Attractors for nonlinear reaction-diffusion systems in unbounded domains via the method of short trajectories

TL;DR

This paper proves well-posedness and the existence of a global attractor for a nonlinear reaction-diffusion system on unbounded domains, using the short trajectory method, and provides entropy bounds in low dimensions.

Contribution

It extends the short trajectory method to unbounded domains for reaction-diffusion equations and establishes the existence of global attractors with entropy estimates in low dimensions.

Findings

Well-posedness of the Cauchy problem in a general functional setting.

Existence of a global attractor for the reaction-diffusion system.

Entropy bounds for the attractor when space dimension ≤ 3.

Abstract

We consider a nonlinear reaction-diffusion equation settled on the whole euclidean space. We prove the well-posedness of the corresponding Cauchy problem in a general functional setting, namely, when the initial datum is uniformly locally bounded in L^2. Then we adapt the short trajectory method to establish the existence of the global attractor and, if the space dimension is at most 3, we also find an upper bound of its Kolmogorov's entropy.