Large-time behavior in a nonlocal heat equation with absorption. The absorption dominated case with fast decaying initial data
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912745698361344 |
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| author | Cortázar, Carmen Quirós, Fernando Wolanski, Noemi |
| author_facet | Cortázar, Carmen Quirós, Fernando Wolanski, Noemi |
| contents | We study the large-time behavior of nonnegative solutions to a nonlocal dispersal equation in $\mathbb R^N$ with an absorption term modeled by $-u^p$, with $1<p<1+\frac2N$. The initial datum $u_0$ is assumed to be bounded, and to satisfy $|x|^{\frac2{p-1}}u_0(x)\to A\ge0$ as $|x|\to\infty$. Under these assumptions, we prove that the decay rate is that of the purely absorbing problem, while the limit profile is a very singular solution to a local diffusion problem with absorption if $A=0$, and a solution to this same local problem with initial datum $A|x|^{-\frac2{p-1}}$ if $A>0$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_03265 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Large-time behavior in a nonlocal heat equation with absorption. The absorption dominated case with fast decaying initial data Cortázar, Carmen Quirós, Fernando Wolanski, Noemi Analysis of PDEs We study the large-time behavior of nonnegative solutions to a nonlocal dispersal equation in $\mathbb R^N$ with an absorption term modeled by $-u^p$, with $1<p<1+\frac2N$. The initial datum $u_0$ is assumed to be bounded, and to satisfy $|x|^{\frac2{p-1}}u_0(x)\to A\ge0$ as $|x|\to\infty$. Under these assumptions, we prove that the decay rate is that of the purely absorbing problem, while the limit profile is a very singular solution to a local diffusion problem with absorption if $A=0$, and a solution to this same local problem with initial datum $A|x|^{-\frac2{p-1}}$ if $A>0$. |
| title | Large-time behavior in a nonlocal heat equation with absorption. The absorption dominated case with fast decaying initial data |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2512.03265 |