Parabolic partial differential equations with discrete state-dependent delay: classical solutions and solution manifold

TL;DR

This paper investigates classical solutions to parabolic PDEs with discrete state-dependent delays, establishing well-posedness and smoothness of evolution operators on a specialized solution manifold.

Contribution

It introduces a solution manifold framework for PDEs with state-dependent delays and proves the smoothness of evolution operators within this setting.

Findings

Well-posedness of solutions in the set $X_F$

Existence of a solution manifold analogous to ODE cases

Evolution operators are $C^1$-smooth on the solution manifold

Abstract

Classical solutions to PDEs with discrete state-dependent delay are studied. We prove the well-posedness in a set $X_F$ which is an analogous to the solution manifold used for ordinary differential equations with state-dependent delay. We prove that the evolution operators are $C^1$-smooth on the solution manifold.