Strong Solutions to a Kind of Cross Diffusion Parabolic System

Abstract

A kind of strongly coupled cross diffusion parabolic system, which can be used as the energy transport model in semiconductor science, is studied in this paper. The existence and uniqueness of strong solution to the initial boundary value problem are obtained, under the condition that the initial data are a small perturbation of an isothermal stationary solution. Based on this result, an application of Newton iteration on solving this problem is also given. In the last decades, more and more strongly coupled parabolic systems with (de- generate or non-degenerate) cross diffusion terms were derived from applied science, for instance, chemotaxis phenomenon in biomathematics, generalized drift diffusion and energy transport model in semiconductor science, separation of granular material, etc. In real applications, due to more information included, such kinds of cross diffu- sion models describe the phenomena more clearly than the classical weakly coupled diffusion systems. But very few theoretical results have been obtained up to now. In the present paper, we consider the following cross diffusion parabolic system.