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

2024/09/10 by Saïd Kouachi, Kouachi, Said
Computer Science · Engineering · Mathematics · #30C80 #35A01 #35K05 #35K57 #Advanced Mathematical Modeling in Engineering #Dynamical Systems (math.DS) #FOS: Mathematics #Numerical methods in inverse problems #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2409.06606

openalex publication_date 2024/09/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Related

• A simple characterization of positivity preserving semi-linear parabolic systems
• Hyperbolic Relaxation of Reaction Diffusion Equations with Dynamic Boundary Conditions
• Blow-up of nonnegative solutions of an abstract semilinear heat equation with convex source
• Reaction-diffusion systems with initial data of low regularity
• Global solutions for semilinear parabolic evolution problems with Hölder continuous nonlinearities