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 $-\mbox{div}\left[(I+\mathbf{m}\otimes \mathbf{m})\nabla p\right]=S(x),\ \ \partial_t\mathbf{m}-D^2Δ\mathbf{m}-E^2(\mathbf{m}\cdot\nabla p)\nabla p+|\mathbf{m}|^{2(γ-1)}\mathbf{m}=0$, 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 $\Vert \mathbf{m}(x,0){\Vert }_{\infty ,\Omega }+\Vert S(x){\Vert }_{\frac{2N}{3},\Omega }$ is made suitably small, where $N$ is the space dimension and $\Vert \cdot {\Vert }_{q,\Omega }$ denotes the norm in $L^q(\Omega )$.