Asymptotic decay for defocusing semilinear wave equations in $\mathbb{R}^{1+1}$
Dongyi Wei Shiwu Yang
math.AP






 
  
  School of Mathematical Sciences, Peking University, Beijing, China E-mail address:  

  
  Beijing International Center for Mathematical Research, Peking University, Beijing, China E-mail address: 
  





Asymptotic decay for defocusing semilinear wave equations in ℝ^1+1
    Dongyi Wei Shiwu Yang
    
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This paper is devoted to the study of  asymptotic behaviors of solutions to the one-dimensional defocusing semilinear wave equation.

We prove that finite energy solution tends to zero in the pointwise sense, hence improving the averaged decay of Lindblad and Tao <cit.>. Moreover, for sufficiently localized data belonging to some weighted energy space, the solution decays in time with an inverse polynomial rate. This confirms a conjecture raised in the mentioned work.

The results are based on new weighted vector fields as multipliers applied to regions enclosed by light rays. The key observation for the first result is an integrated local energy decay for the potential energy, while the second result relies on a type of weighted Gagliardo-Nirenberg inequality.


















§ INTRODUCTION

In this paper, we study the global asymptotic behaviors for solutions of  the following defocusing semilinear wave equation

    ϕ=-∂_t^2ϕ + ∂_x^2ϕ=|ϕ|^p-1ϕ,  ϕ(0, x)=ϕ_0(x),  _tϕ(0, x)=ϕ_1(x)

in ℝ^1+1 with p>1. Due to the dispersive nature of the equation, the associated energy is coercive. Hence the local in time solution in energy space can be extended to the whole spacetime. However nothing too much can be said regarding the global dynamics  except that the solution is uniformly bounded.



In the beautiful work <cit.> by Lindblad and Tao, they showed that finite energy solution enjoys the average decay estimate

    lim_T→+∞1/T∫_0^Tϕ(t)_L_x^∞()dt=0.

A direct and important consequence of this result is that the solution asymptotically does not approach to any linear wave. This is in vast contrast to the situation in higher dimensions, where it has been shown that solutions to the energy subcritical defocusing semilinear wave equations behave like linear waves in certain sense for sufficiently large p. For such results, we refer to <cit.>, <cit.>, <cit.> and references therein.

The key reason underlying this is that higher dimensional free wave decays in time while one-dimensional wave does not. In particular, at least perturbative method can be used to study the asymptotic behaviors for solutions of the nonlinear wave equation (<ref>) in higher dimensions. For the one dimensional case studied in this paper, the above averaged decay estimate can only tell us that the solution does not scatter to linear wave. But it remains unknown what is the solution scatters to  even with small initial data.
Motivated by this, the aim of this paper is to give quantitative asymptotic decay properties for solutions of the nonlinear wave equations (<ref>).

To state our main results, define the weighted energy norm of the initial data

    ℰ_[ϕ]= ∫_(1+|x|)^(|∂_xϕ_0|^2+|ϕ_1|^2+2/p+1|ϕ_0|^p+1)dx,  ∀γ≥ 0.

Our first result is to show that for finite energy solution, the potential energy decays to zero. As a consequence, the solution also decays to zero in the pointwise sense.



Consider the Cauchy problem to the defocusing semilinear wave equation (<ref>) in ℝ^1+1 with finite energy data, that is, ℰ_0[ϕ]<+∞. Then the solution ϕ is globally in time and verifies the following decay properties:
  
    lim_t→∞ϕ(t)_L_x^p+1()=0,  lim_t→∞ϕ(t)_L_x^∞()=0.

  
Due to the energy conservation, the standard Gagliardo-Nirenberg inequality shows that the pointwise decay is a consequence of the potential energy decay.
 This is partly inspired by the related works <cit.>, <cit.>, <cit.> in higher dimensions, in which time
 decay of the potential energy is the first step toward the asymptotic behaviors of the solutions.






However the above improved decay estimates are still far from precisely describing the asymptotics of the solution. As conjectured by Lindblad and Tao in
<cit.>, the solution should decay in time at a polynomial rate instead of merely qualitative decay estimates. We now give this conjecture an
affirmative answer.


Consider the Cauchy problem to the defocusing semilinear wave equation (<ref>). For initial data (ϕ_0,ϕ_1) bounded in ℰ_1[ϕ], the solution ϕ satisfies the following decay estimates:

  * 

  * In the exterior region when |x|≥ |t|, we have

    |ϕ(t,x)|≤ C(1+|t|+|x|)^-1/p+3(1+|x|-|t|)^-1/p+3.


  * In the interior region |x|< |t|, the solution verifies the decay estimate

    |ϕ(t,x)|≤ C(1+|t|+|x|)^1-2/p+3(1+|t|-|x|)^1-2/p+3

for all constants , $̱ such that(1/-1)(1/-1)=4/(p+1)^2,   1/2≤<1.In particular, we have the uniform time decay of the solutionϕ(t, x)_L_x^∞()≤ C(1+|t|)^-p-1/(p+1)^2+4.HereCdepends only onp,,$̱ and the weighted energy ℰ_1[ϕ].


The uniform time decay for the solution follows by taking =̱12. One can also choose

    ==̱p+1/p+3

to conclude that the solution verifies the following decay estimate

    |ϕ(t, x)|≤ C (1+t+|x|)^-p-1/(p+3)^2(1+|t-|x||)^-p-1/(p+3)^2,  ∀ t≥ 0.

This improves the decay rate on the region far away from the light rays.



Although the constant C relies on the power p, it is uniformly bounded in the limit p→ 1 in the exterior region. In particular, the decay estimate in the exterior region is sharp in the sense that it is consistent with the decay of solutions to linear Klein-Gordon equation in the limiting case p=1.  The reason for this is that the vector field method used for studying wave equation also works for Klein-Gordon equations in the exterior region, see for example <cit.>.



It is possible to improve the decay estimates in the interior region. But it seems that there is not too much room for doing this due to the fact that the solution can not decay too fast in view of Theorem <ref>.


Although there is significant difference for the energy subcritical defocusing semilinear equation in one dimension and higher dimensions, a common feature is that the solution exhibits certain decay properties. The key estimate in the work of Lindblad and Tao is the improved potential energy decay

    ∫_t_0-T^t_0+T∫_x_0+vt-R^x_0+vt+R|ϕ(t, x)|^p+1dxdt≤ C(√(RT)+R^-1T),  ∀ T≥ R>0

on parallelogram, derived by using the vector field v_t+_x as multiplier. The averaged decay estimate of the solution then follows by using the classical Rademacher differentiation theorem.


One difficulty to study the asymptotic decay for one dimensional semilinear wave equation is the lack of conformal symmetry. In higher dimensions,
there is a critical power p=d+3/d-1 such that the equation is conformally invariant. And the equation can be further classified into subconformal
case and superconformal case. Since higher dimensional linear wave decays in time, the problem becomes easier for larger p.

The importance of this conformal symmetry is that it directly leads to the time decay of the potential energy

    ∫_ℝ^d|ϕ(t, x)|^p+1dx ≤ C (1+ t)^max{d+1-p(d-1), -2}

by using the conformal Killing vector field as multiplier,
see <cit.>, <cit.>, <cit.> and recent improvements <cit.>, <cit.>. However, this method is only effective when the power p has a lower bound close to the conformal power. It fails for the one dimensional problem studied in this paper when d=1 due to the fact that equation (<ref>) is subconformal.

To overcome these difficulties and partly inspired by the work of Lindblad and Tao, we make use of new vector fields as multipliers to obtain weighted energy estimates. The key observation for the proof of Theorem <ref> is the following integrated local energy decay for the potential energy

    ∫_0^+∞∫_-t-1^t+1((t+1)^2-x^2)|ϕ(t,x)|^p+1/(t+1)^3dxdt≤ C,

derived by using the vector field

    (1+(1+t)^-2x^2)_t+2(1+t)^-1x_x

as multiplier. This estimate plays the role that it indicates that the potential energy concentrates mainly in a neighborhood of the light ray {t=|x|}. On the region close to the light ray, we can bound the potential energy by the standard energy conservation. This immediately leads to the averaged potential energy decay

    lim_T→∞1/T∫_0^T∫_|x|≤ t+R|ϕ|^p+1 dxdt =0,  ∀ R>0.

Then by using the translated scaling vector field (t+R)_t+ x_x as multiplier, we can improve the averaged time decay of the potential energy to be uniform time decay, hence concluding Theorem <ref>. We remark here that since the initial energy on the region {|x|>R} goes to zero as R→∞, in view of finite speed of propagation, the above averaged potential energy decay holds true after taking the limit R→∞. By using the standard energy conservation and Gagliardo-Nirenberg inequality, this gives an alternative proof for the averaged decay of Lindblad and Tao.

For the polynomial decay estimates of Theorem <ref>, we need to use weighted vector fields as multipliers. The main difficulty lies in the necessity that the method should work for all range of p, including the limiting case when p=1, where the nonlinear wave equation degenerates to linear Klein-Gordon equation. Inspired by the vector field method in <cit.> effective both for the wave equation and Klein-Gordon equation in the exterior region {t≤ |x|+1}, we show that the Lorentz rotation vector field x_t+t_x is sufficient to derive the necessary weighted energy estimates through light lines in the exterior region. The inverse polynomial decay of the solution in the exterior region then follows by using a type of weighted Gagliardo-Nirenberg inequality (see Lemma <ref> for details).


The situation inside the light cone {|x|≤ t+1} is more involved. We instead use the following class of new vector fields

    ^̱-1 (1+t-x)^(1+t+x)^-1(_t-_x)+^-1 (1+t-x)^-̱1(1+t+x)^(_t+_x)

as multipliers with positive constants , $̱ verifying the assumptions in Theorem <ref>. Note that1<+<̱2. The weights of these vector fields have order between0and1. The use of such vector fields with low weights as multipliers also appeared in <cit.> for the uniform bound of solution to the three dimensional nonlinear wave equation.




Acknowledgments. S. Yang is partially supported by NSFC-11701017.



§ PRELIMINARY AND ENERGY IDENTITIES

Recall the energy momentum tensor associated to solutionϕof (<ref>)
    𝕋_μν=_μϕ·_νϕ-12 m_μν (^ϕ_ϕ +2/p+1|ϕ|^p+1)

with_1=_x,_0=_tandμ,ν=0, 1. Herem_μνis the flat Minkowski metric onℝ^1+1. In particular, we can compute that

    𝕋_01=𝕋_10=∂_xϕ∂_tϕ,   𝕋_11=|∂_xϕ|^2/2+|∂_tϕ|^2/2-|ϕ|^p+1/p+1,  𝕋_00=|∂_xϕ|^2/2+|∂_tϕ|^2/2+|ϕ|^p+1/p+1.

 Define the null coordinatesu=t+1-x,    v=t+1+xas well as the associated null frameŁ=∂_x+∂_t,   =∂_t-∂_x.In particular we haveŁ u= v=0,   Ł v= u=2.Moreover we can write that

    𝕋_00-𝕋_11=2|ϕ|^p+1/p+1,  𝕋_00-𝕋_01=|ϕ|^2/2+|ϕ|^p+1/p+1,  𝕋_00+𝕋_01=|Łϕ|^2/2+|ϕ|^p+1/p+1.

For solutionϕof the nonlinear wave equation (<ref>), we observe the conservation laws
 
    ∂_t𝕋_00=∂_x𝕋_01,  ∂_t𝕋_01=∂_x𝕋_11.

Applying the first energy conservation to the region bounded by{t=t}and the initial line{t=0}, we derive the
the classical energy conservation

    ∫_𝕋_00(t,x)dx=1/2∫_ℝ|_tϕ(t, x)|^2+|_xϕ(t, x)|^2+2/p+1|ϕ(t, x)|^p+1dx=12 ℰ_0[ϕ],  ∀ t∈ℝ.

We will also use the energy conservation adapted to the region bounded by{t=0},{t=T}and the out going null line{x=t+R}∫_0^T(|Łϕ|^2/2+|ϕ|^p+1/p+1)|_x=t+R dt=∫_t+R^+∞𝕋_00(t,x)dx|_t=T^t=0≤1/2ℰ_0[ϕ]

as well as the associated energy conservation applied to the incoming null line

    ∫_0^T(|ϕ|^2/2+|ϕ|^p+1/p+1)|_-x=t+R dt= ∫^-t-R_-∞𝕋_00(t,x)dx|_t=T^t=0≤1/2ℰ_0[ϕ].

For convenience, we may also carry out the computations under the null frame. First of all, we can rewrite the equation as follows=-∂_t^2+∂_x^2=-Ł,  ϕ=-Łϕ= |ϕ|^p-1ϕ.We therefore can obtain the identities
 
    Ł(ϕ)^2=2(Łϕ)(ϕ)=-2|ϕ|^p-1ϕ·ϕ=-2/p+1 (|ϕ|^p+1) ,
       (Łϕ)^2=2(Łϕ)(Łϕ)=-2|ϕ|^p-1ϕ·Łϕ=-2/p+1Ł(|ϕ|^p+1 ).

The first identity can be viewed as using the vector fieldas multiplier while the second identity is equivalent to take the vector fieldLas multiplier. We finally define a quantity

    Q=(-∂_t^2+∂_x^2)ϕ^2=-2|∂_tϕ|^2+2|∂_xϕ|^2+2|ϕ|^p+1=-2(Łϕ)(ϕ)+2|ϕ|^p+1,

which will be frequently used in the sequel.












The proof for Theorem <ref> relies on weighted energy estimates through null lines obtained by using weighted vector fields as
multipliers.
For the pointwise decay estimates, we will rely on the following type of  
  Gagliardo-Nirenberg  inequality.

Let a_1, a_2, μ_1, μ_2 be constants such that

    a_1≥ 0,    a_2≥ 1,   0≤μ_1≤ 1,  μ_2≥ -μ_1.

Then for a function g(t)∈ H_loc^1([a_1,+∞)), 

it holds that

    (t+a_2)^μ_1+μ_2|g(t)|^p+3   ≤ C   ∫_a_1^∞ (s+a_2)^μ_2|g(s)|^p+1ds  ·∫_a_1^∞(s+a_2)^μ_1|g'(s)|^2ds

for all t≥ a_1 with some constant C independent of t, a_1 and a_2. Here we assume that each integral in the right hand side of the above inequality is finite.

Define the non-negative function
h(t)=(t+a_2)^μ_1+μ_2/2|g(t)|^p+3/2.
 Then we have

    h'(t)   =μ_1+μ_2/2(t+a_2)^μ_1+μ_2/2-1|g(t)|^p+3/2+p+3/2(t+a_2)^μ_1+μ_2/2|g(t)|^p-1/2
    g(t)g'(t)
    
       ≥-p+3/2(t+a_2)^μ_1+μ_2/2|g(t)|^p+1/2|g'(t)|.

Here we note that μ_1+μ_2≥ 0. Now using Hölder inequality we have

    max(-h'(t),0)_L^1([a_1,+∞))   ≤ C(t+a_2)^μ_1+μ_2/2|g(t)|^p+1/2|g'(t)|_L^1([a_1,+∞))
       ≤ C(t+a_2)^μ_2|g|^p+1_L^1([a_1,+∞))^1/2(t+a_2)^μ_1|g'|^2_L^1([a_1,+∞))^1/2.

For t_2≥ t_1≥ a_1 we have

    h(t_1)-h(t_2)=∫_t_1^t_2-h'(t)dt≤∫_t_1^t_2max(-h'(t),0)dt≤max(-h'(t),0)_L^1([a_1,+∞)).

Therefore

    h(t_1)≤inf_t≥ t_1h(t)+max(-h'(t),0)_L^1([a_1,+∞)),  ∀ t_1≥ a_1.

Now for fixed t_1≥ a_1, we claim that inf_t≥ t_1h(t)=0. Otherwise if inf_t≥ t_1h(t)=c_0 for some positive constant c_0, then we derive that

    h(t)=(t+a_2)^μ_1+μ_2/2|g(t)|^p+3/2≥ c_0,   |g(t)|≥ (t+a_2)^-μ_1+μ_2/p+3c_0^2/p+3,
        (t+a_2)^μ_2|g|^p+1≥ (t+a_2)^μ_2-p+1/p+3(μ_1+μ_2)c_0^2(p+1)/p+3,  ∀ t≥ t_1.

As μ_1+μ_2≥ 0 and μ_1≤ 1, we then conclude that

    μ_2-p+1/p+3(μ_1+μ_2)≥μ_2-(μ_1+μ_2)≥ -μ_1≥ -1.

Now from the assumption a_2≥ 1, t_1≥ a_1≥ 0, we show that

    (t+a_2)^μ_2|g|^p+1_L^1([a_1,+∞))   ≥(t+a_2)^μ_2|g|^p+1_L^1([t_1,+∞))
       ≥(t+a_2)^μ_2-p+1/p+3(μ_1+μ_2)_L^1([t_1,+∞))c_0^2(p+1)/p+3
       ≥(t+a_2)^-1_L^1([t_1,+∞))c_0^2(p+1)/p+3=+∞,

which contradicts the assumption that the first integral in the right hand side of the inequality of the Lemma is finite.

 Therefore we must have inf_t≥ t_1h(t)=0, which together with estimates (<ref>) and (<ref>) leads to

    h(t_1)    ≤max(-h'(t),0)_L^1([a_1,+∞))
       ≤ C(t+a_2)^μ_2|g|^p+1_L^1([a_1,+∞))^1/2(t+a_2)^μ_1|g'|^2_L^1([a_1,+∞))^1/2.

The Lemma then holds by definition of h and squaring the above inequality.

For the special case when a_1=μ_1=μ_2=0, we derive from Lemma <ref> that

    |g(t)|^p+3/2   ≤ C|g|^p+1_L^1([0,+∞))^1/2|g'|^2_L^1([0,+∞))^1/2, ∀ t≥ 0.

By symmetry we also have

    |g(t)|^p+3/2   ≤ C|g|^p+1_L^1((-∞,0])^1/2|g'|^2_L^1((-∞,0])^1/2, ∀ t≤ 0.

Now we obtain the classical Gagliardo-Nirenberg inequality

    g_L^∞()^(p+3)/2   ≤ Cg_L^p+1()^(p+1)/2g'_L^2()

with constant C depending only on p.

For finite energy solution of (<ref>), the above classical Gagliardo-Nirenberg inequality together with the energy conservation (<ref>) implies that

    ϕ(t, x)_L_x^∞()^ p+3   ≤ C ∫_ |ϕ(t, x)|^p+1 dx·∫_ |_xϕ(t, x)|^2dx ≤p+1/2C ℰ_0[ϕ]^2.

In particular the finite energy solution verifies the following uniform bound

    |ϕ|≤ C_p ℰ_0[ϕ]^2/p+3.

This uniform bound is crucial during the proof. Moreover we see that the solution decays to0once we have the potential energy decay for finite energy solutions.



Since the wave equation is time reversible, without loss of generality we only prove estimates in the futuret≥0. Hence in the sequel we always assume thatt≥0unless it is specified.





§ PROOF OF THEOREM <REF>

In this section, the implicit constant inABdepends only on the powerpand the standard energyℰ_0[ϕ].


The key new observation to prove Theorem <ref> is the time decay of the potential energy. By using the scaling vector field as multiplier applied to a truncated forward light cone, it follows from a type of weaker averaged decay estimate of the potential energy, which indeed is a consequence of the following the following integrated local potential energy decay estimate.

Let ϕ be a finite energy solution to (<ref>). Then the potential energy verifies the following decay estimate

    ∫_0^+∞∫_-t-1^t+1((t+1)^2-x^2)|ϕ(t,x)|^p+1/(t+1)^3dxdt≤ C < ∞

for some constant C depending only on p and the energy ℰ_0[ϕ].

Such integrated local energy estimate dates back to Morawetz <cit.>, <cit.>. The original method there can easily lead to a stronger estimate than that of the above Proposition in higher dimenions (d≥ 3).
Similar estimates are available for solutions of defocusing nonlinear Klein-Gordon equations in lower dimensions, see <cit.>.

Let t_*=t+1. Recall the null frame  Ł=∂_x+∂_t, =∂_t-∂_x and the associated null coordinates u=t_*-x, v=t_*+x. In particular we have
Ł (u/t_*)=-Ł (v/t_*)=-u/t_*^2,   (v/t_*)=-(u/t_*)=-v/t_*^2 .
Then in view of the equation (<ref>), we have

    Ł(u^2t_*^-2(ϕ)^2)+2/p+1(u^2 |ϕ|^p+1 t_*^-2)   =Ł(u^2t_*^-2)(ϕ)^2+2/p+1(u^2t_*^-2) |ϕ|^p+1
       =-2u^2t_*^-3(ϕ)^2+4/p+1 uv t_*^-3 |ϕ|^p+1.

Similarly by (<ref>), we also have

    (v^2t_*^-2(Łϕ)^2)+2/p+1Ł (t_*^-2v^2|ϕ|^p+1)
    =-2v^2t_*^-3(Łϕ)^2+4/p+1 uvt_*^-3 |ϕ|^p+1.

Combining these two identities, we end up with

    Ł(u^2(ϕ)^2/t_*^2+2v^2|ϕ|^p+1/t_*^2(p+1))
    +(v^2(Łϕ)^2/t_*^2+2u^2|ϕ|^p+1/t_*^2(p+1))
       =-2u^2t_*^-3(ϕ)^2-2v^2t_*^-3(Łϕ)^2+8/p+1 uvt_*^-3 |ϕ|^p+1.

Now we introduce

    P_3:=u^2(ϕ)^2/t_*^2+2v^2|ϕ|^p+1/t_*^2(p+1),    P_4:=v^2(Łϕ)^2/t_*^2+2u^2|ϕ|^p+1/t_*^2(p+1).

And recall Q=ϕ^2 as in (<ref>). Then the previous inequality leads to

    Ł P_3
    + P_4-2uvt_*^-3Q
       =-2u^2t_*^-3(ϕ)^2-2v^2t_*^-3(Łϕ)^2+4uvt_*^-3(Łϕ)(ϕ)-4(p-1)/p+1 uvt_*^-3 |ϕ|^p+1
       =-2t_*^-3|uϕ-vŁϕ|^2-4(p-1)/p+1 uvt_*^-3  |ϕ|^p+1
       ≤ -4(p-1)/p+1 uvt_*^-3  |ϕ|^p+1.

Integrating this inequality on the domain

    Σ_1^T:={(t, x): 0≤ t≤ T,   |x|≤ t+1},   T>0

and using Stokes formula, we obtain that

    p-1/p+1∫_Σ_1^T4uvt_*^-3|ϕ|^p+1dxdt-∫_Σ_1^T2uvt_*^-3Qdxdt
       ≤-∫_Σ_1^T(Ł P_3
    + P_4)dxdt
       =-∫_Σ_1^T(∂_t(P_3+P_4)+∂_x(P_3-P_4)
    )dxdt
       =∫_∂Σ_1^T((P_3+P_4)dx-(P_3-P_4)dt).

Since Q=ϕ^2, by Stokes formula again, we have

    ∫_Σ_1^Tuvt_*^-3Qdxdt=∫_Σ_1^Tuvt_*^-3ϕ^2dxdt=∫_Σ_1^Tϕ^2(uvt_*^-3) dxdt+∫_∂Σ_1^Tω_1,

  where
  
    ω_1:=(uvt_*^-3∂_x (ϕ^2)-ϕ^2∂_x(uvt_*^-3))dt+(uvt_*^-3∂_t (ϕ^2)-ϕ^2∂_t(uvt_*^-3))dx.

Now for |x|≤ t+1=t_*, we compute that

    (uvt_*^-3)=(t_*^-1-x^2t_*^-3)=-4t_*^-3+12x^2t_*^-5≤ 8t_*^-3.

Using the uniform bound (<ref>) of the solution, we conclude that

    ∫_Σ_1^Tϕ^2(uvt_*^-3) dxdt ∫_1^T∫_-T-1^T+1(t+1)^-3dxdt  1.

Now we need to compute the boundary integrals. The boundary Σ_1^T consists of the null segments
Γ_1”={x=t+1, 0≤ t≤ T},  Γ_2”={-x=t+1, 0≤ t≤ T}
 and the constant t-slice
  Γ_3”={t=0, |x|≤ 1},  Γ_4”={t=T, |x|≤ T+1}.
   On the null segment Γ_1”⊂{x=t+1}, we have u=(∂_x+∂_t)u=0  and

    ω_1=-ϕ^2(∂_x+∂_t)(uvt_*^-3)dt=0.

Similarly on  the null segment Γ_2”, we have ω_1=0.
On the constant t-slice Γ_3”, Γ_4”, we have

    ω_1=(uvt_*^-3∂_t (ϕ^2)-ϕ^2∂_t(uvt_*^-3))dx.

Now for |x|≤ t_*=t+1, we in particular have that

    |uvt_*^-3|=|(t_*^2-x^2)t_*^-3|≤ t_*^-1,   |∂_t(uvt_*^-3)|=|t_*^-2-3x^2t_*^-4|≤ 2t_*^-2.

Then in view of the energy conservation (<ref>) and the uniform bound of the solution (<ref>),



we then can bound that

    ∫_-t-1^t+1|uvt_*^-3∂_t (ϕ^2)-ϕ^2∂_t(uvt_*^-3)|dx 
       ∫_-t-1^t+1t_*^-1|∂_t ϕ||ϕ|+t_*^-2|ϕ|^2 dx
         1+t_*^-1∫_-t-1^t+1 |_tϕ| dx
        1+ t_*^-1/2(∫_-t-1^t+1|_tϕ|^2 dx )^1/2
        1,  ∀ t≥ 0.

In particular the integration of ω_1 on Γ_3”, Γ_4” is uniformly bounded by a constant independent of T. Combining this with estimates (<ref>), (<ref>) and (<ref>), we have shown that

    p-1/p+1∫_Σ_1^T4uvt_*^-3|ϕ|^p+1dxdt 1+∫_∂Σ_1^T(P_3+P_4)dx-(P_3-P_4)dt.

Now for the boundary integral on the right hand side, on the null segment Γ_1”⊂{x=t+1}, we have

    (P_3+P_4)dx-(P_3-P_4)dt=2P_4dt,

while on the null segment Γ_2”⊂{-x=t+1}, the integrand becomes

    (P_3+P_4)dx-(P_3-P_4)dt=-2P_3dt.

On the constant t-slice Γ_3”, Γ_4” on which dt=0,
it is clear that

    (P_3+P_4)dx-(P_3-P_4)dt=(P_3+P_4)dx.

Summing up and considering the orientation we can show that
    ∫_∂Σ_1^T((P_3+P_4)dx-(P_3-P_4)dt)
       =∫_-t-1^t+1(P_3+P_4)dx|_t=T^t=0
      +∫_0^T2P_4(t,t+1)dt+∫_0^T2P_3(t,-t-1)dt.

On the outgoing null segment {x=t+1}, by definition, we have

    P_4(t, t+1)= 4|L(ϕ)|^2.

By using the standard energy estimate (<ref>) adapted to the outgoing null line {x=t+1}, we conclude that

    ∫_0^T P_4(t, t+1)dt ≤ 4 ∫_{x=t+1} |L(ϕ)|^2+2|ϕ|^p+1/p+1dt ≤ 8 ℰ_0[ϕ].

 
 
 
 




Similarly on the incoming null lines {x=-t-1}, we also have that

    ∫_0^T P_3(t,-t-1)dt≤ 8ℰ_0[ϕ].

Next on the constant t-slice such that |x|≤ t+1, we show that

    P_3+P_4   =t_*^-2 u^2 |(ϕ)|^2+t_*^-2 v^2|L(ϕ)|^2+2(u^2+v^2)|ϕ|^p+1/t_*^2(p+1)
       ≤ 4(|L(ϕ)|^2+|(ϕ)|^2)+8|ϕ|^p+1/p+1.

Hence by energy conservation (<ref>), we can bound that

    |∫_-t-1^t+1(P_3+P_4)dx|_t=T^t=0|≤ 16ℰ_0[ϕ].

Then in view of inequality (<ref>), we conclude that

 






    p-1/p+1∫_Σ_1^T4uvt_*^-3|ϕ|^p+1dxdt 1.

Recall that t_*=t+1, uv=t_*^2-x^2. By the definition of Σ_1^T, the above estimate implies that

    ∫_0^T∫_-t-1^t+1((t+1)^2-x^2)|ϕ(t,x)|^p+1/(t+1)^3dxdt=∫_Σ_1^Tuvt_*^-3|ϕ|^p+1dxdt 1.

The proposition then follows by letting T→+∞ and the convention that the implicit constant relies only on the power p and the energy ℰ_0[ϕ].


We now use the above integrated local potential energy decay to show the averaged decay for the potential energy.

Let ϕ be a finite energy solution to the defocusing semilinear wave equation (<ref>). Then for fixed R>0, the potential energy verifies the averaged decay estimate

    lim_T→+∞1/T∫_0^T∫_|x|≤ t+R|ϕ(t,x)|^p+1dxdt=0.

For fixed ϵ∈(0,1), by Proposition <ref> there exists T_0>0 such that

    ∫_T_0^+∞∫_-t-1^t+1((t+1)^2-x^2)|ϕ(t,x)|^p+1/(t+1)^3dxdt≤ϵ^2.

Restricting this integral on a smaller region, we conclude that

    ∫_T_0^+∞∫_|x|≤(t+1)(1-ϵ)|ϕ(t,x)|^p+1/(t+1)dxdt≤ϵ

by noting that
 
    (t+1)^2-x^2/(t+1)^3≥(t+1)-|x|/(t+1)^2≥ϵ/(t+1),  for   |x|≤ (t+1)(1-ϵ).

Now for T>T_0, we control the integrated potential energy by three different parts

    ∫_0^T∫_|x|≤ t+R|ϕ(t,x)|^p+1dxdt≤I+II+III,

where

    I=∫_0^T∫_t-T_0-Tϵ^ t+R    |ϕ(t,x)|^p+1dxdt,  II=∫_0^T∫_-t-R^T_0+Tϵ-t|ϕ(t,x)|^p+1dxdt,
      
    III   =∫_T_0^T∫_|x|≤(t+1)(1-ϵ)|ϕ(t,x)|^p+1dxdt.

To see the above claim, for the case when t≤ T_0, we have the inclusion

    {|x|≤ t+R}⊂{ t-T_0-Tϵ≤ x≤ t+R}∪{ -t-R≤ x≤ T_0+Tϵ-t}

as T_0+Tϵ-t≥ 0. For the other case when T_0≤ t≤ T, by symmetry, it suffices to check the relation

    (t+1)(1-ϵ)≥ t-T_0-Tϵ,

which holds as 0<ϵ<1 and t≤ T.






Now by the above estimate (<ref>), we can bound that

    III≤(T+1)∫_T_0^T∫_|x|≤(t+1)(1-ϵ)|ϕ(t,x)|^p+1/(t+1)dxdt≤ϵ(T+1).

To control I, we make use of the standard energy estimate (<ref>) through the out going null lines to deduce that




    I=∫_-T_0-Tϵ^ R∫_0^T|ϕ(t,t+a)|^p+1dtda≤∫_-T_0-Tϵ^ R(p+1)ℰ_0[ϕ]da=(R+T_0+Tϵ)(p+1)ℰ_0[ϕ].

Similarly or by symmetry we also have that

    II≤(R+T_0+Tϵ)(p+1)ℰ_0[ϕ].

Summing up, we have shown that

    ∫_0^T∫_|x|≤ t+R|ϕ(t,x)|^p+1dxdt≤ 2(R+T_0+Tϵ)(p+1)ℰ_0[ϕ]+ϵ(T+1),

which implies
    lim sup_T→+∞1/T∫_0^T∫_|x|≤ t+R|ϕ(t,x)|^p+1dxdt≤2ϵ(p+1)ℰ_0[ϕ]+ϵ,  ∀ ϵ∈(0,1).

This completes the proof by letting ϵ→ 0.


We are now ready to prove the main Theorem <ref>, for which we need to improve the above averaged decay of the potential energy to decay in the pointwise sense.  This is based on weighted energy estimate obtained by using the scaling vector field as multiplier.

 For fixedR>0,in view of the energy identities (<ref>), (<ref>), we have
 
    ∂_t((t+R)𝕋_00+x𝕋_01)-∂_x((t+R)𝕋_01+x𝕋_11)=𝕋_00-𝕋_11
      =2|ϕ|^p+1/p+1.

Integrating this identity on the domain

    Σ_R^T:={(t, x): 0≤ t≤ T,   |x|≤ t+R},   T>0, R>0

 and using Stokes formula, we obtain that
 
    ∫_∂Σ_R^T[((t+R)𝕋_00+x𝕋_01)dx+((t+R)𝕋_01+x𝕋_11)dt]
      =-∫_Σ_R^T2|ϕ|^p+1/p+1dxdt.

The boundaryΣ_R^Tconsists of the null segmentsΓ_1”={x=t+R, 0≤ t≤ T},  Γ_2”={-x=t+R, 0≤ t≤ T}and the constantt-sliceΓ_3”={t=0, |x|≤ R},  Γ_4”={t=T, |x|≤ T+R}.For the null segmentΓ_1”⊂{x=t+R}, we can compute that (using (<ref>))

    ((t+R)𝕋_00+x𝕋_01)dx+((t+R)𝕋_01+x𝕋_11)dt=x(𝕋_00+2𝕋_01+𝕋_11)dt
    =(t+R)|Łϕ|^2dt.

Similarly for the null segmentΓ_2”⊂{-x=t+R}, we instead have

    ((t+R)𝕋_00+x𝕋_01)dx+((t+R)𝕋_01+x𝕋_11)dt=x(𝕋_00-2𝕋_01+𝕋_11)dt
    =-(t+R)|ϕ|^2dt.

Then on the constantt-sliceΓ_3”, Γ_4”, we have

    ((t+R)𝕋_00+x𝕋_01)dx+((t+R)𝕋_01+x𝕋_11)dt=((t+R)𝕋_00+x𝕋_01)dx.

Therefore the above energy identity reads as

    ∫_-t-R^t+R((t+R)𝕋_00(t,x)+x𝕋_01(t,x))dx|_t=0^t=T
    =   ∫_0^T(t+R)|Łϕ(t,t+R)|^2dt+∫_0^T(t+R)|ϕ(t,-t-R)|^2dt+∫_Σ_R^T2|ϕ(t,x)|^p+1/p+1dxdt.

For the first term on the right hand side, we make use of the standard energy conservation (<ref>)
 adapted to the out going null lines


  

 to bound that

    ∫_0^T(t+R)|Łϕ(t,t+R)|^2dt    ≤ (T+R)∫_0^T|Łϕ(t,t+R)|^2dt
       ≤ 2(T+R)∫_t+R^+∞𝕋_00(t,x)dx|_t=T^t=0.

Similarly or by symmetry, we also have

    ∫_0^T(t+R)|ϕ(t,-t-R)|^2dt≤ 2(T+R)∫_-∞^-t-R𝕋_00(t,x)dx|_t=T^t=0.

On the other hand, for|x|≤t+R, we can estimate that

    (t+R)𝕋_00+x𝕋_01≥ (t+R)(𝕋_00-|𝕋_01|)≥(t+R)|ϕ|^p+1/p+1,
       (t+R)𝕋_00+x𝕋_01≤ 2(t+R)𝕋_00.

This implies that

    ∫_-t-R^t+R((t+R)𝕋_00(t,x)+x𝕋_01(t,x))dx|_t=0^t=T
       ≥ (T+R)∫_|x|≤ T+ R|ϕ(T,x)|^p+1/p+1dx-2R∫_|x|≤ R𝕋_00(0,x)dx.

Combining the above estimates, we conclude that

    (T+R)∫_|x|≤ T+ R|ϕ(T,x)|^p+1/p+1dx-2R∫_|x|≤ R𝕋_00(0,x)dx
       ≤ 2(T+R)∫_|x|≥ t+R𝕋_00(t,x)dx|_t=T^t=0+∫_Σ_R^T2|ϕ(t,x)|^p+1/p+1dxdt.

In particular we derive that

    (T+R)∫_|x|≤ T+ R|ϕ(T,x)|^p+1/p+1dx+2(T+R)∫_|x|≥ T+R𝕋_00(T,x)dx
       ≤ 2R∫_|x|≤ R𝕋_00(0,x)dx+2(T+R)∫_|x|≥ R𝕋_00(0,x)dx+∫_Σ_R^T2|ϕ(t,x)|^p+1/p+1dxdt
       = 2R∫_𝕋_00(0,x)dx+2T∫_|x|≥ R𝕋_00(0,x)dx+∫_Σ_R^T2|ϕ(t,x)|^p+1/p+1dxdt.

Now since

    𝕋_00≥1/p+1 |ϕ|^p+1,

the previous estimate leads to

    (T+R)∫_|ϕ(T,x)|^p+1/p+1dx≤ Rℰ_0[ϕ]+2T∫_|x|≥ R𝕋_00(0,x)dx+∫_Σ_R^T2|ϕ(t,x)|^p+1/p+1dxdt.

Dividing byT+Rand taking limit in terms ofT, we end up with

    lim sup_T→+∞∫_|ϕ(T,x)|^p+1dx≤ 2(p+1)∫_|x|≥ R𝕋_00(0,x)dx+lim sup_T→+∞2/T∫_Σ_R^T|ϕ(t,x)|^p+1dxdt.

Now by using Proposition <ref> and the definition ofΣ_R^T , we have

    lim_T→+∞1/T∫_Σ_R^T|ϕ(t,x)|^p+1dxdt=lim_T→+∞1/T∫_0^T ∫_|x|≤ t+R |ϕ(t,x)|^p+1dxdt=0.

Therefore

    lim sup_T→+∞∫_|ϕ(T,x)|^p+1dx≤ 2(p+1)∫_|x|≥ R𝕋_00(0,x)dx,  ∀ R>0.

Since the initial energy is finite

    ∫_|x|≥ R𝕋_00(0, x)dx≤∫_ℝ𝕋_00(0, x)dx=1/2ℰ_0[ϕ]<∞,

by lettingR→+∞, we conclude that

    lim_T→+∞∫_|ϕ(T,x)|^p+1dx=0,  i.e.  lim_t→+∞ϕ(t)_L_x^p+1()^p+1=0.

Once we have this potential energy decay, the pointwise decay estimate follows from Gagliardo-Nirenberg inequality (see Remark <ref>) together with the energy conservation

    ϕ(t)_L_x^∞()^p+3∂_xϕ(t)_L_x^2()^2ϕ(t)_L_x^p+1()^p+1ℰ_0[ϕ]ϕ(t)_L_x^p+1()^p+1.

This in particular implies that

    lim_t→+∞ϕ(t)_L_x^∞()= lim_t→+∞ϕ(t)_L_x^p+1()=0.

By symmetry, this limit is also true in the past ast→-∞. This completes the proof for Theorem <ref>.









§ PROOF FOR THEOREM <REF>


In this section,  we make a convention thatABmeans there is a constantCdepending only  onp,, $̱ and the weighted energy ℰ_1[ϕ] such that A≤ CB. Here the constants , $̱ verify the assumption in Theorem <ref>.




As we have pointed out in the introduction, the improved inverse polynomial decay of the solution follows from weighted energy estimates together with a type of weighted Gagliardo-Nirenberg inequality of Lemma <ref>. In the exterior region|x|≤t, it suffices to make use of the Lorentz vector fieldt_x+x_tas multiplier to deduce weighted energy estimates through out going null lines. This, however, works only in the exterior region. In the interior region{|x|≤t}, we need to use new weighted vector fields as multipliers to derive the necessary weighted energy estimates through out going and incoming null lines.










 §.§ Decay estimates in the exterior region

In view of the energy identity (<ref>), we in particular have that

    ∂_t((x+1)𝕋_00+t𝕋_01)=∂_x((x+1)𝕋_01+t𝕋_11).

Integrate this equality on the domain

    𝒟_a^b:={(t, x): t≥0,   t-x≤ -a,   t+x≤ b},   a<b.

Using Stokes formula, we obtain that

    ∫_∂𝒟_a^b[((x+1)𝕋_00+t𝕋_01)dx+((x+1)𝕋_01+t𝕋_11)dt]=0.

The boundary𝒟_a^bconsists of the null segmentsΓ_1={t-x=-a, 0≤ t≤b-a/2},  Γ_2={t+x=b, 0≤ t≤b-a/2}.and the constantt-sliceΓ_3={t=0, a≤x≤b}.

 Now we compute the boundary integrals. For the outgoing null segmentΓ_1⊂{t-x=-a}, we can compute that

    ((x+1)𝕋_00+t𝕋_01)dx+((x+1)𝕋_01+t𝕋_11)dt
    
    =    ((x+1)𝕋_00+(t+x+1)𝕋_01+t𝕋_11)dt
    
    =   12 ((t+x+1) |Łϕ|^2+2(x+1-t)/p+1  |ϕ|^p+1)dt
    
    =   12 ((2t+a+1) |Łϕ|^2+ 2(a+1)/p+1  |ϕ|^p+1)dt.

Similarly for the incoming null segmentΓ_2⊂{t+x=b}, we show that

    ((x+1)𝕋_00+t𝕋_01)dx+((x+1)𝕋_01+t𝕋_11)dt
    
    =   (-(x+1)𝕋_00+(x+1-t)𝕋_01+t𝕋_11)dt
    
    =   -12 ((x+1-t) |ϕ|^2+2(x+1+t)/p+1 |ϕ|^p+1)dt
    
    =   -12  ((b+1-2t) |ϕ|^2 +2(b+1)/p+1  |ϕ|^p+1)dt.

Finally on the constantt-sliceΓ_3⊂{t=0}, we have

    ((x+1)𝕋_00+t𝕋_01)dx+((x+1)𝕋_01+t𝕋_11)dt=(x+1)𝕋_00dx.

Summing up and considering the orientation, we obtain that





    ∫_0^b-a/2(2(a+1)|ϕ|^p+1/p+1+(2t+a+1) |Łϕ|^2 )|_x=t+adt
      
       +∫_0^b-a/2(2(b+1)|ϕ|^p+1/p+1+(b+1-2t) |ϕ|^2)|_x=b-tdt
    =   
      2∫_a^b(x+1)𝕋_00(0,x)dx≤  2∫_(|x|+1)𝕋_00(0,x)dx=ℰ_1[ϕ].

Ifa≥0then each term on the left hand side is nonnegative. By lettingb→+∞, we conclude that

    ∫_0^+∞(2(a+1)/p+1|ϕ|^p+1+(2t+a+1)|Łϕ|^2)|_x=t+adt≤ℰ_1[ϕ],  ∀ a≥ 0.

By symmetry (changingxto-x), we also have

    ∫_0^+∞(2(a+1)/p+1|ϕ|^p+1+(2t+a+1)|ϕ|^2)|_x=-a-tdt≤ℰ_1[ϕ],  ∀ a≥ 0.

Alternatively one can also subtract the previous energy identity with four times of the standard energy (obtained by using_tas multiplier) and then letb≤0anda→-∞.

Now for fixeda≥0,  defineg(t)=ϕ(t,t+a),a_1=0,a_2=a+1,μ_1=1,μ_2=0. We in particular have thatg'(t)=Łϕ(t,t+a). Then in view of the above estimate (<ref>), we derive that

    (a+1)|g|^p+1_L^1([0,+∞))+(t+a_2)|g'|^2_L^1([0,+∞))≤p+1/2ℰ_1[ϕ].

By Lemma <ref>, we conclude that

    (t+a_2)^1/2|g(t)|^p+3/2≤ C|g|^p+1_L^1([0,+∞))^1/2(t+a_2)|g'|^2_L^1([0,+∞))^1/2 (a+1)^-1/2.

This in particular leads to the pointwise decay estimate

    |ϕ(t,t+a)|
      (t+a+1)^-1/p+3(a+1)^-1/p+3,  ∀ t≥ 0, a≥ 0,

which implies the first inequality of Theorem <ref> for the casex≥t≥0.By symmetry (or changing variablesx→-x) the first inequality holds on the whole exterior region|x|≥|t|≥0.§.§ Decay estimates in the interior region

The decay estimates in the interior region are much more difficult to obtain. We introduce new vector fields as multipliers.


Now foru=t+1-x>0,v=t+1+x>0and constants,$̱ verifying the assumptions in Theorem <ref>, in view of the identity (<ref>), we can compute that

    Ł(u^v^-1(ϕ)^2)+ 2/p+1 (u^v^-1|ϕ|^p+1)    =Ł(u^v^-1)(ϕ)^2+ 2/p+1(u^v^-1)  |ϕ|^p+1
       =2(-1)u^v^-2(ϕ)^2+ 4/p+1u̱^-̱1v^-1  |ϕ|^p+1.

Similarly (from (<ref>)) we also have

    (u^-̱1v^(Łϕ)^2)+ 2/p+1Ł  ( u^-̱1v^|ϕ|^p+1 )
    =2(-̱1)u^-̱2v^(Łϕ)^2+4/p+1 u^-̱1v^-1  |ϕ|^p+1.

Introduce the following quantities

    P_1=u^v^-1(ϕ)^2/+2u^-̱1v^|ϕ|^p+1/(p+1),  
    P_2=u^-̱1v^(Łϕ)^2/+2u^v^-1|ϕ|^p+1/(̱p+1).

Here recall that , $̱ are positive constants.
Combining the above two identities, we end up with







    Ł P_1+ P_2-4/p+1 u^-̱1v^-1 Q 
       =  2(-̱1)/u^-̱2v^(Łϕ)^2+2(-1)/u^v^-2(ϕ)^2+8/p+1 u^-̱1v^-1 (Łϕ)(ϕ)
    
       =-2u^-̱2v^-2|√(^-1(1-)̱)· vŁϕ-√(^̱-1(1-) )· uϕ|^2
       ≤  0.

Here we used the relation

    ^-1(1-)^-1(1-)̱ =4/(p+1)^2,   0<, <̱1

assumed in the theorem andQis defined in (<ref>).
 Integrate the above inequality on the domain

    ℛ_a^b:={(t, x): 0≤ t-x≤ a,   0≤ t+x≤ b},   a>0, b>0.

By using Stokes formula, we obtain that

    4/p+1∫_ℛ_a^bu^-̱1v^-1 Q dxdt   ≥∫_ℛ_a^b(Ł P_1
    + P_2)dxdt=∫_ℛ_a^b(∂_t(P_1+P_2)+∂_x(P_1-P_2)
    )dxdt
       =∫_∂ℛ_a^b((P_1-P_2)dt-(P_1+P_2)dx).

Recall thatQ=ϕ^2. By using Stokes formula again, we can compute that

    ∫_ℛ_a^bu^-̱1v^-1Qdxdt   =∫_ℛ_a^bu^-̱1v^-1ϕ^2dxdt
       =∫_ℛ_a^bϕ^2(u^-̱1v^-1) dxdt+∫_∂ℛ_a^bω,

  in which
  
    ω=(u^-̱1v^-1∂_x (ϕ^2)-ϕ^2∂_x(u^-̱1v^-1))dt+(u^-̱1v^-1∂_t (ϕ^2)-ϕ^2∂_t(u^-̱1v^-1))dx.

Now note that(u^-̱1v^-1)=-Ł(u^-̱1v^-1)=-4(-̱1)(-1)u^-̱2v^-2<0by the assumption that0<, <̱1. In particular this implies that

    ∫_ℛ_a^bϕ^2(u^-̱1v^-1) dxdt≤ 0.

We now compute that boundary terms.
The boundaryℛ_a^bof the domain consists of the four null segments

    Γ_1'    ={t-x=0, 0≤ t≤b/2},  Γ_2'={t+x=b, b/2≤ t≤b+a/2},
    Γ_3'   ={t-x=a, a/2≤ t≤b+a/2},  Γ_4'={t+x=0, 0≤ t≤a/2}.

 For the null segmentΓ_3'⊂{t=x+a}, we can compute that

    ω=   (u^-̱1v^-1(∂_x+∂_t) (ϕ^2)-ϕ^2(∂_x+∂_t)(u^-̱1v^-1))dt=(g_1' g_2-g_1g_2')dt.

Here we denote

    g_1(t) =ϕ^2(t,t-a),   g_2(t):=(u^-̱1v^-1)(t, t-a) =(a+1)^-̱1(2t-a+1)^-1.

Then its integration onΓ_1' becomes

    ∫_Γ_3'ω=∫_a/2^a+b/2(g_1'g_2-g_1g_2')dt=(g_1g_2)|_t=a/2^t=a+b/2-
    2∫_a/2^a+b/2g_1g_2'dt.

Since0<, <̱1,a>0andt≥a/2, we have0<g_2≤ 1,    g_2'<0.Then by using the uniform bound (<ref>) for finite energy solution,
we can bound that

    |∫_Γ_3'ω| 1+∫_a/2^a+b/2|g_2'|dt 1.

By our notation, the implicit constant relies only on the weighted energyℰ_1[ϕ], the powerpand the constants,$̱. In particular, the above uniform bound is independent of a, b. By setting a=0, we also have the uniform bound for the integration of ω on the null segment Γ_1'. In a similar argument (or by symmetry x→ -x), the integration of ω on other incoming null segments Γ_2', Γ_4' is also uniformly bounded by a constant independent of a and b,
that is,

    | ∫_∂ℛ_a^bω| 1.

Combining this estimate with (<ref>), (<ref>) and (<ref>), we conclude that

    ∫_ℛ_a^bu^-̱1v^-1Qdxdt 1,  ∫_∂ℛ_a^b((P_1-P_2)dt-(P_1+P_2)dx) 1.

Now on the outgoing null segments Γ_1' and Γ_3', we compute that

    dx=dt,   (P_1-P_2)dt-(P_1+P_2)dx=-2P_2dt,

while on the incoming null segments Γ_2' and Γ_4', we have

    dx=-dt,   (P_1-P_2)dt-(P_1+P_2)dx=2P_1dt.

Summing up and considering the orientation we have shown that

    -∫_0^b/22P_2(t,t)dt+∫_b/2^b+a/22P_1(t,b-t)dt
      +∫_a/2^b+a/22P_2(t,t-a)dt-∫_0^a/22P_1(t,-t)dt
       =∫_∂ℛ_a^b((P_1-P_2)dt-(P_1+P_2)dx)
        1.

Note that for t≥ 0, we have (u)(t, t)=1 and (v)(t, t)=2t+1≥ 1. Hence we have

    P_2(t,t)   =(u^-̱1v^(Łϕ)^2/+2u^v^-1|ϕ|^p+1/(̱p+1))|_x=t
       =(v^(Łϕ)^2/+2v^-1|ϕ|^p+1/(̱p+1))|_x=t
    
          (2t+1)(Łϕ)^2(t, t)+|ϕ|^p+1(t, t).

In view of estimate (<ref>) with a=0, we derive that

    ∫_0^b/22P_2(t,t)dt∫_0^+∞((2t+1)(Łϕ)^2+|ϕ|^p+1)|_x=tdt 1.

Similarly by using estimate (<ref>) with a=0 (or simply by symmetry again), we also have

    ∫_0^a/22P_1(t,-t)dt 1.

Then from the above inequality (<ref>), we conclude that

    ∫_b/2^b+a/22P_1(t,b-t)dt
      +∫_a/2^b+a/22P_2(t,t-a)dt 1,  ∀ a, b>0.

Since P_1 and P_2 are non-negative, by letting b→+∞, we derive that

    ∫_a/2^+∞2P_2(t,t-a)dt 1.

Note that for t≥ a>a/2>0 it holds that

    (u)(t, t-a)=a+1,   t+1≤ (v)(t, t-a)=2t-a+1≤  2t+1.

This implies that for t≥ a we have the lower bound

    P_2(t,t-a)   =(u^-̱1v^(Łϕ)^2/+2u^v^-1|ϕ|^p+1/(̱p+1))|_x=t-a
       ≥^-1(a+1)^-̱1(t+1)^(Łϕ)^2+2(p+1)^-1^̱-1(a+1)^(2t+1)^-1|ϕ|^p+1.

Here we note that 0<<1.
By restricting the integral on the region where t≥ a, we derive from the previous estimate that

    ∫_a^+∞((a+1)^-̱1(t+1)^(Łϕ)^2+(a+1)^(t+1)^-1|ϕ|^p+1)|_x=t-adt 1.

Now for fixed a> 0,  let
 
    g(t)=ϕ(t,t-a),    a_1=a,   a_2=1,  μ_1=∈[1/2,1),  μ_2=-1.

In particular we have

    μ_1+μ_2=2-1≥ 0,    g'(t)=(Łϕ)(t,t-a).

Therefore the above weighted energy estimate through the outgoing null lines reads as

    (a_1+1)^-̱1∫_a_1^+∞(t+a_2)^μ_1|g'(t)|^2dt+(a_1+1)^∫_a_1^∞ (t+a_2)^μ_2|g|^p+1 dt 1.

Then using Lemma <ref>, we conclude that

    (t+a_2)^μ_1+μ_2 |g(t)|^p+3≤ C  ∫_a_1^∞ (s+a_2)^μ_2 |g|^p+1 ds·∫_a_1^∞(s+a_2)^μ_1|g'|^2ds  (a_1+1)^1-2.

By definition, this implies that





    |ϕ(t,t-a)|
     (t+1)^1-2/p+3(a+1)^1-2/p+3,   ∀ t≥ a> 0.

Hence the pointwise decay estimate of Theorem <ref> in the interior region holds for the case when t≥ x≥ 0. By symmetry (changing x to -x) again, the same estimate is valid for -t≤ x≤ 0.

Finally to show the uniform inverse polynomial time decay of the solution, in the exterior region where t≤ |x|, the decay estimate in the exterior region in the previous subsection implies that

    |ϕ(t,x)| (t+1)^-1/p+3
       (1+|t|)^-p-1/(p+1)^2+4.

In the interior region |x|≤ t, take the constants , $̱ as follows

    =̱1/2,  ^-1=1+4/(p+1)^2<2.

This verifies the condition in Theorem <ref>. Then the pointwise decay estimate in the interior region shows that

    |ϕ(t,x)|
     (t+1)^1-2/p+3=(1+t)^-p-1/(p+1)^2+4.

Combining this with the above estimate in the exterior region, we therefore have shown that on the whole future spacetime, it always holds that

    |ϕ(t,x)|
     (t+1)^1-2/p+3=(1+t)^-p-1/(p+1)^2+4.

This completes the proof for Theorem <ref>.t_*=t+1, u=t_*-x, v=t_*+x, Λ=t_*^2-x^2,r̊_̊1̊s_1
    ∂_t(u^v^(t_*𝕋_00+x𝕋_01))-∂_x(u^v^(t_*𝕋_01+x𝕋_11))
      
    =   u^v^(𝕋_00-𝕋_11)- u^-̊1v^-1((v̊+ u)t_*𝕋_00+(v̊^2- u^2)𝕋_01+(v̊- u)x𝕋_11) 
    =   (1-γ)Λ^-γ(𝕋_00-𝕋_11)-γΛ^-γ-1((t_*^2+x^2)(𝕋_00+𝕋_11)+
      4t_*x𝕋_01).

    (v̊+ u)t_*𝕋_00+(v̊^2- u^2)𝕋_01+(v̊- u)x𝕋_11
    =   (v̊^2+ u^2)(𝕋_00+𝕋_11)/2 +(+̊)uv(𝕋_00-𝕋_11)/2+(v̊^2- u^2)𝕋_01
    =   |(√() v-√() u)∂_tϕ+(√() v+√() u)∂_xϕ|^2/2+√()uv(|∂_tϕ|^2-|∂_xϕ|^2)+(+̊)uv|ϕ|^p+1/p+1
    =   |(√() v-√() u)∂_tϕ+(√() v+√() u)∂_xϕ|^2/2-√()uvQ/2+(+̊/p+1+√())uv|ϕ|^p+1.

    ∂_t(u^v^(t_*𝕋_00+x𝕋_01))-∂_x(u^v^(t_*𝕋_01+x𝕋_11))-√()u^v^Q/2
      
    =   u^v^(𝕋_00-𝕋_11)- u^-̊1v^-1|(√() v-√() u)∂_tϕ+(√() v+√() u)∂_xϕ|^2/2
       +(+̊/p+1+√())u^v^|ϕ|^p+1
    =   (2--̊/p+1-√())u^v^|ϕ|^p+1-u^-̊1v^-1|(√() v-√() u)∂_tϕ+(√() v+√() u)∂_xϕ|^2/2.

    Q:=(-∂_t^2+∂_x^2)ϕ^2=-2|∂_tϕ|^2+2|∂_xϕ|^2+2|ϕ|^p+1.

    (t_*^2+x^2)(𝕋_00+𝕋_11)+4t_*x𝕋_01=(t_*^2+x^2)(|∂_tϕ|^2+|∂_xϕ|^2)+4t_*x∂_xϕ∂_tϕ.

    (t_*^2+x^2)(𝕋_00+𝕋_11)+4t_*x𝕋_01+Λ Q/2=2|x∂_tϕ+t_*∂_xϕ|^2+Λ|ϕ|^p+1.

    ∂_t(Λ^-γ(t_*𝕋_00+x𝕋_01))-∂_x(Λ^-γ(t_*𝕋_01+x𝕋_11))-
      γΛ^-γQ/2
      
    =   (1-γ)Λ^-γ2|ϕ|^p+1/p+1-γΛ^-γ-1(2|x∂_tϕ+t_*∂_xϕ|^2+Λ|ϕ|^p+1)
      
    =   (2-(p+3)γ)Λ^-γ|ϕ|^p+1/p+1-2γΛ^-γ-1|x∂_tϕ+t_*∂_xϕ|^2.
Now we take γ=2/p+3then
    ∂_t(Λ^-γ(t_*𝕋_00+x𝕋_01))-∂_x(Λ^-γ(t_*𝕋_01+x𝕋_11))-
      γΛ^-γQ/2
      
    =   -2γΛ^-γ-1|x∂_tϕ+t_*∂_xϕ|^2≤ 0.

    ∫_a/2^a+b/2Λ^-γ(t_*𝕋_00+(x+t_*)𝕋_01+x𝕋_11)|_x=t-adt
      
    +   ∫_b/2^a+b/2Λ^-γ(t_*𝕋_00+(x-t_*)𝕋_01-x𝕋_11)|_x=b-tdt
      
    -   ∫_0^b/2Λ^-γ(t_*𝕋_00+(x+t_*)𝕋_01+x𝕋_11)|_x=tdt
      
    -   ∫_0^a/2Λ^-γ(t_*𝕋_00+(x-t_*)𝕋_01-x𝕋_11)|_x=-tdt
      
    -   ∫_ℛ_a^b(γΛ^-γQ/2)dxdt=∫_ℛ_a^b-2γΛ^-γ-1|x∂_tϕ+t_*∂_xϕ|^2
     dxdt≤ 0.
Forx=t≥0we have Λ≥1∫_0^b/2[(x+1)𝕋_00+(t+x+1)𝕋_01+t𝕋_11]|_x=tdt
      
    +   ∫_0^b/2[(x+1)𝕋_00+(t-x-1)𝕋_01-t𝕋_11]|_x=b-tdt
    =   ∫_0^b(x+1)𝕋_00(0,x)dx,

    ∫_0^b/2Λ^-γ(t_*𝕋_00+(x+t_*)𝕋_01+x𝕋_11)|_x=tdt
      
    ≤   ∫_0^b/2(t_*𝕋_00+(x+t_*)𝕋_01+x𝕋_11)|_x=tdt
      
    =   ∫_0^b/2[(x+1)𝕋_00+(t+x+1)𝕋_01+t𝕋_11]|_x=tdt
    ≤   ∫_0^b(x+1)𝕋_00(0,x)dx≤ C,
similarly 
    ∫_0^a/2Λ^-γ(t_*𝕋_00+(x-t_*)𝕋_01-x𝕋_11)|_x=-tdt≤ C.
Forx=t-a, t≥a/2, t_*=t+1we have x+t_*=2t-a+1>2t-a≥0, t_*-x=a+1>0and
    t_*𝕋_00+(x+t_*)𝕋_01+x𝕋_11=(x+t_*)(𝕋_00+𝕋_11+2𝕋_01)/2+
      (t_*-x)(𝕋_00-𝕋_11)/2≥ 0,

    ∫_a/2^a+b/2Λ^-γ(t_*𝕋_00+(x+t_*)𝕋_01+x𝕋_11)|_x=t-adt≥ 0,
similarly 
    ∫_b/2^a+b/2Λ^-γ(t_*𝕋_00+(x-t_*)𝕋_01-x𝕋_11)|_x=b-tdt≥ 0.
Therefore
    ∫_a/2^a+b/2Λ^-γ(t_*𝕋_00+(x+t_*)𝕋_01+x𝕋_11)|_x=t-adt
      ≤ C+∫_ℛ_a^b(γΛ^-γQ/2)dxdt.

    ∫_ℛ_a^bΛ^-γQdxdt=∫_ℛ_a^bΛ^-γ(-∂_t^2+∂_x^2)ϕ^2dxdt
      
    =   ∫_ℛ_a^b(-∂_t^2+∂_x^2)(Λ^-γ)ϕ^2dxdt+
      ∫_a/2^a+b/2(Ł(Λ^-γ)ϕ^2-Λ^-γŁϕ^2)|_x=t-adt
    
      +   ∫_b/2^a+b/2((Λ^-γ)ϕ^2-Λ^-γϕ^2)|_x=b-tdt-
      ∫_0^b/2(Ł(Λ^-γ)ϕ^2-Λ^-γŁϕ^2)|_x=tdt
      
    -   ∫_0^a/2((Λ^-γ)ϕ^2-Λ^-γϕ^2)|_x=-tdt.
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