Nonlocal solutions of parabolic equations with strongly elliptic differential operators

TL;DR

This paper develops a unifying topological method to analyze the existence and localization of solutions for nonlocal parabolic equations with strongly elliptic operators, incorporating boundary conditions like Cauchy multipoint and mean value conditions.

Contribution

It introduces a novel Lyapunov-like bounding function theory in infinite dimensions, enabling degree-based existence proofs for nonlocal parabolic problems.

Findings

Established existence and localization results for solutions

Unified approach for local and nonlocal boundary conditions

Developed a new bounding function theory in infinite dimensions

Abstract

The paper deals with second order parabolic equations on bounded domains with Dirichlet conditions in arbitrary Euclidean spaces. Their interest comes from being models for describing reaction-diffusion processes in several frameworks. A linear diffusion term in divergence form is included which generates a strongly elliptic differential operator. A further linear part, of integral type, is present which accounts of nonlocal diffusion behaviours. The main result provides a unifying method for studying the existence and localization of solutions satisfying nonlocal associated boundary conditions. The Cauchy multipoint and the mean value conditions are included in this investigation. The problem is transformed into its abstract setting and the proofs are based on the homotopic invariance of the Leray-Schauder topological degree. A bounding function (i.e. Lyapunov-like function) theory is developed, which is new in this infinite dimensional context. It allows that the associated vector fields have no fixed points on the boundary of their domains and then it makes possible the use of a degree argument.