A Priori Estimates for a Nonlinear System with Some Essential Symmetrical Structures

Abstract

In this paper, we are concerned with a nonlinear system containing some essential symmetrical structures (e.g., cross-diffusion) in the two-dimensional setting, which is proposed to model the biological transport networks. We first provide an a priori blow-up criterion of strong solution of the corresponding Cauchy problem. Based on this, we also establish a priori upper bounds to strong solution for all positive times.