Reaction-diffusion systems with supercritical nonlinearities revisited

TL;DR

This paper thoroughly investigates the analytic properties and long-term dynamics of reaction-diffusion systems with supercritical nonlinearities, emphasizing minimal restrictions and contrasting with subcritical cases.

Contribution

It provides a comprehensive analysis of existence, uniqueness, smoothing, and attractors for supercritical reaction-diffusion systems with minimal nonlinearity restrictions.

Findings

Global existence and uniqueness of solutions

Smoothing properties and asymptotic compactness

Existence of global and exponential attractors

Abstract

We give a comprehensive study of the analytic properties and long-time behavior of solutions of a reaction-diffusion system in a bounded domain in the case where the nonlinearity satisfies the standard monotonicity assumption. We pay the main attention to the supercritical case, where the nonlinearity is not subordinated to the linear part of the equation trying to put as small as possible amount of extra restrictions on this nonlinearity. The properties of such systems in the supercritical case may be very different in comparison with the standard case of subordinated nonlinearities. We examine the global existence and uniqueness of weak and strong solutions, various types of smoothing properties, asymptotic compactness and the existence of global and exponential attractors.