What is it about?
The authors consider a parabolic PDE of the form $$ \rho_2(x)\,u_t = u^q div(\rho_1(x) u^{m-1} |\nabla u^k|^{p-2} \nabla u) - \rho_3(x) t^l u^{\beta} $$ where each $\rho_i(x)$ is a space‐dependent weight. They prove that, despite the double nonlinearities and lack of homogeneity, one can still construct a unique weak solution and derive its basic regularity and decay properties.
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Why is it important?
Nonlinear diffusion–absorption equations describe heat or fluid transport with sinks (e.g. chemical reactions, phase changes) in media whose properties (density, permeability) change from point to point. Guaranteeing well‑posedness ensures that physical predictions (temperature profiles, concentration fronts) are mathematically sound and stable under small perturbations of data.
Perspectives
Optimal decay rates: refine estimates of how fast solutions vanish or approach steady states. Heterogeneous anisotropy: generalize to tensor‐valued weights \rho(x) for directional diffusion. Singular limits: study fast‑diffusion or slow‑diffusion limits as exponents approach critical thresholds. Computational experiments: validate analytical predictions and explore pattern formation in layered or randomly varying media.
Mr. Makhmud Bobokandov
National University of Uzbekistan
Read the Original
This page is a summary of: The Cauchy problem for a double nonlinear time-dependent parabolic equation with absorption in a non-homogeneous medium, Carpathian Mathematical Publications, June 2025, Vasyl Stefanyk Precarpathian National University,
DOI: 10.15330/cmp.17.1.277-291.
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