Qualitative properties of solutions to parabolic anisotropic equations: Part II -- The anisotropic Trudinger's equation
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866909697999634432 |
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| author | Ciani, Simone Henriques, Eurica Savchenko, Mariia O. Skrypnik, Igor I. |
| author_facet | Ciani, Simone Henriques, Eurica Savchenko, Mariia O. Skrypnik, Igor I. |
| contents | In this paper we study the local regularity properties of weak solutions to a special class of anisotropic doubly nonlinear parabolic operators, whose prototype is the anisotropic Trudinger's equation $$ u_t- \sum\limits_{i=1}^N D_i\Big(u^{2-p_i}|D_i u|^{p_i-2} D_i u\Big)=0,\quad u\geqslant 0. $$ We prove a parabolic Harnack inequality for nonnegative local weak solutions, without any restrictions on the sparseness of the exponents $p_i$s. Moreover, for a restricted range of diffusion exponents, we prove that solutions are Hölder continuous. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_15730 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Qualitative properties of solutions to parabolic anisotropic equations: Part II -- The anisotropic Trudinger's equation Ciani, Simone Henriques, Eurica Savchenko, Mariia O. Skrypnik, Igor I. Analysis of PDEs 35B45, 35B65, 35K55 In this paper we study the local regularity properties of weak solutions to a special class of anisotropic doubly nonlinear parabolic operators, whose prototype is the anisotropic Trudinger's equation $$ u_t- \sum\limits_{i=1}^N D_i\Big(u^{2-p_i}|D_i u|^{p_i-2} D_i u\Big)=0,\quad u\geqslant 0. $$ We prove a parabolic Harnack inequality for nonnegative local weak solutions, without any restrictions on the sparseness of the exponents $p_i$s. Moreover, for a restricted range of diffusion exponents, we prove that solutions are Hölder continuous. |
| title | Qualitative properties of solutions to parabolic anisotropic equations: Part II -- The anisotropic Trudinger's equation |
| topic | Analysis of PDEs 35B45, 35B65, 35K55 |
| url | https://arxiv.org/abs/2507.15730 |