Qualitative properties of solutions to parabolic anisotropic equations: Part I -- Expansion of positivity
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866915379328057344 |
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| author | Ciani, Simone Henriques, Eurica Savchenko, Mariia Skrypnik, Igor I. |
| author_facet | Ciani, Simone Henriques, Eurica Savchenko, Mariia Skrypnik, Igor I. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_06714 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Qualitative properties of solutions to parabolic anisotropic equations: Part I -- Expansion of positivity Ciani, Simone Henriques, Eurica Savchenko, Mariia Skrypnik, Igor I. Analysis of PDEs 35B45, 35B65, 35K55, 35K65, 35K67 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. |
| title | Qualitative properties of solutions to parabolic anisotropic equations: Part I -- Expansion of positivity |
| topic | Analysis of PDEs 35B45, 35B65, 35K55, 35K65, 35K67 |
| url | https://arxiv.org/abs/2507.06714 |