Global solutions and asymptotic behavior for a parabolic degenerate coupled system arising from biology

TL;DR

This paper proves global existence, uniqueness, and analyzes the long-term behavior of solutions for a biological parabolic degenerate coupled system in two dimensions.

Contribution

It establishes the first rigorous results on global solutions and their asymptotics for this specific biological model involving degenerate parabolic equations.

Findings

Global existence and uniqueness of solutions are proved.

Solutions are shown to be nonnegative and H"older continuous.

Asymptotic behavior of solutions is characterized.

Abstract

In this paper we will focus on a parabolic degenerate system with respect to unknown functions u and w on a bounded domain of the two-dimensional Euclidean space. This system appears as a mathematical model for some biological processes. Global existence and uniqueness of a nonnegative classical H\"older continuous solution are proved. The last part of the paper is devoted to the study of the asymptotic behavior of the solutions.