Topologies of continuity for Carath\'eodory parabolic PDEs from a dynamical perspective

TL;DR

This paper develops topological frameworks for analyzing the continuous dependence of solutions to non-autonomous Carathéodory parabolic PDEs on initial conditions and nonlinear terms, enhancing understanding of their dynamical behavior.

Contribution

It introduces new topologies on Lipschitz Carathéodory maps that ensure solution continuity with respect to nonlinearities and initial data under various bounds.

Findings

Established topologies guarantee continuous dependence of solutions.

Extended the analysis to nonlinearities with different bound conditions.

Provided a dynamical systems perspective on PDE solution behavior.

Abstract

Systems of non-autonomous parabolic partial differential equations over a bounded domain with nonlinear term of Carath\'eodory type are considered. Appropriate topologies on sets of Lipschitz Carath\'eodory maps are defined in order to have a continuous dependence of the mild solutions with respect to the variation of both the nonlinear term and the initial conditions, under different assumptions on the bound-maps of the nonlinearities.