Heat Equation driven by mixed local-nonlocal operators with non-regular space-dependent coefficients

TL;DR

This paper investigates the well-posedness of a heat equation driven by a mixed local-nonlocal operator with irregular, space-dependent coefficients, extending classical solutions to very weak frameworks.

Contribution

It establishes existence, uniqueness, and continuous dependence for solutions with irregular coefficients, extending classical theory to distributional coefficients.

Findings

Proved classical well-posedness for bounded, measurable coefficients.

Extended solution concepts to distributional coefficients via regularisation.

Ensured consistency with classical solutions when coefficients are regular.

Abstract

In this paper, we study the Cauchy problem for a heat equation governed by a mixed local--nonlocal diffusion operator with spatially irregular coefficients. We first establish classical well-posedness in an energy framework for bounded, measurable coefficients that satisfy uniform positivity, and we derive an a priori estimate ensuring uniqueness and continuous dependence on the initial data. We then extend the notion of solution to distributional coefficients and initial data by a Friedrichs-type regularisation procedure. Within this very weak framework, we establish the existence and uniqueness of solution nets and prove consistency with the classical weak solution whenever the coefficients are regular.