Self-similar extinction for a fast diffusion equation with weighted absorption
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914338655174656 |
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| author | Iagar, Razvan Gabriel Munteanu, Diana-Rodica |
| author_facet | Iagar, Razvan Gabriel Munteanu, Diana-Rodica |
| contents | Finite time extinction of any bounded solution to the fast diffusion equation with spatially inhomogeneous absorption $$ \partial_tu=Δu^m-|x|^σu^p, \quad (x,t)\in\mathbb{R}^N\times(0,\infty), $$ with $N\geq1$ and exponents $$ p>1, \quad m_c=\frac{(N-2)_+}{N}<m<1, \quad σ>σ_*:=\frac{2(p-1)}{1-m}, $$ is established. Moreover, the existence of self-similar solutions of the form $$ U(x,t)=(T-t)^αf(|x|(T-t)^β), \quad α=\frac{σ+2}{(1-m)(σ-σ_*)}, \ β=\frac{p-m}{(1-m)(σ-σ_*)}, $$ with $f(0)>0$, $f'(0)=0$ and $$ \lim\limits_{ξ\to\infty}ξ^{(σ+2)/(p-m)}f(ξ)=L\in(0,\infty). $$ is proved, together with some unbounded self-similar solutions as well. The property of finite time extinction is in striking contrast to the standard fast diffusion equation with absorption (that is, $σ=0$), where the strict positivity of solutions for any $t\in(0,\infty)$ is well-known. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_16940 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Self-similar extinction for a fast diffusion equation with weighted absorption Iagar, Razvan Gabriel Munteanu, Diana-Rodica Analysis of PDEs Dynamical Systems Finite time extinction of any bounded solution to the fast diffusion equation with spatially inhomogeneous absorption $$ \partial_tu=Δu^m-|x|^σu^p, \quad (x,t)\in\mathbb{R}^N\times(0,\infty), $$ with $N\geq1$ and exponents $$ p>1, \quad m_c=\frac{(N-2)_+}{N}<m<1, \quad σ>σ_*:=\frac{2(p-1)}{1-m}, $$ is established. Moreover, the existence of self-similar solutions of the form $$ U(x,t)=(T-t)^αf(|x|(T-t)^β), \quad α=\frac{σ+2}{(1-m)(σ-σ_*)}, \ β=\frac{p-m}{(1-m)(σ-σ_*)}, $$ with $f(0)>0$, $f'(0)=0$ and $$ \lim\limits_{ξ\to\infty}ξ^{(σ+2)/(p-m)}f(ξ)=L\in(0,\infty). $$ is proved, together with some unbounded self-similar solutions as well. The property of finite time extinction is in striking contrast to the standard fast diffusion equation with absorption (that is, $σ=0$), where the strict positivity of solutions for any $t\in(0,\infty)$ is well-known. |
| title | Self-similar extinction for a fast diffusion equation with weighted absorption |
| topic | Analysis of PDEs Dynamical Systems |
| url | https://arxiv.org/abs/2602.16940 |