Parabolic Differential Equations with Bounded Delay

TL;DR

This paper establishes the continuous dependence of solutions to linear nonautonomous second order parabolic PDEs with bounded delay on their coefficients and delay, under very weak convergence assumptions, aiding Lyapunov exponent applications.

Contribution

It proves continuous dependence of solutions on coefficients and delay with minimal assumptions, advancing the theory of PDEs with delay.

Findings

Solutions depend continuously on coefficients and delay

Weak-* convergence suffices for stability analysis

Supports applications to Lyapunov exponents

Abstract

We show the continuous dependence of solutions of linear nonautonomous second order parabolic partial differential equations (PDEs) with bounded delay on coefficients and delay. The assumptions are very weak: only convergence in the weak-* topology of delay coefficients is required. The results are important in the applications of the theory of Lyapunov exponents to the investigation of PDEs with delay.