Decay of mass for a semilinear heat equation with mixed local-nonlocal operators
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866916562501369856 |
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| author | Kirane, Mokhtar Fino, Ahmad Z. Ayoub, Alaa |
| author_facet | Kirane, Mokhtar Fino, Ahmad Z. Ayoub, Alaa |
| contents | In this paper, we are concerned with the Cauchy problem for the reaction-diffusion equation $\partial_t u+t^β\mathcal{L} u= - h(t)u^p$ posed on $\mathbb{R}^N$, driven by the mixed local-nonlocal operator $\mathcal{L}=-Δ+(-Δ)^{α/2}$, $α\in(0,2)$, and supplemented with a nonnegative integrable initial data, where $p>1$, $β\geq 0$, and $h:(0,\infty)\to(0,\infty)$ is a locally integrable function. We study the large time behavior of non-negative solutions and show that the nonlinear term determines the large time asymptotic for $p\leq 1+α/{N(β+1)},$ while the classical/anomalous diffusion effects win if $p>1+α/{N(β+1)}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_02827 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Decay of mass for a semilinear heat equation with mixed local-nonlocal operators Kirane, Mokhtar Fino, Ahmad Z. Ayoub, Alaa Analysis of PDEs Functional Analysis 35K57, 35B40, 35B51, 26A33, 35A01, 35B33 In this paper, we are concerned with the Cauchy problem for the reaction-diffusion equation $\partial_t u+t^β\mathcal{L} u= - h(t)u^p$ posed on $\mathbb{R}^N$, driven by the mixed local-nonlocal operator $\mathcal{L}=-Δ+(-Δ)^{α/2}$, $α\in(0,2)$, and supplemented with a nonnegative integrable initial data, where $p>1$, $β\geq 0$, and $h:(0,\infty)\to(0,\infty)$ is a locally integrable function. We study the large time behavior of non-negative solutions and show that the nonlinear term determines the large time asymptotic for $p\leq 1+α/{N(β+1)},$ while the classical/anomalous diffusion effects win if $p>1+α/{N(β+1)}$. |
| title | Decay of mass for a semilinear heat equation with mixed local-nonlocal operators |
| topic | Analysis of PDEs Functional Analysis 35K57, 35B40, 35B51, 26A33, 35A01, 35B33 |
| url | https://arxiv.org/abs/2501.02827 |