E}, s, q} - \theta_{\mathcal{E}', \beta s, q'} ) (x,y,z) = (\delta \phi)(x,y,z)$, for some $\phi : \wedge^2 \mathfrak{g} \rightarrow M$. Hence, $[\theta_{\mathcal{E}, s, q}] = [\theta_{\mathcal{E}', \beta s, q'}]$ in $H^3_{CE} (\mathfrak{g}, M)$. Moreover, from the first part, we have $[\theta_{\mathcal{E}', \beta s, q'}] = [\theta_{\mathcal{E}',  s', q'}]$. Hence the class $[\theta_{\mathcal{E}, s, q}]$ does not depend on the sections $s$ and $q$. We denote the corresponding class by $[\theta_\mathcal{E}]$. Therefore, the map
\begin{align*}
 \psi : \text{Cross} (\mathfrak{g}, M) \rightarrow H^3_{CE} (\mathfrak{g}, M), ~ \mathcal{E} \rightarrow [\theta_\mathcal{E}]
\end{align*}
is well-defined. The surjectivity of $\psi$ follows from the next observation.
\end{proof}

Let $\mathfrak{g}$ be a Lie algebra and $0 \rightarrow M \rightarrow M' \xrightarrow{\pi} M'' \rightarrow 0$ an exact sequence of $\mathfrak{g}$-modules. Given an abelian extension
\begin{align}\label{abel-extn}
0 \rightarrow M'' \rightarrow \mathfrak{e} \rightarrow \mathfrak{g} \rightarrow 0
\end{align}
of $\mathfrak{g}$ by $M''$, we consider the Yoneda product
\begin{align}\label{yoneda}
 0 \rightarrow M \rightarrow M' \xrightarrow{\mu} \mathfrak{e} \rightarrow \mathfrak{g} \rightarrow 0.
\end{align}
Writing $\mathfrak{e} = M'' \oplus \mathfrak{g}$ as a vector space, we have $\mu (m') = (\pi (m'), 0)$. An $\mathfrak{e}$-action on $M'$ is induced from the $\mathfrak{g}$-action on $M'$, namely,
\begin{align*}
[ (m'' , x) , m' ] = [x, m' ], ~~~~ \text{ for } (m'' , x) \in \mathfrak{e} \text{ and } m' \in M'.
\end{align*}
It is easy to see that (\ref{yoneda}) defines a crossed module of $\mathfrak{g}$ by $M$. It has been shown in \cite{wagemann} that the corresponding third cohomology class in $H^3_{CE} (\mathfrak{g}, M)$ as constructed above is the image of the second cohomology class in $H^2_{CE} (\mathfrak{g}, M'')$ (defined by the abelian extension (\ref{abel-extn})) under the connecting homomorphism $\partial$ in the cohomology long exact sequence
\begin{align*}
\cdots \rightarrow H^2_{CE} (\mathfrak{g}, M) \rightarrow H^2_{CE} (\mathfrak{g}, M') \rightarrow H^2_{CE} (\mathfrak{g}, M'') \xrightarrow{\partial} H^3_{CE} (\mathfrak{g}, M) \rightarrow \cdots .\\
\end{align*}

\noindent {\em Surjectivity of the map $\psi$ (of Theorem \ref{cross-3}).} The surjectivity of $\psi$ can be shown as follows \cite{wagemann}. As the category $\mathfrak{g}\text{-mod}$ possesses enough injectives, we can choose an injective $\mathfrak{g}$-module $I$ and a monomorphism $i : M \rightarrow I$. Consider the short exact sequence of $\mathfrak{g}$-modules
\begin{align}\label{inj-short}
0 \rightarrow M \xrightarrow{i} I \rightarrow M'' \rightarrow 0
\end{align}
where $M''$ is the cokernel of the map $i$. Since $I$ is injective, the cohomology long exact sequence yields $H^2_{CE} (\mathfrak{g}, M'') \cong H^3_{CE} (\mathfrak{g}, M).$ This isomorphism is given by the connecting homomorphism in the long exact sequence of cohomology.  Thus, a cohomology class $[\gamma] \in H^3_{CE} (\mathfrak{g}, M)$ corresponds to a class $[\alpha] \in H^2_{CE} (\mathfrak{g}, M'')$. We now consider an abelian extension $0 \rightarrow M'' \rightarrow \mathfrak{e}  \rightarrow \mathfrak{g} \rightarrow 0$ corresponding to the cohomology class $[\alpha] \in H^2_{CE} (\mathfrak{g}, M'')$ . It follows from the above discussion that the Yoneda product of (\ref{inj-short}) and the above abelian extension  gives rise to a crossed module whose associated third cohomology class is given by $[\gamma]$.

\section{Crossed $n$-fold extensions of Lie algebras}\label{sec-4}
Using the notion of crossed module of previous section, we introduce crossed $n$-fold extensions of a Lie algebra $\mathfrak{g}$ by a module $M$. We show that there is an abelian group structure on $\text{Opext}^n(\mathfrak{g}, M)$ of equivalence classes of crossed $n$-fold extensions of $\mathfrak{g}$ by $M$, for $n \geq 2$. The construction is similar to the case of associative algebra \cite{baues-minian}. Finally, we show that such extensions represent cohomology classes in $H^{n+1}_{CE} (\mathfrak{g}, M).$

Let $\mathfrak{g}$ be a Lie algebra and $M$ be a $\mathfrak{g}$-module. Let $n \geq 2$.

\begin{defn}
 A crossed $n$-fold extension of $\mathfrak{g}$ by $M$ is an exact sequence
\begin{align*}
0 \rightarrow M \xrightarrow{f} M_{n-1} \xrightarrow{\partial_{n-1}} \cdots \cdot \xrightarrow{\partial_2} M_1  \xrightarrow{\partial_1} L \xrightarrow{\pi} \mathfrak{g} \rightarrow 0
\end{align*}
of $\mathbb{K}$-vector spaces with the following properties:
\begin{itemize}
\item $(M_1 , L , \partial_1)$ is a crossed module with coker$(\partial_1) = \mathfrak{g},$
\item for $1 < i \leq n-1$, $M_i$ is a $\mathfrak{g}$-module and $\partial_i,~ f$ are morphisms of $\mathfrak{g}$-modules.
\end{itemize}
\end{defn}

\begin{defn}
Let
\begin{align*}
\mathcal{E} := \big( 0 \rightarrow M \xrightarrow{f} M_{n-1} \xrightarrow{\partial_{n-1}} \cdots \cdot \xrightarrow{\partial_2} M_1  \xrightarrow{\partial_1} L \xrightarrow{\pi} \mathfrak{g} \rightarrow 0  \big)
\end{align*}
and
\begin{align*}
\mathcal{E}' := \big( 0 \rightarrow M' \xrightarrow{f'} M'_{n-1} \xrightarrow{\partial'_{n-1}} \cdots \cdot \xrightarrow{\partial'_2} M'_1  \xrightarrow{\partial'_1} L' \xrightarrow{\pi'} \mathfrak{g} \rightarrow 0  \big)
\end{align*}
be two crossed $n$-fold extensions of $\mathfrak{g}$ by $M$ and $M'$, respectively. A morphism between them consists of maps $\alpha : M \rightarrow M'$, $\beta : L \rightarrow L'$ and $\delta_i : M_i \rightarrow M_i'$ for $1 \leq i \leq n-1$ such that
\begin{itemize}
\item all squares commute (see figure (\ref{cross-n-mor})),
\item $\alpha,~ \delta_i$ ($2 \leq i \leq n-1$) are morphism of $\mathfrak{g}$-modules,
\item $(\delta_1, \beta) : (M_1 , L, \partial_1) \rightarrow (M'_1 , L', \partial_1')$ is a map of crossed modules inducing the identity map on $\mathfrak{g}.$
\end{itemize}
\begin{align}\label{cross-n-mor}
\xymatrix{
 \quad  0 \ar[r] & M \ar[r]^{f} \ar@{=}[d] & M_{n-1} \ar[r]^{\partial_{n-1}} \ar[d]_{\alpha} & M_{n-2} \ar[d]_{\delta_{n-2}} \ar[r] & \ar@{.}[r] & \ar[r] & M_1 \ar[r]^{\partial_1} \ar[d]_{\delta_1} &L \ar[r]^{\pi} \ar[d]^{\beta} & \mathfrak{g} \ar[r] \ar@{=}[d] & 0 \\
 \quad  0 \ar[r] & M \ar[r]_{f'} & M'_{n-1} \ar[r]_{\partial'_{n-1}} &M_{n-2} \ar[r] &  \ar@{.}[r] &  \ar[r] & M_1' \ar[r]_{\partial'} & L' \ar[r]_{\pi'} & \mathfrak{g} \ar[r] & 0
}
\end{align}
\end{defn}

Let $\text{Opext}^n (\mathfrak{g}, M)$ be the equivalence classes of crossed $n$-fold extensions of $\mathfrak{g}$ by $M$. Then it follows that $\text{Opext}^2(\mathfrak{g}, M) = \text{Cross} (\mathfrak{g}, M).$ In the next, we show that $\text{Opext}^n (\mathfrak{g}, M)$ has a natural abelian group structure. To do that, we start with the following proposition.

\begin{prop}\label{cross-induced-by-morphism}
Let $\mathcal{E} \in \text{Opext}^n (\mathfrak{g}, M)$ be an $n$-fold extension of $\mathfrak{g}$ by $M$ and $\alpha : M \rightarrow M'$ be an $\mathfrak{g}$-module map. Then there exists an $n$-fold extension $\alpha \mathcal{E} \in \text{Opext}^n (\mathfrak{g}, M')$ and a morphism of the form $(\alpha, \delta_{n-1}, \ldots, \delta_1, \beta)$ from $\mathcal{E}$ to $\alpha \mathcal{E}$. Moreover, $\alpha \mathcal{E} \in \text{Opext}^n (\mathfrak{g}, M')$ is the unique $n$-fold extension with this property.
\end{prop}

\begin{proof}
The proof is based on pushout of $\mathfrak{g}$-modules. Let
$\mathcal{E} := (0 \rightarrow M \xrightarrow{f} M_{n-1} \xrightarrow{\partial_{n-1}} \cdots \xrightarrow{\partial_2} M_1 \xrightarrow{\partial_1} L \xrightarrow{\pi} \mathfrak{g} \rightarrow 0) \in \text{Opext}^n (\mathfrak{g}, M)$ be a crossed $n$-fold extension. We consider the $n$-fold extension $\alpha \mathcal{E} \in \text{Opext}^n (\mathfrak{g}, M)$ as
\begin{align*}
\alpha \mathcal{E} := (   0 \rightarrow M' \rightarrow \overline{M_{n-1}} \rightarrow M_{n-2} \xrightarrow{\partial_{n-2}} \cdots \xrightarrow{\partial_2} M_1 \xrightarrow{\partial_1} L \xrightarrow{\pi} \mathfrak{g} \rightarrow 0 )
\end{align*}
where $M' \rightarrow \overline{M_{n-1}} \rightarrow M_{n-2}$ is obtained from the following pushout of $\mathfrak{g}$-modules
\[
\xymatrix{
M \ar[r] \ar[d]_{\alpha} & M_{n-1} \ar[d]^{i} \ar@/^/[ddr] &\\
M' \ar[r] \ar@/_/[drr]_0 & \overline{M_{n-1}} \ar@{.>}[rd]& \\
 & & M_{n-2}.
}
\]
Moreover $(\alpha, i, \text{id}, \ldots, \text{id})$ defines a morphism from $\mathcal{E}$ to $\alpha \mathcal{E}$.

Finally, let $\mathcal{E}' \in \text{Opext}^n (\mathfrak{g}, M')$ be an $n$-fold extension and there is a morphism $(\alpha, \delta_{n-1}, \ldots, \delta_1, \beta) : \mathcal{E} \rightarrow \mathcal{E}'$. Then by properties of pushout, one obtains a map $(1, j, \delta_{n-2}, \ldots, \delta_1, \beta) : \alpha \mathcal{E} \rightarrow \mathcal{E}'$. This shows that the class of $\alpha \mathcal{E}$ and $\mathcal{E}'$ is same in $\text{Opext}^n (\mathfrak{g}, M)$.
\end{proof}

Thus, it follows from the above proposition that a $\mathfrak{g}$-module map $\alpha : M \rightarrow M'$ induces a well-defined map
\begin{align*}
\alpha_* : \text{Opext}^n (\mathfrak{g}, M) \rightarrow \text{Opext}^n (\mathfrak{g}, M'),~ \mathcal{E} \mapsto \alpha \mathcal{E}.
\end{align*}

\begin{defn}
let $\mathcal{E} = \big( 0 \rightarrow M \xrightarrow{f} M_{n-1} \xrightarrow{\partial_{n-1}} \cdots \cdot \xrightarrow{\partial_2} M_1  \xrightarrow{\partial_1} L \xrightarrow{\pi} \mathfrak{g} \rightarrow 0  \big) \in \text{Opext}^n (\mathfrak{g}, M)$ and $\mathcal{E}' = \big( 0 \rightarrow M' \xrightarrow{f'} M'_{n-1} \xrightarrow{\partial'_{n-1}} \cdots \cdot \xrightarrow{\partial'_2} M'_1  \xrightarrow{\partial'_1} L' \xrightarrow{\pi'} \mathfrak{g} \rightarrow 0  \big) \in \text{Opext}^n (\mathfrak{g}, M)$ be two crossed $n$-fold extensions of $\mathfrak{g}$ by $M$ and $M'$, respectively. The sum of $\mathcal{E}$ and $\mathcal{E}'$ over $\mathfrak{g}$ is denoted by $\mathcal{E} \oplus_\mathfrak{g} \mathcal{E}'$ and is given by the following $n$-fold extension
\begin{align*}
0 \rightarrow M \oplus M' \rightarrow M_{n-1} \oplus M'_{n-1} \rightarrow \cdots \cdot  \rightarrow M_1 \oplus M_1' \xrightarrow{ (\partial_1, \partial'_1)} L \times_\mathfrak{g} L' \xrightarrow{q} \mathfrak{g} \rightarrow 0.
\end{align*}
Here the Lie algebra structure on $L \times_\mathfrak{g} L'$ is given by
\begin{align*}
[(x, x'), (y, y')] = ([x,y], [x',y']), ~~~ \text{ for } (x, x'), (y, y') \in L \times_\mathfrak{g} L'.
\end{align*}
The projection $q : L \times_{\mathfrak{g}} L' \rightarrow \mathfrak{g}$ is the obvious one. The action of the Lie algebra $L \times_{\mathfrak{g}} L'$ on $M_1 \oplus M_1'$ is defined coordinatewise. It is easy to check that $(M_1 \oplus M_1' ,  L \times_\mathfrak{g} L',  (\partial_1, \partial'_1))$ defines a crossed module.
\end{defn}

\begin{defn} (Baer sum) Let $\mathcal{E}, \mathcal{E}' \in \text{Opext}^n (\mathfrak{g}, M)$ with $n \geq 3$. Then the Baer sum $\mathcal{E} + \mathcal{E}' \in \text{Opext}^n (\mathfrak{g}, M)$ is defined by
\begin{align*}
\mathcal{E} + \mathcal{E}' = \nabla_M (\mathcal{E} \oplus_\mathfrak{g} \mathcal{E}')
\end{align*}
where $\nabla_M : M \oplus M \rightarrow M, ~ (m_1 , m_2) \mapsto m_1 + m_2$ is the codiagonal map.
\end{defn}

\begin{defn}\label{zero-element} (Zero extension) Let $n \geq 3$. Then
\begin{align*}
0 \rightarrow M = M \rightarrow \underbrace{0 \rightarrow \cdots \cdot \rightarrow 0}_{n-2} \rightarrow \mathfrak{g} \rightarrow \mathfrak{g} \rightarrow 0
\end{align*}
is a crossed $n$-fold extension of $\mathfrak{g}$ by $M$. We define $0 \in \text{Opext}^n (\mathfrak{g}, M)$ to be the class of this extension.
\end{defn}

\begin{remark}\label{cross-rep-zero}
Let $\mathcal{E} := ( 0 \rightarrow M \xrightarrow{f} M_{n-1} \xrightarrow{\partial_{n-1}} \cdots \xrightarrow{\partial_2} M_1 \xrightarrow{\partial_1} L \xrightarrow{\pi} \mathfrak{g} \rightarrow 0 ) \in \text{Opext}^n (\mathfrak{g}, M)$ be a crossed $n$-fold extension, for $n \geq 3$. Suppose there is a map $g : M_{n-1} \rightarrow M$ of $\mathfrak{g}$-modules satisfying $gf = \text{id}_M$. Then there is a morphism
\begin{align*}
\xymatrix{
 \quad  0 \ar[r] & M \ar[r]^{f} \ar@{=}[d] & M_{n-1} \ar[r]^{\partial_{n-1}} \ar[d]_{g} & M_{n-2} \ar[d]_{0} \ar[r]^{\partial_{n-2}} & \ar@{.}[r] & \ar[r]^{\partial_2} & M_1 \ar[r]^{\partial_1} \ar[d]_{0} &L \ar[r]^{\pi} \ar[d]^{\pi} & \mathfrak{g} \ar[r] \ar@{=}[d] & 0 \\
 \quad  0 \ar[r] & M \ar@{=}[r] & M \ar[r]_{0} & 0 \ar[r] &  \ar@{.}[r] &  \ar[r] & 0 \ar[r]_{0} & \mathfrak{g} \ar@{=}[r] & \mathfrak{g} \ar[r] & 0
}
\end{align*}
of crossed $n$-fold extensions. This shows that the class of $\mathcal{E}$ defines $0 \in \text{Opext}^n (\mathfrak{g}, M)$.
\end{remark}

\begin{remark}
Let $\mathcal{E} := ( 0 \rightarrow M \xrightarrow{f} M_{n-1} \xrightarrow{\partial_{n-1}} \cdots \xrightarrow{\partial_2} M_1 \xrightarrow{\partial_1} L \xrightarrow{\pi} \mathfrak{g} \rightarrow 0 ) \in \text{Opext}^n (\mathfrak{g}, M)$ be a crossed $n$-fold extension, for $n \geq 3$. It follows from Proposition \ref{cross-induced-by-morphism} that $f \mathcal{E} \in \text{Opext}^n (\mathfrak{g}, M_{n-1})$ and is given by the bottom row of the following diagram
\begin{align*}
\xymatrix{
 \quad  0 \ar[r] & M \ar[r]^{f} \ar[d]_{f} & M_{n-1} \ar[r]^{\partial_{n-1}} \ar[d]_{(id, \partial_{n-1})} & M_{n-2} \ar@{=}[d] \ar[r]^{\partial_{n-2}} & \ar@{.}[r] &  \ar[r]^{\partial_1}  &L \ar[r]^{\pi} \ar@{=}[d] & \mathfrak{g} \ar[r] \ar@{=}[d] & 0 \\
 \quad  0 \ar[r] & M_{n-1} \ar[r]_{(id, 0)} & M_{n-1} \oplus M_{n-2} \ar[r]_{pr_2} & M_{n-2} \ar[r] &  \ar@{.}[r] & \ar[r]_{\partial_1} & L \ar[r]_{\pi} & \mathfrak{g} \ar[r] & 0.
}
\end{align*}
Observe that, in $f \mathcal{E}$, there is a map $g = pr_1 : M_{n-1} \oplus M_{n-2} \rightarrow M_{n-1}$ which satisfies $g \circ (id, 0) = id_{M_{n-1}}$. Hence, it follows from Remark \ref{cross-rep-zero} that $f \mathcal{E} = 0$ in $\text{Opext}^n (\mathfrak{g}, M_{n-1}).$
\end{remark}

\begin{defn}\label{inverse-element} (Inverse of an extension) Let
\begin{align*}
\mathcal{E} := \big( 0 \rightarrow M \xrightarrow{f} M_{n-1} \xrightarrow{\partial_{n-1}} \cdots \cdot \xrightarrow{\partial_2} M_1  \xrightarrow{\partial_1} L \xrightarrow{\pi} \mathfrak{g} \rightarrow 0  \big) \in \text{Opext}^n (\mathfrak{g}, M)
\end{align*}
be an extension. Then
\begin{align*}
- \mathcal{E} := \big( 0 \rightarrow M \xrightarrow{- f} M_{n-1} \xrightarrow{\partial_{n-1}} \cdots \cdot \xrightarrow{\partial_2} M_1  \xrightarrow{\partial_1} L \xrightarrow{\pi} \mathfrak{g} \rightarrow 0  \big) \in \text{Opext}^n (\mathfrak{g}, M)
\end{align*}
defines a new crossed $n$-extension of $\mathfrak{g}$ by $M$.
\end{defn}

The proof of the following theorem is straightforward.
\begin{thm}\label{thm-abelian-grp}
Let $n \geq 3$. Then the set $\text{Opext}^n (\mathfrak{g}, M) $ eqipped with the Baer sum is an abelian group. The zero element and inverse elements of this group are given by Definitions \ref{zero-element} and \ref{zero-element}, respectively.

Moreover, if $\alpha : M \rightarrow M'$ is a morphism of $\mathfrak{g}$-modules, the map $\alpha_* : \text{Opext}^n (\mathfrak{g}, M) \rightarrow \text{Opext}^n (\mathfrak{g}, M')$ is a morphism of groups.
\end{thm}

\begin{remark}
The set $\text{Opext}^2 (\mathfrak{g}, M)$ also inherits the structure of an abelian group. However, one can easily feel that the structure of this abelian group must be different than the one defined in Theorem \ref{thm-abelian-grp}.

Note that an element in $\text{Opext}^2 (\mathfrak{g}, M)$ is a equivalence class of crossed modules. We denote $0 \in \text{Opext}^2 (\mathfrak{g}, M)$ to be the class of the crossed module
$ 0 \rightarrow M = M \xrightarrow{0}  \mathfrak{g} = \mathfrak{g} \rightarrow 0$. The addition in $\text{Opext}^2 (\mathfrak{g}, M)$ is given as follows. Let $\mathcal{E} := (0 \rightarrow M \xrightarrow{i} V \xrightarrow{\partial} L \xrightarrow{\pi} \mathfrak{g} \rightarrow 0)$ and
 $\mathcal{E}' := (0 \rightarrow M \xrightarrow{i'} V' \xrightarrow{\partial'} L' \xrightarrow{\pi'} \mathfrak{g} \rightarrow 0)$ be two crossed modules in $\text{Opext}^2 (\mathfrak{g}, M)$. Then the Baer sum $\mathcal{E} + \mathcal{E}'$ is the class of the extension
 \begin{align*}
 0 \rightarrow M \xrightarrow{j} V + V' \xrightarrow{\tilde{\partial}} L \times_\mathfrak{g} L' \xrightarrow{q} \mathfrak{g} \rightarrow 0.
 \end{align*}
 Here $V+ V'$ is the pushout of $\mathbb{K}$-vector spaces
 \[
\xymatrix{
M \oplus M \ar[r]^{i \oplus i'} \ar[d]_{\nabla_M} & V \oplus V' \ar[d]^{r} \ar@/^/[ddr]^{(\partial, \partial')} &\\
M \ar[r]_{j} \ar@/_/[drr]_0 & V + V' \ar@{.>}[rd]^{\tilde{\partial}} & \\
 & & L \times_\mathfrak{g} L'.
}
\]
The $(L \times_\mathfrak{g} L')$-module structure on $V + V'$ is induced from the module structure on $V \oplus V'$ via the quotient map $r : V \oplus V' \rightarrow V+ V'$. Namely, the action is given by $[(x, x'), r (v,v')] = r ([x,v], [x',v'])$, for $(x, x') \in L \times_\mathfrak{g} L'$ and $(v, v') \in V \oplus V'$. With this module structure on $V + V'$, it is easy to show that $(V + V' , L \times_{\mathfrak{g}} L', \tilde{\partial})$ is a crossed module.
\end{remark}

Next we associate a long exact sequence to any short exact sequence of $\mathfrak{g}$-modules. This will help us to prove our main theorem. Let
\begin{align*}
0 \rightarrow M \xrightarrow{\alpha} M'  \xrightarrow{\beta} M'' \rightarrow 0
\end{align*}
be a short exact sequence of $\mathfrak{g}$-modules. For any $n \geq 2$, we define a homomorphism
\begin{align*}
\delta : \text{Opext}^n (\mathfrak{g}, M'') \rightarrow \text{Opext}^{n+1} (\mathfrak{g}, M)
\end{align*}
as follows. Given an extension
\begin{align*}
\mathcal{E} = \big(  0 \rightarrow M'' \xrightarrow{f} M_{n-1} \xrightarrow{\partial_{n-1}} \cdots \cdot \xrightarrow{\partial_2} M_1 \xrightarrow{\partial_1} L \xrightarrow{\pi} \mathfrak{g} \rightarrow 0 \big) \in \text{Opext}^n (\mathfrak{g}, M''),
\end{align*}
we define $\delta \mathcal{E} \in \text{Opext}^{n+1} (\mathfrak{g}, M)$ to be the class of the extension
\begin{align*}
0 \rightarrow M \xrightarrow{\alpha} M' \xrightarrow{f \beta} & M_{n-1} \xrightarrow{\partial_{n-1}} \cdots \cdot \xrightarrow{\partial_2} M_1 \xrightarrow{\partial_1} L \xrightarrow{\pi} \mathfrak{g} \rightarrow 0.
\end{align*}
Then $\delta$ is a well-defined homomorphism  for $n \geq 2$.

The proof of the following theorem is similar to the proof of Baues and Minian for associative algebra case \cite{baues-minian}, hence we omit the details.
\begin{thm}\label{short-long-seq}
A short exact sequence
\begin{align*}
0 \rightarrow M \xrightarrow{\alpha} M'  \xrightarrow{\beta} M'' \rightarrow 0
\end{align*}
of $\mathfrak{g}$-modules induces a long exact sequence of abelian groups ($n \geq 2$)
\begin{align*}
\text{Opext}^n (\mathfrak{g}, M) \xrightarrow{\alpha_*} \text{Opext}^n (\mathfrak{g}, M') \xrightarrow{\beta_*} \text{Opext}^n (\mathfrak{g}, M'') \xrightarrow{\delta} \text{Opext}^{n+1} (\mathfrak{g}, M) \rightarrow \cdots .
\end{align*}
\end{thm}

Now we are in a position to prove our main theorem.
\begin{thm}
For any $n \geq 2$, there exists an isomorphism of abelian groups
\begin{align*}
\text{Opext}^n (\mathfrak{g}, M) \cong H^{n+1}_{CE} (\mathfrak{g}, M).
\end{align*}
\end{thm}

\begin{proof}
The result is true for $n = 2$ (see Theorem \ref{cross-3}). Let $n \geq 3$. We consider a short exact sequence
\begin{align*}
0 \rightarrow M \xrightarrow{\alpha} M'  \xrightarrow{\beta} M'' \rightarrow 0
\end{align*}
of $\mathfrak{g}$-modules with $M'$ injective. Then by Theorem \ref{short-long-seq}, it follows that
\begin{align*}
\text{Opext}^n (\mathfrak{g}, M'') \cong \text{Opext}^{n+1} (\mathfrak{g}, M).
\end{align*}
Moreover, it follows from the cohomology long exact exact sequence that $H^{n+1}_{CE} (\mathfrak{g}, M'') \cong H^{n+2}_{CE} (\mathfrak{g}, M)$. Therefore,
\begin{align*}
\text{Opext}^3 (\mathfrak{g}, M) \cong~ \text{Opext}^2 (\mathfrak{g}, M'')
\cong~& H^3_{CE} (\mathfrak{g}, M'') \qquad(\text{by Theorem }\ref{cross-3})\\
\cong~& H^4_{CE} (\mathfrak{g}, M).
\end{align*}
We conclude the result by using the induction on $n$.
\end{proof}

\begin{remark}
Lie-Rinehart algebras are algebraic analogue of Lie algebroids and closely related to Lie algebras. These algebras pay special attention due to its connection Poisson geometry \cite{hueb2}. In \cite{casas-ladra-pira} the authors studied crossed modules for Lie-Rinehart algebras and classify them by the third cohomology of Lie-Rinehart algebras. However, their crossed modules are similar to the traditional one for Lie algebras. It would be interestiong to study crossed extensions of Lie-Rinehart algebras and their classification in terms of higher cohomologies of Lie-Rinehart algebras.
\end{remark}

\section{The case of a Leibniz algebra}
Leibniz algebra was introduced by Loday in connection with cyclic homology and Hochschild homology of matrix algebras \cite{loday}. The cohomology theory of Liebniz algebras was introduced by the same author. In this section, we outline that Leibniz cohomology groups are also represented by crossed extensions of Leibniz algebras.

A (right) Leibniz algebra is a $\mathbb{K}$-vector space $\mathfrak{h}$ together with a $\mathbb{K}$-bilinear map $[~,~] : \mathfrak{h} \times \mathfrak{h} \rightarrow \mathfrak{h}$ satisfying
\begin{align*}
[x,[y,z]] = [[x,y],z] - [[x,z],y], ~~~ \text{ for all } x, y, z \in \mathfrak{h}.
\end{align*}

Let $(\mathfrak{h}, [~,~])$ be a Leibniz algebra. An $\mathfrak{h}$-module is a vector space $M$ together with two bilinear maps (both of them denoted by $[~,~]$) $\mathfrak{h} \times M \rightarrow M$ and $M \times \mathfrak{h} \rightarrow M$ satisfying
\begin{align*}
[x,[y,z]] = [[x,y],z] - [[x,z],y],
\end{align*}
whenever one of the variable is from $M$ and the others two are from $\mathfrak{h}$.

The cohomology of the Leibniz algebra $\mathfrak{h}$ with coefficients in $M$  is the cohomology of the complex $(C^n_{Leib} (\mathfrak{h}, M), \delta)_{n \geq 0}$ where $C^0_{Leib} (\mathfrak{h}, M) = M$ and $C^n_{Leib} (\mathfrak{h}, M) = Hom_{\mathbb{K}} (\mathfrak{h}^{\otimes n} , M)$, for $n \geq 1$. The differential $\delta$ is given by $(\delta m)(x) = [x,m]$, for $m \in M$, $x \in \mathfrak{h}$ and
\begin{align*}
(\delta f) (x_1, \ldots, x_{n+1}) =~& [x_1, f(x_2, \ldots, x_{n+1}) ] + \sum_{i=2}^{n+1} (-1)^i [ f(x_1, \ldots, \widehat{x_i}, \ldots, x_{n+1}), x_i ] \\
~& + \sum_{1 \leq i < j \leq n+1} (-1)^{j+1} f(x_1, \ldots, x_{i-1}, [x_i, x_j], x_{i+1}, \ldots, \widehat{x_j}, \ldots, x_n).
\end{align*}
Similar to the case of Chevalley-Eilenberg cohomology, the lower degree Leibniz cohomology has the following interpretations. The zero-th cohomology group $H^0_{Leib} (\mathfrak{h}, M)$ is the submodule of invariants on $M$ and the first cohomology group
\begin{align*}
H^1_{Leib} (\mathfrak{h}, M) = \frac{\{ f : \mathfrak{h} \rightarrow M|~ f[x,y] = [x,f(y)] + [f(x), y], \forall x , y \in \mathfrak{h}\}}{\{\delta m |~ m \in M \}}
\end{align*}
is the space of outer derivations. The second cohomology group $H^2_{Leib} (\mathfrak{h}, M)$ classifies the equivalence classes of extensions of the Leibniz algebra $\mathfrak{h}$ by $M$.

Crossed modules over a Leibniz algebra can be defined in a similar way. One only needs to careful about non skew-symmetric version of the identities used in the case of Lie algebra.

A crossed module over a Leibniz algebra is a triple $(V, L, \partial)$ in which $L$ is a Leibniz algebra, $V$ is a $L$-module and $\partial : V \rightarrow L$ is a map of $L$-modules satisfying
\begin{align*}
[\partial v, w ] = [ v, \partial w], ~~~~ \text{ for all } v, w \in L.
\end{align*}
In a similar way, a crossed module yields a Leibniz algebra structure on $\mathfrak{h}= \text{coker} (\partial)$ and there is an exact sequence
\begin{align*}
0 \rightarrow M \rightarrow V \xrightarrow{\partial} L \xrightarrow{\pi} \mathfrak{h} \rightarrow 0
\end{align*}
where $M = \text{ker}(\partial)$. Moreover, there is a Leibniz algebra action of $\mathfrak{h}$ on $M$. A morphism of crossed modules over Leibniz algebras can be defined in a similar way.

In this case, one can also prove that equivalence classes of crossed modules with cokernel $\mathfrak{h}$ and kernel $M$ are in one-to-one correspondence with $H^3_{\text{Leib}} (\mathfrak{h}, M)$. The proof is similar to the Lie algebra case (Theorem \ref{cross-3}). However, in this case, the Leibniz $3$-cocycle $\theta_{\mathcal{E}, s, q}: \mathfrak{h}^{\otimes 3} \rightarrow M$ is given by
\begin{align*}
\theta_{\mathcal{E}, s, q} (x,y,z) = [s(x), g (y,z)] + [ g(x,z), s(y)] - [g(x,y), s(z)] - g([x,y], z) + g([x,z], y) + g (x, [y,z]).
\end{align*}
Associated to any third cohomology class in $H^3_{\text{Leib}} (\mathfrak{h}, M)$ the existence of the corresponding crossed module can be shown by the way that have been described in Theorem \ref{cross-3}.

Moreover, one can define crossed $n$-fold extension of a Leibniz algebra $\mathfrak{h}$ by a module $M$. Along the lines of Section \ref{sec-4}, we obtain the following theorem for Leibniz algebras.

\begin{thm}
The set $\text{Opext}^n (\mathfrak{h}, M)$ of equivalence classes of crossed $n$-fold extensions of $\mathfrak{h}$ by $M$ forms an abelian group, for $n \geq 2$. Moreover, there exists an isomorphism of abelian groups $\text{Opext}^n (\mathfrak{h}, M) \cong H^{n+1}_{\text{Leib}} (\mathfrak{h}, M).$
\end{thm}

\medskip

\noindent {\bf Acknowledgement.} The research is supported by the fellowship
of Indian Institute of Technology, Kanpur (India). The author would like to thank the Institute for their support.


\section{Introduction}
In this paper, we consider
the initial-boundary value problem of the semilinear damped wave equation
\begin{equation}\label{eq:ndw}
\begin{cases}
\partial_t^2u(x,t)-\Delta u(x,t) + \partial_tu(x,t) = f\big(u(x,t)\big),
& (x,t) \in \Omega\times (0,T),
\\
u(x,t)=0,
& (x,t) \in \partial\Omega\times (0,T),
\\
u(x,0)=u_0(x),
& x \in \Omega,
\\
\partial_tu(x,0)=u_1(x),
& x \in \Omega,
\end{cases}
\end{equation}
where $\partial_t=\frac{\partial}{\partial t}$, $\Delta=\sum_{j=1}^N\frac{\partial^2}{\partial x_j^2}$
and $\Omega\subset\mathbb{R}^N$ $(N\in \mathbb{N}, N\geq 2)$
is an exterior domain (that is, $\mathbb{R}^N\setminus \Omega$ is bounded)
having a smooth boundary $\partial\Omega$.
The function $u:\overline{\Omega}\times [0,T)\to \mathbb{R}$ is unknown,
the pair $(u_0,u_1)\in H^1_0(\Omega)\times L^2(\Omega)$
is given
and $f\in C^1(\mathbb{R})$ satisfies that
there exist constants $p>1$ and $C_f\geq 0$ such that
\begin{align}\label{ass.f}
f(0)=0, \quad |f(\xi)-f(\eta)|\leq C_f(|\xi|+|\eta|)^{p-1}|\xi-\eta|, \quad \xi,\eta\in\mathbb{R}.
\end{align}
The damped wave equation was introduced
by Cattaneo \cite{Cattaneo} and Vernotte \cite{Vernotte}
to discuss an model of heat conduction with finite propagation property.
The equation is derived by combining
``balance law'' $u_t = {\rm div} q$ and
``time-delayed Fourier law'' $\tau q_t + q = \nabla u$, where $q$ is the heat
flux and $\tau$ is small enough.
Therefore one can expect that the behavior of solution to \eqref{eq:ndw}
can be approximated by the one of solution to heat equation.

The aim of this study is to give
global existence of solutions to \eqref{eq:ndw}
under the smallness of initial data in the sense of
weighted norm
\[
\int_{\Omega}\Big(|\nabla u_0|^2+|u_0|^2+|u_1|^2\Big)\lr{x}^{2\lambda}\,dx
\]
for fixed $\lambda\geq 0$, by using the idea of
weighted energy estimates including Kummer's confluent hypergeometric
functions, originated in \cite{SoWa4}.

For the case $\Omega=\mathbb{R}^N$, there are many previous works
dealing with the analysis of critical exponent
in the following sense: $p_c$ is the critical exponent
if $1<p<p_c$, then a blowup solution with sufficiently small initial data exists,
on the other hand,
if $p>p_c$, then smallness of initial data provides the existence
of global-in-time solutions.
The critical exponent for compactly supported initial data
was given by Todorova--Yordanov \cite{ToYo01} with $p_c=1+\frac{2}{N}$,
by introducing the energy estimates with exponential type weight function.
Similar philosophy can be found in Ikehata--Tanizawa \cite{IkTa05}
for non-compactly supported initial data.

It should be noticed that the critical exponent
is exactly the same as Fujita exponent for
semilinear heat equation found in the pioneering work in Fujita \cite{Fujita66}.
The critical case $p=p_c$ was discussed in Zhang \cite{Zhang01}
and blowup result for small initial data is proved.

For the framework of weak solutions in $H^1$ with non-compactly supported initial data,
Nakao--Ono \cite{NaOn93} found that
the existence of global-in-time solutions with sufficiently small initial data
when $1+\frac{4}{N}\leq p\leq \frac{N+2}{N-2}$.
Later,
Ikehata--Ohta \cite{IkOh02} discussed the critical exponent of \eqref{eq:ndw}
with initial data $(u_0,u_1)\in (H^1\cap L^r)\times (L^2\cap L^r)$.
It is proved that the critical exponent for this problem
is $p_c=1+\frac{2r}{N}$:
if $1<p<1+\frac{2r}{N}$, then nonexistence of global-in-time solutions
occurs, and
if $1+\frac{2r}{N}<p<\frac{N+2}{N-2}$, then global-in-time solutions
exist for sufficiently small initial data.

The initial data in the class
$(H^{\alpha,0}\cap H^{0,\delta})\times (H^{\alpha-1,0}\cap H^{0,\delta})$
are discussed by Hayashi--Kaikina--Naumkin \cite{HaKaNa04DIE}, where
$H^{\ell,m}=\{\phi\in L^2\;;\;\|\lr{x}^{m}\lr{i\partial_x}^\ell \phi\|_{L^2}<\infty\}$
with the Fourier multiplier $\lr{i\partial_x}=\mathcal{F}^{-1}\lr{\xi}\mathcal{F}$.
They proved the existence of global-in-time solutions (in $L^1$)
to \eqref{eq:ndw} with $p>1+\frac{2}{N}$ and
and a heat-like asymptotic profile of solutions.
The analysis of \cite{HaKaNa04}
is generalized by Ikeda--Inui--Wakasugi \cite{IkInWa17}
in the framework of $(H^{\alpha,0}\cap H^{0,\delta})\times (H^{\alpha-1,0}\cap H^{0,\delta})$
which can be embedded into $L^r$-space ($r\in (1,2]$).
In their paper the critical exponent is determined as
$1+\frac{2r}{N}$ which is the same as \cite{IkOh02}.
Recently, Inui--Ikeda--Okamoto--Wakasugi \cite{IkInOkWa_arxiv}
discussed the critical case $p=1+\frac{2r}{N}$ under some restriction
on $r$, which is required by a derivative loss of $L^p$-$L^q$ estimates
for high frequency part of solutions to the linear damped wave equation.
We note that
for the analysis of the Cauchy problem of
the equation $\partial_t^2u-\Delta u + \frac{\mu}{(1+t)^{\beta}}\partial_tu=f(u)$
(with time-dependent damping term),
a similar study can be found in
the literature (see e.g.,
Wirth \cite{Wirth04,Wirth06,Wirth07},
Nishihara \cite{Nishihara11},
Lin--Nishihara--Zhai \cite{LiNiZh12},
Wakasugi \cite{Wakasugi14},
Lai--Takamura--Wakasa \cite{LaTaWa17},
and Ikeda--Sobajima \cite{IkSo2} and their reference therein).

For the case of damped wave equation in an exterior domain,
Ono \cite{Ono03} discussed the existence of global-in-time solutions to
\eqref{eq:ndw} under $2\leq N\leq 6, 1+4/N+2<p<1+2/(N-2)$
by using the result of Dan--Shibata \cite{DaSh95}.
On the one hand,
Ikehata \cite{Ikehata04,Ikehata05}
proved the existence of global-in-time solutions
for $N=2, 2<p<\infty$ by using weighted energy estimates.
Takeda--Ogawa \cite{TaOg09} proved non-existence
of global-in-time solutions
to \eqref{eq:ndw} when $N\geq 2, 1<p<1+2/N$
by employing the method of Kaplan \cite{Kaplan63} and Fujita \cite{Fujita66}.
Note that
in the analysis of weighted energy estimates
of the linear problem
with a class of space-dependent damping term
in exterior domain
can be found in
Ikehata \cite{Ikehata05-wee},
Todorova--Yordanov \cite{ToYo09},
Radu--Todorova--Yordanov \cite{RaToYo09,RaToYo10}
and
Wakasugi--Sobajima \cite{SoWa1,SoWa2},
however,
their weight function forms $e^{c(1+|x|^2)^{2-\alpha}/(1+t)^{1+\beta}}$
and therefore
the initial data must have an exponential decay property.

Recently, Wakasugi--Sobajima \cite{SoWa4}
found a framework of weighted energy estimates
with a weight function of polynomial type.
In \cite{SoWa4},
the weight function is taken as
the inverse of the positive solution of heat equation $\partial_t\Phi=\Delta\Phi$
including the Kummer confluent hypergeometric function (see Section 2.1 below).
This enables us to obtain the weighted energy estimate
of polynomial type.

The purpose of the present paper is
to discuss the nonlinear problem of damped wave equation
in exterior domain in view of weighted energy estimate
of polynomial type introduced by \cite{SoWa4}.

To state the main result, we first give
the definition of
the solutions to \eqref{eq:ndw} in this paper.
\begin{definition}[Weak solution]
\label{def:sol}
The function $u:\overline{\Omega}\times [0,T)\to \mathbb{R}$ is called
a weak solution of \eqref{eq:ndw} in $(0,T)$
if $u$ belongs to the class
\[
S_T=
\{u\in C([0,T);H^1_0(\Omega))\cap C^1([0,T);L^2(\Omega)) \;;\;f(u(\cdot))\in C([0,T);L^2(\Omega))\}
\]
and
$U=(u(t),\partial_tu(t))$ satisfies the  following
integral equation in $\mathcal{H}=H^1_0(\Omega)\times L^2(\Omega)$:
\begin{align*}
U(t)=e^{-t\mathcal{A}}U_0+\int_0^te^{-(t-s)\mathcal{A}}[\mathcal{N}(U(s))]\,ds,
\quad t\in [0,T),
\end{align*}
where $U_0=(u_0,u_1)$
$\mathcal{A}=\begin{pmatrix}0&-1\\-\Delta&1\end{pmatrix}$
with domain $D(\mathcal{A})=(H^2(\Omega)\times H^1_0(\Omega))\times H^1_0(\Omega)$
and $\mathcal{N}(u,v)=(0,f(u))$.
\end{definition}

The existence of local-in-time solutions to
\ref{eq:ndw} is well-known (see e.g., Ikawa \cite{Ikawa68} and Cazenave--Haraux \cite{CazenaveBOOK}).

\begin{proposition}\label{prop:LWP}
Assume that $f$ satisfies \eqref{ass.f} with $1<p\leq \frac{N}{N-2}$.
Then for every $(u_0,u_1)\in H^1_0(\Omega)\times L^2(\Omega)$,
there exists a positive constant $T>0$ depending only on
$N,p,C_0,\|u_0\|_{H^1},\|u_1\|_{L^2}$
such that there exists a unique weak solution $u$ in $(0,T)$.
\end{proposition}

The notion of lifespan is the following.
\begin{definition}[Lifespan]
\label{def:lifespan}
For the solution $u$ of \eqref{eq:ndw} with initial data
$(u_0,u_1)\in H^1_0(\Omega)\times L^2(\Omega)$, we define lifespan
$T_* = T_*(u_0,u_1)>0$
(the maximal existence time of solution $u$) is as follows:
\[
T_* = \sup \{T>0\;;\; \text{\eqref{eq:ndw} has a unique weak solution in }(0,T)\}.
\]
\end{definition}

Now we are in a position to state the main result of the present paper.

\begin{theorem}\label{main}
Assume that $f$ satisfies \eqref{ass.f} with $1<p\leq \frac{N}{N-2}$.
Then for every $\lambda \in [0,\frac{N}{2})$,
there exists a positive constant $\delta_{\lambda}^*>0$ such that
the following assertion holds:

For every $(u_0,u_1)\in H^1_0(\Omega)\times L^2(\Omega)$ satisfying
\[
\int_{\Omega}\Big(|\nabla u_0(x)|^2+|u_0(x)|^2+|u_1(x)|^2\Big)\lr{x}^{2\lambda}\,dx\leq \delta_\lambda^*,
\]
one has $T_*(u_0,u_1)=\infty$ when $p\geq 1+\frac{4}{N+2\lambda}$. Namely, there exists a global weak solution $u$ of \eqref{eq:ndw}
with initial data $(u_0,u_1)$.
Moreover, $u$ satisfies the following weighted estimates:
there exists a positive constant $M_{\lambda}^*>0$ such that
\begin{gather*}
\int_{\Omega}\left(|\nabla u(x,t)|^2+|\partial_tu(x,t)|^2\right)\left(1+t+|x|^{2}\right)^{\lambda}\,dx
\leq \frac{M_{\lambda}^*}{1+t},
\\
\int_{\Omega}|u_1(x,t)|^2\left(1+t+|x|^{2}\right)^{\lambda}\,dx
\leq M_{\lambda}^*,
\\
\int_0^\infty\!\!
\int_{\Omega}|\nabla u(x,s)|^2\left(1+s+|x|^{2}\right)^{\lambda}\,dx\,ds
\leq M_{\lambda}^*,
\\
\int_0^\infty(1+s)
\int_{\Omega}|\partial_t u(x,s)|^2\left(1+s+|x|^{2}\right)^{\lambda}\,dx\,ds
\leq M_{\lambda}^*.
\end{gather*}
On the other hand, for the case $1<p<1+\frac{4}{N+2\lambda}$,
one has the following lower estimate of lifespan
\[
T_*(\varepsilon u_0,\varepsilon u_1)\geq C\varepsilon^{-(\frac{1}{p-1}-\frac{N+2\lambda}{4})^{-1}}
\]
for some $C>0$ (independent of $\varepsilon$) and sufficiently small $\varepsilon>0$.
\end{theorem}
\begin{remark}
In the case $\Omega=\mathbb{R}^N$, the global-in-time solution
of \ref{eq:ndw}
with slowly decaying initial data (like $\lr{x}^{-\mu}$)
was constructed in Hayashi--Kaikina--Naumkin \cite{HaKaNa04DIE}
(for $p>1+\frac{2}{N}$) under a weaker assumption than ours.
In the case of exterior domain,
it is already discussed when $\lambda=1$.
However, the case $\lambda\in (0,\frac{N}{2})$
is not dealt with so far.
The global existence for weighted-$L^2$-type initial data
is now established.
\end{remark}

\begin{remark}
For $L^r$-type initial data,
Ikeda--Inui--Okamoto--Wakasugi \cite{IkInOkWa_arxiv}
proved the case $p=1+\frac{2r}{N}$ under some restriction on $r$,
which is critical in this situation and
related to our critical case $p=1+\frac{4}{N+2\lambda}$.
Although their aspect is quite far form ours,
it should be noticed that
the framework in \cite{IkInOkWa_arxiv}
is difficult to apply to the case of exterior domain
because of the use of a deep Fourier analysis.
\end{remark}

\begin{remark}
For the lifespan estimates,
Ikeda--Inui--Wakasugi \cite{IkInWa17}
provided upper bound of lifespan estimates
with a specific situation
\[
\Omega=\mathbb{R}^N, \quad
u_0+u_1 \geq \max\{1,|x|\}^{-\frac{N}{2}-\lambda}, \quad \lambda<\frac{2}{p-1}-\frac{N}{2}, \quad
f(u)=|u|^p.
\]
They proved $T_*(\varepsilon u_0,\varepsilon u_1)\geq C\varepsilon^{-(\frac{1}{p-1}-\frac{N+2\lambda}{4})^{-1}}$.
Combining their result, we can assert that
the lower estimate in Theorem \ref{main} is almost sharp.
\end{remark}
As a corollary of Theorem \ref{main}, we can deduce
the existence of global-in-time solutions to
\eqref{eq:ndw} with $p>1+\frac{2}{N}$ for polynomially decaying initial data.

\begin{corollary}\label{cor}
Assume that $f$ satisfies \eqref{ass.f}.
Then for every $1+\frac{2}{N}<p\leq \frac{N}{N-2}$,
there exists a positive constant $\delta_{p}^{**}>0$ such that
the following assertion holds:
For every $(u_0,u_1)\in H^1_0(\Omega)\times L^2(\Omega)$ satisfying
\[
\int_{\Omega}\Big(|\nabla u_0(x)|^2+|u_0(x)|^2+|u_1(x)|^2\Big)\lr{x}^{N}\,dx\leq \delta_p^{**},
\]
one has $T_*(u_0,u_1)=\infty$.
\end{corollary}

Of course we can proceed the same argument
in the one-dimensional case $\Omega=\mathbb{R}$.
However, the lack of the validity of
(weighted) Hardy's inequality causes, and some difficulty appears.
To avoid the use of Hardy's inequality,
we use a solution of heat equation with some modification.
As a result, we lose the result of the critical situation $p=1+\frac{4}{1+2\lambda}$.
The precise statement for the case $\Omega=\mathbb{R}$ is written in the end of the last section.

This paper is organized as follows.
In Section \ref{sec:pre},
we state the properties of a family of
self-similar solutions to the heat equation
including Kummer's Confluent hypergeometric  functions
and collect some functional inequalities
we need in the derivation of weight energy estimates.
Section \ref{sec:wee} is devoted to prove Theorem \ref{main}.
Finally, we give a remark about the weighted energy estimates
and global existence for one-dimensional case
in Section \ref{sec:1d}.

\section{Preliminaries}\label{sec:pre}
\subsection{The weight functions including confluent hypergeometric functions}
For $t_0\geq 1$ and $\beta\geq 0$, define
\[
\Phi_{\beta}(x,t:t_0)=(t_0+t)^{-\beta}
e^{-\frac{|x|^2}{4(t_0+t)}}
M\left(\frac{N}{2}-\beta,\frac{N}{2};\frac{|x|^2}{4(t_0+t)}\right),
\quad
(x,t)\in \mathbb{R}^N\times[0,\infty),
\]
where $M(a,c;z)$ is Kummer's confluent hypergeometric function
defined as
\[
M(a,c,z)=\sum_{n=0}^\infty\frac{(a)_n}{(c)_n}\frac{z^n}{n!}
\]
with the Pochhammer symbol $(d)_0=1$ and $(d)_n=\prod_{k=0^{n-1}}(d+k-1)$.
 These functions are given by Sobajima--Wakasugi \cite{SoWa4}
as a family of self-similar solution of
linear heat equation $\partial_t\Phi-\Delta\Phi=0$.
Then we have the following lemma.
\begin{lemma}[{\cite{SoWa4}}]\label{Phi}
\begin{itemize}
\item[\bf (i)]for every $\beta\geq 0$,
$\partial_t\Phi_\beta(x,t)=\Delta \Phi_\beta(x,t)$
for $(x,t)\in \mathbb{R}^N\times[0,\infty)$,
\item[\bf (ii)]for every $\beta\geq 0$,
$\partial_t\Phi_\beta(x,t)=-\beta\Phi_{\beta+1}(x,t)$ for $(x,t)\in \mathbb{R}^N\times[0,\infty)$,
\item[\bf (iii)] for every $\beta\geq 0$, there exists a positive constant $C_\beta$ such that
\[
|\Phi_\beta(x,t)|\leq C_\beta\left(t_0+t+\frac{|x|^2}{4}\right)^{-\beta},
\]
\item[\bf (iv)] for every $0\leq \beta<\frac{N}{2}$, there exists a positive constant $c_\beta$ such that
\[
\Phi_\beta(x,t)\geq c_\beta \left(t_0+t+\frac{|x|^2}{4}\right)^{-\beta}.
\]
\end{itemize}
\end{lemma}
\subsection{Functional inequalities with weights}
In view of Lemma \ref{Phi}, for the same constant $t_0\geq 1$ as $\Phi_\beta$,
we also introduce
\[
\Psi(x,t:t_0)=t_0+t+\frac{|x|^2}{4},
\quad (x,t)\in \mathbb{R}^N\times [0,\infty).
\]
The following Hardy type inequality with $\Psi$
is also needed.
\begin{lemma}\label{lem:hardy}
Let $\lambda>-\frac{N-2}{2}$.
For every $w\in C_c^1(\Omega)$,
\begin{equation}\label{hardy}
4K(\lambda)^2
\int_{\Omega}
  |w|^2\Psi^{\lambda-1}
\,dx
\leq
\int_{\Omega}
  |\nabla w|^2\Psi^{\lambda}
\,dx
\end{equation}
with $K(\lambda)=\min\{\frac{N}{2}+\lambda-1,\frac{N}{2}\}$.
That is, if $N\geq 2$, then \eqref{hardy} holds for every $\lambda>0$
and every $w\in H^1_0(\Omega)$ satisfying $w\Psi^{\frac{\lambda}{2}}, (\nabla w)\Psi^{\frac{\lambda}{2}}\in L^2(\Omega)$.
\end{lemma}
\begin{proof}
Noting that
\begin{align*}
{\rm div} \left(\frac{x}{2}\Psi^{\lambda-1}\right)
=
\frac{N}{2}\Psi^{\lambda-1}+ (\lambda-1)\frac{|x|^2}{4}\Psi^{\lambda-2}
\geq
K\Psi^{\lambda-1}
\end{align*}
with $K(\lambda)=\min\{\frac{N}{2}+\lambda-1,\frac{N}{2}\}>0$,
we see from integration by parts and
H\"older inequality that
\begin{align*}
K(\lambda)\int_{\mathbb{R}^N}|w|^2\Psi^{\lambda-1}\,dx
&
\leq
\int_{\mathbb{R}^N}|w|^2{\rm div}\left(\frac{x}{2}\Psi^{\lambda-1}\right)\,dx
\\
&
\leq
-2\int_{\mathbb{R}^N}w\left(\nabla w\cdot\frac{x}{2}\right)\Psi^{\lambda-1}\,dx
\\
&
\leq
2
\left(\int_{\mathbb{R}^N}|w|^2\Psi^{\lambda-1}\,dx\right)^{\frac{1}{2}}
\left(\int_{\mathbb{R}^N}|\nabla w|^2\frac{|x|^2}{4}\Psi^{\lambda-1}\,dx\right)^{\frac{1}{2}}
\\
&
\leq
2
\left(\int_{\mathbb{R}^N}|w|^2\Psi^{\lambda-1}\,dx\right)^{\frac{1}{2}}
\left(\int_{\mathbb{R}^N}|\nabla w|^2\Psi^{\lambda}\,dx\right)^{\frac{1}{2}}.
\end{align*}
The last assertion can be verified by the standard approximation argument
with the molifier and cut-off functions.
\end{proof}

The following lemma is well-known
Gagliardo--Nirenberg inequality.

\begin{lemma}[Gagliardo--Nirenberg inequality]
\label{lem:GNineq}
If $1<p\leq \frac{N+2}{N-2}$, then there exists a constant $C_{GN,p}>0$ such that
for every $w\in H^1_0(\Omega)$,
\begin{align*}
\|w\|_{L^{p+1}(\Omega)}\leq
C_{GN,p}
\|w\|_{L^2(\Omega)}^{1-\frac{N(p-1)}{2(p+1)}}
\|\nabla w\|_{L^2(\Omega)}^{\frac{N(p-1)}{2(p+1)}}.
\end{align*}
\end{lemma}

Next we give a weight version of Gagliardo-Nirenberg inequality,
which we will exactly need in the treatment of nonlinear term $f(u)$ in \eqref{eq:ndw}.

\begin{lemma}\label{lem:GNtype}
If $N\geq 2$ and $1<p\leq \frac{N+2}{N-2}$ and $\lambda>0$, then there exists a constant $\widetilde{C}_p>0$ such that
for every $w\in H^1_0(\Omega)$ satisfying $v\Psi^{\lambda},(\nabla v)\Psi^\lambda\in L^2(\Omega)$,
\begin{align*}
\|v\Psi^{\frac{\lambda}{p+1}}\|_{L^{p+1}(\Omega)}
\leq
\widetilde{C}_p
(t_0+t)^{-\frac{\lambda(p-1)}{2(p+1)}}
\|v\Psi^{\frac{\lambda}{2}}\|_{L^2(\Omega)}^{1-\frac{N(p-1)}{2(p+1)}}
\|(\nabla v)\Psi^{\frac{\lambda}{2}}\|_{L^2(\Omega)}^{\frac{N(p-1)}{2(p+1)}}.
\end{align*}
\end{lemma}
\begin{proof}
Note that by assumption, we have
$v\Psi^{\frac{\lambda}{p+1}}\in H_0^1(\Omega)$. Therefore
applying Lemma \ref{lem:GNineq} to $w=v\Psi^{\frac{\lambda}{p+1}}$
and using $|\nabla \Psi|\leq \Psi^{\frac{1}{2}}$
imply
\begin{align*}
\|v\Psi^{\frac{\lambda}{p+1}}\|_{L^{p+1}(\Omega)}
&
\leq
C_{GN,p}
\|v\Psi^{\frac{\lambda}{p+1}}\|_{L^2(\Omega)}^{1-\frac{N(p-1)}{2(p+1)}}
\|\nabla (v\Psi^{\frac{\lambda}{p+1}})\|_{L^2(\Omega)}^{\frac{N(p-1)}{2(p+1)}}.
\\
&
\leq
C_{GN,p}
\|v\Psi^{\frac{\lambda}{p+1}}\|_{L^2(\Omega)}^{1-\frac{N(p-1)}{2(p+1)}}
\left(
\|(\nabla v)\Psi^{\frac{\lambda}{p+1}}\|_{L^2(\Omega)}
+
\frac{\lambda}{p+1}\|v\Psi^{\frac{\lambda}{p+1}-\frac{1}{2}}\|_{L^2(\Omega)}
\right)^{\frac{N(p-1)}{2(p+1)}}.
\end{align*}
Combining the above inequality with Lemma \ref{lem:hardy} with $v$ and $\frac{\lambda}{p+1}$, we have
\begin{align*}
\|v\Psi^{\frac{\lambda}{p+1}}\|_{L^{p+1}(\Omega)}
\leq
C_{GN,p}\left(1+\frac{\lambda K(\frac{\lambda}{p+1})}{p+1}\right)^{\frac{N(p-1)}{2(p+1)}}
\|v\Psi^{\frac{\lambda}{p+1}}\|_{L^2(\Omega)}^{1-\frac{N(p-1)}{2(p+1)}}
\|(\nabla v)\Psi^{\frac{\lambda}{p+1}}\|_{L^2(\Omega)}^{\frac{N(p-1)}{2(p+1)}}.
\end{align*}
Using the inequality $\Psi^{-1}\leq (t_0+t)^{-1}$, we deduce the
desired inequality.
\end{proof}

Thirdly,
we give an inequality related to
integration by parts formula with non-uniform weight.
Although its proof is essentially stated in \cite{SoWa4},
we would give a proof for reader's convenience.

\begin{lemma}\label{lem:ibp}
Assume that $\Phi\in C^2(\overline{\Omega})$ is positive and $\delta\in (0,\frac{1}{2})$.
Then for every $u\in H^1_0(\Omega)$ having a compact support,
\[
\int_{\Omega}
u\Delta u
\Phi^{-1+2\delta}
\,dx
\leq
-
\frac{\delta}{1-\delta}
\int_{\Omega}
|\nabla u|^2
\Phi^{-1+2\delta}
\,dx
+
\frac{1-2\delta}{2}
\int_{\mathbb{R}^N}
u^2(\Delta \Phi)
\Phi^{-2+2\delta}
\,dx
\]
\end{lemma}
\begin{proof}
Set $v=\Phi^{-1+\delta}u$. Then we have
\begin{align*}
\int_{\Omega}
|\nabla u|^2
\Phi^{-1+2\delta}
\,dx
=
\int_{\Omega}
|\nabla v|^2
\Phi
\,dx
+2(1-\delta)
\int_{\Omega}
v(\nabla v\cdot\nabla \Phi)
\,dx
+
(1-\delta)^2\int_{\Omega}
v^2\frac{|\nabla \Phi|^2}{\Phi}
\,dx.
\end{align*}
We see from integration by parts that
\[
\int_{\Omega}
|\nabla u|^2
\Phi^{-1+2\delta}
\,dx
\geq
-(1-\delta)
\int_{\Omega}
u^2(\Delta \Phi)\Phi^{-2+2\delta}
\,dx
+
(1-\delta)^2\int_{\Omega}u^2|\nabla \Phi|^2\Phi^{-3+2\delta}
\,dx.
\]
Using the above inequality with integration by parts twice, we deduce
\begin{align*}
\int_{\Omega}
u\Delta u
\Phi^{-1+2\delta}
\,dx
&
=
-
\int_{\Omega}
|\nabla u|^2
\Phi^{-1+2\delta}
\,dx
+
(1-2\delta)\int_{\Omega}
u(\nabla u\cdot\nabla\Phi)
\Phi^{-2+2\delta}
\,dx
\\
&
=
-
\int_{\Omega}
|\nabla u|^2
\Phi^{-1+2\delta}
\,dx
-\frac{1-2\delta}{2}\int_{\Omega}
u^2(\Delta\Phi)
\Phi^{-2+2\delta}
\,dx
\\
&\quad
+
(1-\delta)(1-2\delta)
\int_{\Omega}
u^2|\nabla\Phi|^2\Phi^{-3+2\delta}
\,dx
\\
&\leq
\left(
\frac{1-2\delta}{1-\delta}-1\right)
\int_{\Omega}
|\nabla u|^2
\Phi^{-1+2\delta}
\,dx
+\frac{1-2\delta}{2}\int_{\Omega}
u^2(\Delta\Phi)
\Phi^{-2+2\delta}
\,dx.
\end{align*}
The proof is complete.
\end{proof}

\section{Proof of main theorem (Theorem \ref{main})}\label{sec:wee}

Since the weak solution $u$ of \eqref{eq:ndw} can be
approximated by the one with smooth compactly supported initial data,
in this section
we may assume that $u_0\in H^2(\Omega)\cap H^1_0(\Omega)$ and $u_1\in H^1_0(\Omega)$ are compactly supported
without loss of generality.
By finite propagation property, we also can assume that
the solution $u(t)\in H^2(\Omega)\cap H^1_0(\Omega)$ $(t\geq 0)$
is also compactly supported for every $t\in [0,T_*)$.

The proof of Theorem \ref{main}
is based on the following proposition
which is well-known, and so-called blowup alternative.

\begin{proposition}\label{prop:alt}
Assume that $f$ satisfies \eqref{ass.f} with $1<p\leq \frac{N}{N-2}$.
Let $u$ be the weak solution of \eqref{eq:ndw} in $(0,T_*)$ with the corresponding lifespan $T_*$.
If $T_*<\infty$, then one has
\[
\lim_{t\uparrow T_*}(\|u\|_{H^1_0(\Omega)}+\|u\|_{L^2(\Omega)})=\infty.
\]
\end{proposition}

For $t_0\geq 1$ and $\lambda>0$, define
the following weighted energy functional
for the weak solution $u$ as follows:
\begin{equation}
\label{eq:E_lambda_1}
E_\lambda(t:t_0)
:=
(t_0+t)
\int_{\Omega}\Big(|\nabla u(x,t)|^2+|\partial_tu(x,t)|^2\Big)\Psi(x,t:t_0)^{\lambda}\,dx,
\quad
t\geq 0.
\end{equation}
Then the following lemma holds.
\begin{lemma}\label{lem:energy-1}
Let $E_\lambda$ be given in \eqref{eq:E_lambda_1}.
Then for every $t_0\geq 1$ and $\lambda\geq 0$,
\begin{align*}
\frac{d}{dt}E_\lambda(t:t_0)
&\leq
(\lambda^2+\lambda+1)\int_{\Omega}|\nabla u|^2\Psi^{\lambda}\,dx
+
(\lambda+1-t_0-t)\int_{\Omega}|\partial_tu|^2\Psi^{\lambda}\,dx
\\
&\quad+
\frac{d}{dt}
\left[
2(t_0+t)\int_{\Omega}
F(u)\Psi^{\lambda}
\,dx
\right]
+
2(\lambda+1)\int_{\Omega}
F(u)\Psi^{\lambda}
\,dx, \quad\text{a.a.}\ t\in (0,T_*),
\end{align*}
where $F(\xi)=\int_{0}^\xi f(\eta)\,d\eta$ for $\xi\in \mathbb{R}$.
\end{lemma}
\begin{proof}
Observe that $\partial_tu(t)\in H^1_0(\Omega)$ and $\Psi=t_0+t+\frac{|x|^2}{4}$. By integration by parts we have
\begin{align*}
\frac{d}{dt}E_\lambda(t:t_0)
&=
\int_{\Omega}\Big(|\nabla u|^2+|\partial_tu|^2\Big)[\Psi^{\lambda}+(t_0+t)\lambda\Psi^{\lambda-1}]\,dx
\\
&\quad
+
2(t_0+t)\int_{\Omega}\Big(\nabla \partial_tu\cdot\nabla u+\partial_tu\partial_t^2u\Big)\Psi^{\lambda}\,dx
\\
&\leq
(\lambda+1)
\int_{\Omega}\Big(|\nabla u|^2+|\partial_tu|^2\Big)\Psi^{\lambda}\,dx
\\
&\quad
+
2(t_0+t)
\int_{\mathbb{R}^N}\partial_tu(-\Delta u+\partial_t^2u)\Psi^{\lambda}
-
\lambda \partial_tu\left(\nabla u\cdot\nabla\Psi\right)\Psi^{\lambda-1}\,dx.
\end{align*}
Since $u$ satisfies \eqref{eq:ndw},
the Schwarz inequality and the inequality $(t_0+t)^{\frac{1}{2}}|\nabla \Psi|\leq \Psi$ yield
\begin{align*}
\frac{d}{dt}E_\lambda(t:t_0)
&\leq
(\lambda+1)
\int_{\mathbb{R}^N}\Big(|\nabla u|^2+|\partial_tu|^2\Big)\Psi^{\lambda}\,dx
-
2
(t_0+t)
\int_{\mathbb{R}^N}|\partial_tu|^2\Psi^{\lambda}\,dx
\\
&\quad
+
2
(t_0+t)
\int_{\mathbb{R}^N}\partial_tu(f(u))\Psi^{\lambda}\,dx
\\
&\quad+
2
\lambda
(t_0+t)^{\frac{1}{2}}
\left(
\int_{\mathbb{R}^N}
|\partial_tu|^2\Psi^{\lambda}\,dx
\right)^{\frac{1}{2}}
\left(
\int_{\mathbb{R}^N}
|\nabla u|^2\Psi^{\lambda}\,dx
\right)^{\frac{1}{2}}.
\end{align*}
Noting that
\begin{align*}
\int_{\mathbb{R}^N}\partial_tu(f(u))\Psi^{\lambda}\,dx
&=
\frac{d}{dt}
\left[
(t_0+t)\int_{\mathbb{R}^N}F(u)\Psi^{\lambda}\,dx
\right]
+
\int_{\mathbb{R}^N}F(u)[\Psi^{\lambda}+(t_0+t)\lambda \Psi^{\lambda-1}]\,dx
\\
&\leq
\frac{d}{dt}
\left[
(t_0+t)\int_{\mathbb{R}^N}F(u)\Psi^{\lambda}\,dx
\right]
+
(\lambda+1)\int_{\mathbb{R}^N}F(u)\Psi^{\lambda}\,dx,
\end{align*}
we deduce the desired inequality.
\end{proof}
Next we assume $\lambda\in [0,\frac{N}{2})$.
Set $\delta=\frac{N-2\lambda}{4N}\in (0,\frac{1}{4})$.
Define
\[
\beta:=\frac{\lambda}{1-2\delta}=\frac{2\lambda N}{N+2\lambda}\in \left(\lambda,\frac{N}{2}\right)
\]
and
\begin{equation}
\label{eq:E_lambda_2}
\widetilde{E}_\lambda(t:t_0)
:=
\int_{\mathbb{R}^N}\Big(2u(x,t)\partial_tu(x,t)+|u(x,t)|^2\Big)\Phi_\beta(x,t:t_0)^{-1+2\delta}\,dx,
\quad
t\in [0,T_*).
\end{equation}
Then the following inequality holds.
\begin{lemma}\label{lem:energy-2}
Let $\widetilde{E}_{\lambda}$ be as in \eqref{eq:E_lambda_2}.
Then for every $t_0\geq 1$ and $\lambda\in [0,\frac{N}{2})$
with $\delta=\frac{N-\lambda}{4N}$ and $\beta=\frac{2\lambda N}{N+2\lambda}$, one has
\begin{align*}
\frac{d}{dt}\widetilde{E}_\lambda(t:t_0)
&\leq
\frac{1}{c_{\beta}^{1-2\delta}}\left(
2
+\frac{(1-2\delta)\beta C_{\beta+1}}{c_\beta K(\lambda)}
\right)
\int_{\mathbb{R}^N}|\partial_tu|^2\Psi^{\lambda}\,dx
\\
&\quad+
\left(
\frac{(1-2\delta)\beta C_{\beta+1}}{c_\beta^{2-2\delta}t_0}
-
\frac{2\delta}{(1-\delta)C_\beta^{1-2\delta}}
\right)
\int_{\mathbb{R}^N}|\nabla u|^2\Psi^\lambda\,dx
+\frac{2}{c_{\beta}^{1-2\delta}}
\int_{\mathbb{R}^N}|uf(u)|\Psi^\lambda\,dx.
\end{align*}
\end{lemma}

\begin{proof}
Since $u$ is a solution of \eqref{eq:ndw},
we have
\begin{align*}
\frac{d}{dt}\widetilde{E}_\lambda(t:t_0)
&=
2\int_{\mathbb{R}^N}|\partial_tu|^2\Phi_\beta^{-1+2\delta}\,dx
+2\int_{\mathbb{R}^N}u(\partial_t^2u+\partial_tu)\Phi_\beta^{-1+2\delta}\,dx
\\
&\quad -(1-2\delta)\int_{\mathbb{R}^N}\Big(2u\partial_tu+|u|^2\Big)\Phi_\beta^{-2+2\delta}\partial_t\Phi_\beta\,dx
\\
&=
2\int_{\mathbb{R}^N}|\partial_tu|^2\Phi_\beta^{-1+2\delta}\,dx
+2\int_{\mathbb{R}^N}u(\Delta u)\Phi_\beta^{-1+2\delta}\,dx
+2\int_{\mathbb{R}^N}uf(u)\Phi_\beta^{-1+2\delta}\,dx
\\
&
\quad
-2(1-2\delta)\int_{\mathbb{R}^N}u\partial_tu\Phi_\beta^{-2+2\delta}\partial_t\Phi_\beta\,dx
-(1-2\delta)\int_{\mathbb{R}^N}|u|^2\Phi_\beta^{-2+2\delta}\partial_t\Phi_\beta\,dx.
\end{align*}
Employing Lemma \ref{lem:ibp} with $\Phi=\Phi_\beta$,  we have
\begin{align*}
\frac{d}{dt}\widetilde{E}_\lambda(t:t_0)
&\leq
2\int_{\mathbb{R}^N}|\partial_tu|^2\Phi_\beta^{-1+2\delta}\,dx
-\frac{2\delta}{1-\delta}\int_{\mathbb{R}^N}|\nabla u|^2\Phi_\beta^{-1+2\delta}\,dx
+2\int_{\mathbb{R}^N}uf(u)\Phi_\beta^{-1+2\delta}\,dx
\\
&
\quad
-2(1-2\delta)\int_{\mathbb{R}^N}u\partial_tu\Phi_\beta^{-2+2\delta}\partial_t\Phi_\beta\,dx
-(1-2\delta)\int_{\mathbb{R}^N}|u|^2\Phi_\beta^{-2+2\delta}(\partial_t\Phi_\beta-\Delta\Phi_\beta)\,dx.
\end{align*}
Here we use the profile of $\Phi_\beta$ stated in Lemma \ref{Phi}.
Then
\begin{align*}
\frac{d}{dt}\widetilde{E}_\lambda(t:t_0)
&\leq
\frac{2}{c_\beta^{1-2\beta}}\int_{\mathbb{R}^N}|\partial_tu|^2\Psi^{\lambda}\,dx
-
\frac{2\delta}{(1-\delta)C_\beta^{1-2\delta}}\int_{\mathbb{R}^N}|\nabla u|^2\Psi^\lambda\,dx
+\frac{2}{c_\beta^{1-2\beta}}\int_{\mathbb{R}^N}|uf(u)|\Psi^\lambda\,dx
\\
&
\quad
+\frac{2(1-2\delta)\beta C_{\beta+1}}{c_\beta^{2-2\delta}}
\int_{\Omega}|u|\,|\partial_tu|\Psi^{\lambda-1}\,dx.
\end{align*}
By Lemma \ref{lem:hardy},
the last term on the right-hand side of the above inequality
can be estimated as follows:
\begin{align*}
\int_{\Omega}|u|\,|\partial_tu|\Psi^{\lambda-1}\,dx
&\leq
\left(\int_{\Omega}|u|^2\Psi^{\lambda-1}\,dx\right)^{\frac{1}{2}}
\left(\int_{\Omega}|\partial_tu|^2\Psi^{\lambda-1}\,dx\right)^{\frac{1}{2}}
\\
&\leq
\frac{1}{K(\lambda)(t_0+t)^{\frac{1}{2}}}
\left(\int_{\Omega}|\nabla u|^2\Psi^{\lambda}\,dx\right)^{\frac{1}{2}}
\left(\int_{\Omega}|\partial_tu|^2\Psi^{\lambda}\,dx\right)^{\frac{1}{2}}.
\end{align*}
Hence we obtain
the desired inequality.
\end{proof}

To the end of this section
we will give an estimate for the following weighted total energy functional:
\begin{align}
\nonumber
m_\lambda(t:t_0)
&:=
(t_0+t)\int_{\mathbb{R}^N}\Big(|\nabla u(x,t)|^2+|\partial_tu(x,t)|^2\Big)\Psi(x,t:t_0)^{\lambda}\,dx
\\
\label{eq:total}
&\quad+\int_{\mathbb{R}^N}|u(x,t)|^2\Psi(x,t:t_0)^{\lambda}\,dx,
\quad
t\in [0,T_*).
\end{align}
\begin{lemma}\label{lem:equiv}
for every $\nu>0$ there exists positive constants $\gamma_\nu>0$ and $\Gamma_\nu>0$
such that
if $t_0\geq c_\beta^{-1+2\delta}\nu$, then
\[
\gamma_\nu
m_\lambda(t:t_0)
\leq
E_\lambda(t:t_0)
+
\nu
\widetilde{E}_\lambda(t:t_0)
\leq
\gamma_\nu
m_\lambda(t:t_0),
\quad t\in (0,T_*).
\]
\end{lemma}
\begin{proof}
By the Schwarz inequality and Lemma \ref{Phi} {\bf (iv)}
we see that
\begin{align*}
\left|
\int_{\Omega}
  u\partial_tu\Phi_\beta^{-1+2\delta}
\,dx
\right|
&\leq
\left(
\int_{\Omega}
  |\partial_tu|^2\Phi_\beta^{-1+2\delta}
\,dx
\right)^{\frac{1}{2}}
\left(
\int_{\Omega}
  |u|^2\Phi_\beta^{-1+2\delta}
\,dx
\right)^{\frac{1}{2}}
\\
&\leq
\frac{1}{c_\beta^{\frac{1}{2}-\delta}(t_0+t)^{\frac{1}{2}}}
E_\lambda(t:t_0)^{\frac{1}{2}}
\left(
\int_{\Omega}
  |u|^2\Phi_\beta^{-1+2\delta}
\,dx
\right)^{\frac{1}{2}}
\\
&\leq
\frac{1}{c_\beta^{1-2\delta}t_0}
E_\lambda(t:t_0)
+
\frac{1}{4}
\int_{\Omega}
  |u|^2\Phi_\beta^{-1+2\delta}
\,dx
\end{align*}
and therefore
\begin{align*}
E_\lambda(t:t_0)
+\nu \widetilde{E}_\lambda(t:t_0)
&\geq
\left(1-
\frac{\nu}{c_\beta^{1-2\delta}t_0}\right)
E_\lambda(t:t_0)
+
\frac{\nu}{2}
\int_{\Omega}
  |u|^2\Phi_\beta^{-1+2\delta}
\,dx
\\
E_\lambda(t:t_0)
+\nu \widetilde{E}_\lambda(t:t_0)
&\leq
\left(1+
\frac{\nu}{c_\beta^{1-2\delta}t_0}\right)
E_\lambda(t:t_0)
+
\frac{3\nu}{2}
\int_{\Omega}
  |u|^2\Phi_\beta^{-1+2\delta}
\,dx.
\end{align*}
In view of Lemma \ref{Phi} {\bf (iii)},
this means that the assumption $t_0\geq c_\beta^{-1+2\delta}\nu$
implies the assertion of this lemma.
\end{proof}
Furthermore we set
\begin{align*}
Y_\lambda(t:t_0)
&:=
\int_0^t\!\!\int_{\Omega}|\nabla u(x,s)|^2\Psi(s:t_0)^\lambda\,dx\,ds, \quad t\in [0,T_*)
\\
Z_\lambda(t:t_0)
&:=
\int_0^t(t_0+s)\int_{\Omega}|\partial_tu(x,s)|^2\Psi(s:t_0)^\lambda\,dx\,ds,
\quad t\in [0,T_*).
\end{align*}
\begin{proposition}\label{prop:WEE}
There exists positive constants  $t_0^*\geq 1$ and $\eta^*>0$
such that
\begin{align*}
&\eta^*
\left(
m_\lambda(t:t_0^*)
+
Y_\lambda(t:t_0^*)
+
Z_\lambda(t:t_0^*)
\right)
\\
&\leq
m_\lambda(0:t_0^*)
+\int_{\mathbb{R}^N}|F(u_0)|\Psi_*(0)^\lambda\,dx
\\
&\quad
+(t_0^*+t)\int_{\mathbb{R}^N}|F(u(t))|\Psi_*(t)^\lambda\,dx
+
\int_0^t\!\!
\int_{\mathbb{R}^N}
(|F(u(s))|+|u(s)f(u(s))|)\Psi_*(s)^{\lambda}
\,dx\,ds,
\end{align*}
where $\Psi_*(\cdot,t)=\Psi(\cdot,t:t_0^*)$.
\end{proposition}
\begin{remark}
If $f\equiv 0$, then \eqref{eq:ndw} is linear problem of
damped wave equation in exterior domain.
In this case Proposition \ref{prop:WEE} provides the following energy decay estimates
\begin{gather*}
\int_{\mathbb{R}^N}|\nabla u(t)|^2\,dx
+\int_{\mathbb{R}^N}|\partial_t u(t)|^2\,dx
\leq
C(1+t)^{-\lambda-1},
\\
\int_{\mathbb{R}^N}|u(t)|^2\,dx
\leq
C(1+t)^{-\lambda}
\end{gather*}
under the assumption $(|\nabla u_0|^2+|u_0|^2+|u_1|^2)\lr{x}^{\lambda}\in L^1(\Omega)$ with $\lambda\in [0,\frac{N}{2})$,
which is slightly weaker than that of \cite{SoWa4}.
\end{remark}
\begin{proof}
We see from Lemmas \ref{lem:energy-1} and \ref{lem:energy-2} that
if there exists a constant $t_1>0$ such that if
$t_0\geq t_1$, then we have
\begin{align*}
\frac{d}{dt}
E_\lambda(t:t_0)
&\leq
(\lambda^2+\lambda+1)\int_{\Omega}|\nabla u|^2\Psi^{\lambda}\,dx
-\frac{1}{2}(t_0+t)\int_{\Omega}|\partial_tu|^2\Psi^{\lambda}\,dx
\\
&\quad
+
\frac{d}{dt}
\left[
2(t_0+t)\int_{\Omega}
F(u)\Psi^{\lambda}
\,dx
\right]
+
2(\lambda+1)\int_{\Omega}
F(u)\Psi^{\lambda}
\,dx
\\
c_{\beta}^{1-2\delta}
\frac{d}{dt}
\widetilde{E}_\lambda(t:t_0)
&\leq
\left(
2
+\frac{(1-2\delta)\beta C_{\beta+1}}{c_\beta K(\lambda)}
\right)
\int_{\mathbb{R}^N}|\partial_tu|^2\Psi^{\lambda}\,dx
-
\frac{\delta c_\beta^{1-2\delta}}{(1-\delta)C_\beta^{1-2\delta}}
\int_{\mathbb{R}^N}|\nabla u|^2\Psi^\lambda\,dx
\\
&\quad
+2
\int_{\mathbb{R}^N}|uf(u)|\Psi^\lambda\,dx.
\end{align*}
Therefore by choosing $\nu=\delta^{-1}(1-\delta)C_\beta^{1-2\delta}(\lambda^2+\lambda+2)$
and $t_0^*$ sufficiently large,
we have
\begin{align*}
\frac{d}{dt}\Big[
E_\lambda(t:t_0)
+\nu \widetilde{E}_\lambda(t:t_0)
\Big]
&\leq
\left[
\nu
\left(
2
+\frac{(1-2\delta)\beta C_{\beta+1}}{c_\beta K(\lambda)}
\right)
-\frac{1}{2}(t_0+t)
\right]
\int_{\Omega}|\partial_tu|^2\Psi^{\lambda}\,dx
\\
&\quad
-\int_{\Omega}|\nabla u|^2\Psi^{\lambda}\,dx
+
\frac{d}{dt}
\left[
2(t_0+t)\int_{\Omega}
F(u)\Psi^{\lambda}
\,dx
\right]
\\
&\quad
+
2(\lambda+1)\int_{\Omega}
F(u)\Psi^{\lambda}
\,dx
+
2\nu
\int_{\mathbb{R}^N}|uf(u)|\Psi^\lambda\,dx
\\
&\leq
-\frac{1}{4}(t_0+t)
\int_{\Omega}|\partial_tu|^2\Psi^{\lambda}\,dx
-\int_{\Omega}|\nabla u|^2\Psi^{\lambda}\,dx
\\
&\quad
+
\frac{d}{dt}
\left[
2(t_0+t)\int_{\Omega}
F(u)\Psi^{\lambda}
\,dx
\right]
\\
&\quad
+
2(\lambda+1)\int_{\Omega}
F(u)\Psi^{\lambda}
\,dx
+
2\nu
\int_{\mathbb{R}^N}|uf(u)|\Psi^\lambda\,dx
\end{align*}
Integrating it over $[0,t]$ and applying Lemma \ref{lem:equiv}, we have
\begin{align*}
&
\gamma_\nu m_\lambda(t:t_0)
+
\int_0^t\!\!\int_{\Omega}|\nabla u(s)|^2\Psi(s)^\lambda\,dx\,ds
+
\frac{1}{4}
\int_0^t(t_0^*+s)\int_{\Omega}|\partial_tu(s)|^2\Psi(s)^\lambda\,dx\,ds
\\
&\leq
\Gamma_\nu m_\lambda(0:t_0)
-
2t_0\int_{\Omega}
F(u_0)\Psi(0)^{\lambda}
\,dx
\\
&\quad
+
2(t_0+t)\int_{\Omega}
F(u(t))\Psi(t)^{\lambda}
\,dx
+
\int_0^t\!\!\int_{\Omega}
\Big(2(\lambda+1)F(u(s))+2\nu |u(s)f(u(s))|)\Big)\Psi(s)^{\lambda}
\,dx\,ds,
\end{align*}
where $\Psi_*(t)=\Psi(\cdot,t:t_0^*)$.
This yields the desired inequality.
\end{proof}

\begin{proof}[Proof of Theorem \ref{main}]
Put $m_{\lambda}(t)=m_\lambda(t:t_0^*)$ and $Y_{\lambda}(t)=Y_\lambda(t:t_0^*)$.
By Proposition \ref{prop:WEE}
and \eqref{ass.f}, we deduce
\begin{align}
\nonumber
\eta^*
\Big(m_\lambda(t)+Y_\lambda(t)\Big)
&
\leq
m_\lambda^*+\frac{C_f}{p+1}(t_0^*+t)\int_{\mathbb{R}^N}|u(t)|^{p+1}\Psi_*(t)^\lambda\,dx
\\
\label{eq:proofmain}
&\quad+
\frac{(p+2)C_f}{p+1}
\int_0^t\!\!
\int_{\Omega}
|u(s)|^{p+1}\Psi_*(s)^{\lambda}
\,dx\,ds,
\end{align}
where
\[
m_{\lambda}^*:=
m_\lambda(0)+\frac{C_f}{p+1}\int_{\mathbb{R}^N}|u_0|^{p+1}\Psi_*(0)^\lambda\,dx<\infty.
\]
{\bf (The supercritical case $1+\frac{4}{N+2\lambda}<p\leq \frac{N}{N-2}$)} Observe that Lemma \ref{lem:GNtype} that
\begin{align*}
&\int_{\Omega}|u(t)|^{p+1}\Psi_*(t)^\lambda\,dx
\\
&\leq
\widetilde{C}_p^{p+1}(t_0^*+t)^{-\frac{\lambda}{2}(p-1)}
\left(
\int_{\Omega}|u(t)|^{2}\Psi_*(t)^\lambda\,dx
\right)^{\frac{p+1}{2}-\frac{N(p-1)}{4}}
\left(
\int_{\Omega}|\nabla u(t)|^{2}\Psi_*(t)^\lambda\,dx
\right)^{\frac{N(p-1)}{4}}
\\
&\leq
\widetilde{C}_p^{p+1}(t_0^*+t)^{-\frac{N+2\lambda}{4}(p-1)}
m_{\lambda}(t)^{\frac{p+1}{2}}.
\end{align*}
Therefore from \eqref{eq:proofmain}
we obtain the following integral inequality for $m_{\lambda}(t)$:
\begin{align*}
\eta^*
m_\lambda(t)
&\leq
m_\lambda^*
+
\frac{C_f\widetilde{C}_p^{p+1}}{p+1}
(t_0^*+t)^{1-\frac{(N+2\lambda)(p-1)}{4}}m_\lambda(t)^{\frac{p+1}{2}}
\\
&\quad+
\frac{(p+2)C_f\widetilde{C}_p^{p+1}}{p+1}
\int_0^t\!\!
(t_0^*+t)^{-\frac{(N+2\lambda)(p-1)}{4}}m_\lambda(s)^{\frac{p+1}{2}}
\,ds, \quad t\in [0,T_*).
\end{align*}
Consequently, setting
\[
M_\lambda(t):=\sup_{0\leq \tau\leq t}m_\lambda(\tau), \quad t\in [0,T_*)
\]
we see from the assumption $p>1+\frac{4}{N+2\lambda}$ that $\frac{(N+2\lambda)(p-1)}{4}>1$
\[
\eta^*	M_{\lambda}(t)
\leq
m_{\lambda}^*
+
\frac{C_f\widetilde{C}_p^{p+1}}{p+1}
\left(
1
+
\frac{p+2}{1-\frac{(N+2\lambda)(p-1)}{4}}
\right)M_\lambda(t)^{\frac{p+1}{2}}.
\]
It is worth noticing that $M_\lambda$ is continuous.
This implies that there exist constants $\delta_\lambda\*>0$
and $M_\lambda^*>0$ such that
if $m_{\lambda}^*\leq \delta_\lambda^*$, then $M_\lambda(t)\leq M_\lambda^*$ for every $t\in [0,T_*)$, that is, we obtain
\[
(t_0+t)
\int_{\mathbb{R}^N}\Big(|\nabla u(x,t)|^2+|\partial_tu(x,t)|^2\Big)\Psi(x,t)^{\lambda}\,dx
+
\int_{\mathbb{R}^N}|u(x,t)|^2\Psi(x,t)^{\lambda}\,dx
\leq M_\lambda^*, \quad t\in(0,T_*).
\]
{\bf (The critical case $p=1+\frac{4}{N+2\lambda}$)} In this case, $Y_\lambda$ plays an important role.
Note that H\"older inequality yields
\begin{align*}
\int_{\Omega}|u|^{p+1}\Psi^{\lambda}\,dx
\leq
\left(
\int_{\Omega}|u|^2\Psi^{\lambda-1}\,dx
\right)^{1-\theta}
\left(
\int_{\Omega}|u|^{q+1}\Psi^{\frac{q+1}{2}\lambda}\,dx
\right)^{\theta}
\end{align*}
with $\theta=\frac{N}{N+2\lambda}\in(0,1)$ and
$q=1+\frac{4}{N}$.  By Lemma \ref{lem:GNineq}
with $p=q$ and Lemma \ref{lem:hardy}, we deduce
\[
\int_{\Omega}|u|^{q+1}\Psi^{\frac{q+1}{2}\lambda}\,dx
\leq
C_*
\int_{\Omega}|\nabla u|^2\Psi^{\lambda}\,dx
\left(\int_{\Omega}|u|^{2}\Psi^{\lambda}\,dx\right)^{\frac{2}{N}}.
\]
Combining the above two estimates and using Lemma \ref{lem:hardy} again, we have
\begin{align*}
\int_{\Omega}|u|^{p+1}\Psi^{\lambda}\,dx
&\leq
C_*^\theta
\left(
\int_{\Omega}|\nabla u|^2\Psi^{\lambda}\,dx
\right)^{1-\theta+\theta}
\left(\int_{\Omega}|u|^{2}\Psi^{\lambda}\,dx\right)^{\frac{2\theta}{N}}
\\
&=
C_*^\theta
\int_{\Omega}|\nabla u|^2\Psi^{\lambda}\,dx
\left(\int_{\Omega}|u|^{2}\Psi^{\lambda}\,dx\right)^{\frac{p-1}{2}}.
\end{align*}
Therefore we see from \ref{eq:proofmain} that
\begin{align}
\nonumber
\eta^*
\Big(m_\lambda(t)+Y_\lambda(t)\Big)
&
\leq
m_\lambda^*+\frac{C_*^\theta C_f}{p+1}m_\lambda(t)^{\frac{p+1}{2}}
\\
\label{eq:proofmain2}
&\quad+
\frac{(p+2)C_*^\theta C_f}{p+1}
\int_0^t m_{\lambda}(s)^{\frac{p-1}{2}}
\int_{\Omega}
|\nabla u(s)|^{2}\Psi_*(s)^{\lambda}
\,dx\,ds,
\end{align}
Choosing
\[
\widetilde{M}_{\lambda}(t):=M_{\lambda}(t)+Y_{\lambda}(t), \quad t\in [0,T_*)
\]
and noting that
\[
\int_0^t m_{\lambda}(s)^{\frac{p-1}{2}}
\int_{\Omega}
|\nabla u(s)|^{2}\Psi_*(s)^{\lambda}
\,dx\,ds
\leq
M_{\lambda}(t)^{\frac{p-1}{2}}
Y_{\lambda}(t)
\leq
\widetilde{M}_{\lambda}(t)^{\frac{p+1}{2}},
\]
we have
\[
\widetilde{M}_{\lambda}(t)
\leq m_{\lambda}^*
+
\frac{(p+3)C_*^\theta C_f}{p+1}
\widetilde{M}_{\lambda}(t)^{\frac{p+1}{2}}.
\]
The rest of the proof is exactly the same as the supercritical case.
The proof is complete.
\end{proof}
\begin{remark}
Similar argument as the critical case also works
when
$1+\frac{4}{N+2\lambda}<p<\{\frac{N}{N-2}, 1+\frac{2}{\lambda}\}$.
\end{remark}

\section{Remark on one-dimensional case}\label{sec:1d}
In this section we consider the one-dimensional case
\begin{equation}\label{eq:ndw_1d}
\begin{cases}
\partial_t^2u(x,t)-\partial_x^2 u(x,t) + \partial_tu(x,t) = f\big(u(x,t)\big),
& (x,t) \in \mathbb{R}\times (0,T),
\\
u(x,0)=u_0(x),
& x \in \mathbb{R},
\\
\partial_tu(x,0)=u_1(x),
& x \in \mathbb{R}.
\end{cases}
\end{equation}
In this case, we can also discuss
the weighted energy estimate of polynomial type for $\lambda\in [0,\frac{1}{2})$.
However, the lack of validity of Hardy type inequality (Lemma \ref{lem:hardy}),
we take a small modification of the weight function in \eqref{eq:E_lambda_2} as follows:
\begin{align*}
\widetilde{\Phi}_{\beta}(x,t:t_0)=\left(2-\frac{1}{t_0+t}\right)\Phi_\beta(x,t:t_0)
\end{align*}
with $\delta=\frac{1-2\lambda}{4}$ and $\beta=\frac{2\lambda}{1+2\lambda}$.
Then by virtue of the properties of $\Phi_\beta$ in Lemma \ref{Phi},
we have
\[
\partial_t\widetilde{\Phi}_{\beta}\geq \Delta\widetilde{\Phi}_{\beta}+\frac{1}{(1+t)^2}\widetilde{\Phi}_{\beta}
\]
and hence
we can proceed an argument similar to the one in Section \ref{sec:wee} with $p>1+\frac{4}{N+2\lambda}=1+\frac{4}{1+2\lambda}$.
It should be noticed that
the case $p=1+\frac{4}{1+2\lambda}$ cannot be treated because of
the lack of the validity of weighted Hardy inequality.
Therefore we have the following estimate
\[
\eta^* M_{\lambda}(t)\leq
m_{\lambda}^*+\widetilde{C}_\varepsilon M_{\lambda}(t)^{\frac{p+1}{2}}\quad
t\in [0,T_*).
\]
Consequently, we can obtain
\begin{theorem}\label{main_1d}
Assume that $N=1$ and $f$ satisfies \eqref{ass.f} with $1<p<\infty$.
Then for every $\lambda \in [0,\frac{1}{2})$,
there exists a positive constant $\delta_{\lambda}^*>0$ such that
the following assertion holds:
For every $(u_0,u_1)\in H^2(\mathbb{R})\times H^1(\mathbb{R})$ satisfying
\[
\int_{\mathbb{R}}\Big(|\nabla u_0|^2+|u_0|^2+|u_1|^2\Big)\lr{x}^{2\lambda}\,dx\leq \delta_\lambda^*,
\]
one has $T_*(u_0,u_1)=\infty$ when $p>1+\frac{4}{1+2\lambda}$. Namely, there exists a global weak solution $u$ of \eqref{eq:ndw_1d}
with initial data $(u_0,u_1)$.
Moreover, $u$ satisfies the following weighted energy estimate:
there exists a positive constant $M_{\lambda}^*>0$ such that
\begin{gather*}
\int_{\mathbb{R}}\left(|\nabla u(t)|^2+|\partial_tu(t)|^2\right)\left(1+t+|x|^{2}\right)^{\lambda}\,dx
\leq \frac{M_{\lambda}^*}{1+t},
\\
\int_{\mathbb{R}}|u_1(t)|^2\left(1+t+|x|^{2}\right)^{\lambda}\,dx
\leq M_{\lambda}^*.
\end{gather*}
\end{theorem}

\subsection*{Acknowedgements}
This work is partially supported
by Grant-in-Aid for Young Scientists Research
No.18K134450.
The author also thanks Prof.\ Yuta Wakasugi
for giving a valuable comment for the result,
which helps the author to accomplish to close the paper.