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arXiv · 2507.06714

Qualitative properties of solutions to parabolic anisotropic equations: Part I -- Expansion of positivity

Simone Ciani·Eurica Henriques·Mariia Savchenko·Igor I. Skrypnik
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Abstract

We prove expansion of positivity and reduction of the oscillation results to the local weak solutions to a doubly nonlinear anisotropic class of parabolic differential equations with bounded and measurable coefficients, whose prototype is \begin{equation*} u_t-\sum\limits_{i=1}^N \left( u^{(m_i-1)(p_i-1)} \ |u_{x_i}|^{p_i-2} \ u_{x_i} \right)_{x_i}=0 , \end{equation*} for a restricted range of $p_i$s and $m_i$s, that reflects their competition for the diffusion. The positivity expansion relies on an exponential shift and is presented separately for singular and degenerate cases. Finally we present a study of the local oscillation of the solution for some specific ranges of exponents, within the singular and degenerate cases.

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Simone Ciani, Eurica Henriques, Mariia Savchenko, Igor I. Skrypnik. 2025-07-09. Qualitative properties of solutions to parabolic anisotropic equations: Part I -- Expansion of positivity. https://arxiv.org/abs/2507.06714

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