A comparison principle for nonlinear parabolic equations with nonlocal source and gradient absorption
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| Format: | Preprint |
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2025
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| author | Amirzhankyzy, Zhaniya Yessirkegenov, Nurgissa |
| author_facet | Amirzhankyzy, Zhaniya Yessirkegenov, Nurgissa |
| contents | This paper investigates the initial-boundary value problem for a nonlinear parabolic equation involving the $p$-Laplacian operator, nonlocal source terms, gradient absorption, and various nonlinearities: \[ \frac{\partial u}{\partial t} - \text{div}(|\nabla u|^{p-2} \nabla u ) = α|u|^{k-1}u \int_Ω|u|^s \, dx - β|u|^{l-1}u |\nabla u|^q + γu^m + μ|\nabla u|^r - ν|u|^{σ-1}u, \] where $ Ω$ is a bounded domain in $\mathbb{R}^N$, $N \geq 1$, with a smooth boundary $\partial Ω$. The parameters satisfy $ α, l, σ> 0 $, $ β, ν\geq 0 $, $ k, m, s \geq 1 $, $ r \geq p - 1 \geq \frac{p}{2}$, and $γ, μ\in \mathbb{R}$.
We establish a comparison principle for this problem. Using this principle, we derive blow-up results as well as global-in-time boundedness of solutions. Our results extend and unify previous studies in the literature. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_08753 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A comparison principle for nonlinear parabolic equations with nonlocal source and gradient absorption Amirzhankyzy, Zhaniya Yessirkegenov, Nurgissa Analysis of PDEs 35K55, 35B09, 35B51 This paper investigates the initial-boundary value problem for a nonlinear parabolic equation involving the $p$-Laplacian operator, nonlocal source terms, gradient absorption, and various nonlinearities: \[ \frac{\partial u}{\partial t} - \text{div}(|\nabla u|^{p-2} \nabla u ) = α|u|^{k-1}u \int_Ω|u|^s \, dx - β|u|^{l-1}u |\nabla u|^q + γu^m + μ|\nabla u|^r - ν|u|^{σ-1}u, \] where $ Ω$ is a bounded domain in $\mathbb{R}^N$, $N \geq 1$, with a smooth boundary $\partial Ω$. The parameters satisfy $ α, l, σ> 0 $, $ β, ν\geq 0 $, $ k, m, s \geq 1 $, $ r \geq p - 1 \geq \frac{p}{2}$, and $γ, μ\in \mathbb{R}$. We establish a comparison principle for this problem. Using this principle, we derive blow-up results as well as global-in-time boundedness of solutions. Our results extend and unify previous studies in the literature. |
| title | A comparison principle for nonlinear parabolic equations with nonlocal source and gradient absorption |
| topic | Analysis of PDEs 35K55, 35B09, 35B51 |
| url | https://arxiv.org/abs/2505.08753 |