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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2411.04783 |
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| _version_ | 1866910687908855808 |
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| author | König, Tobias Yu, Meng |
| author_facet | König, Tobias Yu, Meng |
| contents | Let $s \in (0, 1]$ and $N > 2s$. It is known that positive solutions to the (fractional) fast diffusion equation $\partial_t u + (-Δ)^s (u^\frac{N-2s}{N+2s}) = 0$ on $(0, \infty) \times \mathbb R^N$ with regular enough initial datum extinguish after some finite time $T_* > 0$. More precisely, one has $\frac{u(t,\cdot)}{U_{T_*, z, λ}(t,\cdot)} - 1 =o(1)$ as $t \to T_*^-$ for a certain extinction profile $U_{T_*, z, λ}$, uniformly on $\mathbb R^N$. In this paper, we prove the quantitative bound $ \frac{u(t,\cdot)}{U_{T_*, z, λ}(t,\cdot)} - 1 = \mathcal O( (T_*-t)^\frac{N+2s}{N-2s+2})$, in a natural weighted energy norm. The main point here is that the exponent $\frac{N+2s}{N-2s+2}$ is sharp. This is the analogue of a recent result by Bonforte and Figalli (CPAM, 2021) valid for $s = 1$ and bounded domains $Ω\subset \mathbb R^N$. Our result is new also in the local case $s = 1$. The main obstacle we overcome is the degeneracy of an associated linearized operator, which generically does not occur in the bounded domain setting.
For a smooth bounded domain $Ω\subset \mathbb R^N$, we prove similar results for positive solutions to $\partial_t u + (-Δ)^s (u^m) = 0$ on $(0, \infty) \times Ω$ with Dirichlet boundary conditions when $s \in (0,1)$ and $m \in (\frac{N-2s}{N+2s}, 1)$, under a non-degeneracy assumption on the stationary solution. An important step here is to prove the convergence of the relative error, which is new for this case. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_04783 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Sharp extinction rates for positive solutions of fast diffusion equations König, Tobias Yu, Meng Analysis of PDEs Let $s \in (0, 1]$ and $N > 2s$. It is known that positive solutions to the (fractional) fast diffusion equation $\partial_t u + (-Δ)^s (u^\frac{N-2s}{N+2s}) = 0$ on $(0, \infty) \times \mathbb R^N$ with regular enough initial datum extinguish after some finite time $T_* > 0$. More precisely, one has $\frac{u(t,\cdot)}{U_{T_*, z, λ}(t,\cdot)} - 1 =o(1)$ as $t \to T_*^-$ for a certain extinction profile $U_{T_*, z, λ}$, uniformly on $\mathbb R^N$. In this paper, we prove the quantitative bound $ \frac{u(t,\cdot)}{U_{T_*, z, λ}(t,\cdot)} - 1 = \mathcal O( (T_*-t)^\frac{N+2s}{N-2s+2})$, in a natural weighted energy norm. The main point here is that the exponent $\frac{N+2s}{N-2s+2}$ is sharp. This is the analogue of a recent result by Bonforte and Figalli (CPAM, 2021) valid for $s = 1$ and bounded domains $Ω\subset \mathbb R^N$. Our result is new also in the local case $s = 1$. The main obstacle we overcome is the degeneracy of an associated linearized operator, which generically does not occur in the bounded domain setting. For a smooth bounded domain $Ω\subset \mathbb R^N$, we prove similar results for positive solutions to $\partial_t u + (-Δ)^s (u^m) = 0$ on $(0, \infty) \times Ω$ with Dirichlet boundary conditions when $s \in (0,1)$ and $m \in (\frac{N-2s}{N+2s}, 1)$, under a non-degeneracy assumption on the stationary solution. An important step here is to prove the convergence of the relative error, which is new for this case. |
| title | Sharp extinction rates for positive solutions of fast diffusion equations |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2411.04783 |