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arXiv · 2512.12612

Parabolic Equations with Singular Coefficients and Boundary Data: Analysis and Numerical Simulations

Arshyn Altybay·Alibek Yeskermessuly
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Abstract

We investigate linear parabolic equations in divergence form with singular coefficients and nonsmooth initial--boundary data. When the diffusion, drift, or potential terms, as well as the source term and boundary conditions, are distributions rather than functions, classical and weak solution concepts become inadequate, since products involving distributions are not well defined in general. To address this difficulty, we introduce a framework of very weak solutions based on regularisation procedures and the theory of moderate nets. Under the stated moderateness and uniform-ellipticity assumptions, we establish existence of very weak solutions and prove uniqueness via negligibility arguments. Moreover, in the regular-data regime, we show consistency with classical weak solutions. Finally, we present numerical experiments illustrating the behaviour of regularised solutions for highly singular inputs, including delta-type potentials and distributional boundary traces.

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Arshyn Altybay, Alibek Yeskermessuly. 2026-09-15. Parabolic Equations with Singular Coefficients and Boundary Data: Analysis and Numerical Simulations. https://arxiv.org/abs/2512.12612

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