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| Main Authors: | , , |
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| Format: | Preprint |
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2021
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2106.11620 |
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| _version_ | 1866910106057179136 |
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| author | Razik, Belhaoues Biccari, Umberto Rahmoune, Abita |
| author_facet | Razik, Belhaoues Biccari, Umberto Rahmoune, Abita |
| contents | In this paper, we consider an initial-boundary value problem for the following mixed pseudo-parabolic $p(.)$-Laplacian type equation with logarithmic nonlinearity: $$ u_t-Δu_t-\mbox{div}\left(\left\vert \nabla u\right\vert^{p(.)-2}\nabla u\right) =|u|^{q(.)-2}u\ln(|u|), \quad (x,t)\inΩ\times(0,+\infty),$$ where $Ω\subset\mathbb{R}^n$ is a bounded and regular domain, and the variable exponents $p(.)$ and $q(.)$ satisfy suitable regularity assumptions. By adapting the first-order differential inequality method, we establish a blow-up criterion for the solutions and obtain an upper bound for the blow-up time. In a second moment, we show that blow-up may be prevented under appropriate smallness conditions on the initial datum, in which case we also establish decay estimates in the $H_0^1(Ω)$-norm as $t\to+\infty$. This decay result is illustrated by a two-dimensional numerical example. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2106_11620 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Blow-up results for a logarithmic pseudo-parabolic $p(.)$-Laplacian type equation Razik, Belhaoues Biccari, Umberto Rahmoune, Abita Analysis of PDEs In this paper, we consider an initial-boundary value problem for the following mixed pseudo-parabolic $p(.)$-Laplacian type equation with logarithmic nonlinearity: $$ u_t-Δu_t-\mbox{div}\left(\left\vert \nabla u\right\vert^{p(.)-2}\nabla u\right) =|u|^{q(.)-2}u\ln(|u|), \quad (x,t)\inΩ\times(0,+\infty),$$ where $Ω\subset\mathbb{R}^n$ is a bounded and regular domain, and the variable exponents $p(.)$ and $q(.)$ satisfy suitable regularity assumptions. By adapting the first-order differential inequality method, we establish a blow-up criterion for the solutions and obtain an upper bound for the blow-up time. In a second moment, we show that blow-up may be prevented under appropriate smallness conditions on the initial datum, in which case we also establish decay estimates in the $H_0^1(Ω)$-norm as $t\to+\infty$. This decay result is illustrated by a two-dimensional numerical example. |
| title | Blow-up results for a logarithmic pseudo-parabolic $p(.)$-Laplacian type equation |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2106.11620 |