Global existence for reaction diffusion systems with strict balance Law and nonlinearities with non constant-sign and unlimited polynomial growth

TL;DR

This paper proves the global existence of solutions for reaction-diffusion systems with nonlinearities that change sign and grow polynomially, using a Lyapunov functional approach under quasi-positivity and balance law conditions.

Contribution

It introduces a method to establish global solutions for complex reaction-diffusion systems with sign-changing and unbounded nonlinearities, expanding the scope of existing results.

Findings

Global existence of solutions established

Lyapunov functional effectively handles polynomial growth

Sign-changing nonlinearities managed through reaction sign fixing

Abstract

The purpose of this paper is to prove global existence of solutions for general systems of reaction diffusion equations with nonlinearities for which only two main proprieties hold: Quasi-Positivity and balance law but with two difficulties: they change sign and with unlimited polynomial growth. We overcome the first difficulty by fixing the reaction sign after some time and the second one by using a judicious polynomial Lyapunov functional.