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

Arshyn Altybay; Michael Ruzhansky

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arXiv · 2602.16348

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

Arshyn Altybay·Michael Ruzhansky

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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.

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Arshyn Altybay, Michael Ruzhansky. 2026-02-18. Heat Equation driven by mixed local-nonlocal operators with non-regular space-dependent coefficients. https://arxiv.org/abs/2602.16348

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