Finite-dimensional global attractors for parabolic nonlinear equations with state-dependent delay

TL;DR

This paper establishes the existence of finite-dimensional global attractors for a class of parabolic nonlinear PDEs with state-dependent delay, including models relevant to biology, using quasi-stability methods.

Contribution

It proves well-posedness and the existence of compact global and exponential attractors for these delay PDEs, extending attractor theory to closed evolution operators.

Findings

Existence of global attractors with finite fractal dimension.

Well-posedness in Lipschitz function spaces.

Extension of attractor theory to closed evolution operators.

Abstract

We deal with a class of parabolic nonlinear evolution equations with state-dependent delay. This class covers several important PDE models arising in biology. We first prove well-posedness in a certain space of functions which are Lipschitz in time. We show that the model considered generates an evolution operator semigroup $S_t$ on a space $CL$ of Lipschitz type functions over delay time interval. The operators $S_t$ are closed for all $t\ge 0$ and continuous for $t$ large enough. Our main result shows that the semigroup $S_t$ possesses compact global and exponential attractors of finite fractal dimension. Our argument is based on the recently developed method of quasi-stability estimates and involves some extension of the theory of global attractors for the case of closed evolutions.