Non-local PDEs with discrete state-dependent delays: well-posedness in a metric space

TL;DR

This paper establishes the well-posedness and existence of a global attractor for non-local PDEs with discrete state-dependent delays, expanding the understanding of their dynamical behavior in a suitable metric space.

Contribution

It proves existence and uniqueness of solutions in a broad space and constructs a dynamical system using a solution manifold analogue for these PDEs.

Findings

Existence and uniqueness of solutions in a wider linear space

Construction of a dynamical system on a solution manifold

Existence of a compact global attractor

Abstract

Partial differential equations with discrete (concentrated) state-dependent delays are studied. The existence and uniqueness of solutions with initial data from a wider linear space is proven first and then a subset of the space of continuously differentiable (with respect to an appropriate norm) functions is used to construct a dynamical system. This subset is an analogue of \textit{the solution manifold} proposed for ordinary equations in [H.-O. Walther, The solution manifold and $C\sp 1$-smoothness for differential equations with state-dependent delay, J. Differential Equations, {195}(1), (2003) 46--65]. The existence of a compact global attractor is proven.