A surprising regularizing effect of the nonlinear semigroup associated to the semilinear heat equation and applications to reaction diffusion systems

TL;DR

This paper demonstrates that positive weak solutions to certain quasilinear parabolic equations become classical and global if the reaction term maintains a consistent sign after some time, with applications to reaction diffusion systems.

Contribution

It establishes a novel regularizing effect for solutions of reaction-diffusion equations under sign conditions on reactions, without growth restrictions.

Findings

Positive weak solutions become classical and global under sign conditions.

Application of the result to reaction diffusion systems shows global existence.

The proof relies on the maximum principle.

Abstract

In this paper we prove that positive weak solutions for quasilinear parabolic equations on bounded domains subject to homogenous Neumann boundary conditions becme classical and global under the unique condition that the reaction doesn't change sign after certain positive time. We apply this result to reaction diffusion systems and prove global existence of theirs positive weak solutions under the same condition on theirs reactions. The nonlinearities growth isn't taken in consideration. The proof is based on the maximum principle.