Partial regularity of weak solutions and life-span of smooth solutions to a biological network formulation model

Abstract

In this paper we study partial regularity of weak solutions to the initial boundary value problem for the system $$-\text {div}\left[ (I+\mathbf{m}\otimes \mathbf{m})\nabla p\right] =S(x), \partial _t\mathbf{m}-D^2\Delta \mathbf{m}-E^2(\mathbf{m}\cdot \nabla p)\nabla p+|\mathbf{m}|^{2(\gamma -1)}{} \mathbf{m}=0$$ , 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. The mathematical difficulty is due to the fact that the system in the model features both a quadratic nonlinearity and a cubic nonlinearity. The regularity issue seems to have a connection to a conjecture by De Giorgi (Congetture sulla continuita delle soluzioni di equazioni lineari ellittiche autoaggiunte a coefficienti illimitati, Unpublished, 1995). We also 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 $$L^q(\Omega )$$ .