Title: Partial regularity of weak solutions and life-span of smooth solutions to a biological network formulation model,
Abstract: In this paper we first study partial regularity of weak solutions to the
initial boundary value problem for the system
$-\mbox{div}\left[(I+\mathbf{m}\otimes \mathbf{m})\nabla p\right]=S(x),\ \
\partial_t\mathbf{m}-D^2\Delta \mathbf{m}-E^2(\mathbf{m}\cdot\nabla p)\nabla
p+|\mathbf{m}|^{2(\gamma-1)}\mathbf{m}=0$, where $S(x)$ is a given function and
$D, E, \gamma$ are given numbers. This problem has been proposed as a PDE model
for biological transportation networks. Mathematically, it seems to have a
connection to a conjecture by De Giorgi \cite{DE}. Then we investigate the
life-span of classical solutions. Our results show that local existence of a
classical solution can always be obtained and the life-span of such a solution
can be extended as far away as one wishes as long as the term $\|{\bf
m}(x,0)\|_{\infty, \Omega}+\|S(x)\|_{\frac{2N}{3}, \Omega}$ is made suitably
small, where $N$ is the space dimension and $\|\cdot\|_{q,\Omega}$ denotes the
norm in $L^q(\Omega)$.