Global solutions versus finite time blow-up for the supercritical fast diffusion equation with inhomogeneous source
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866916603654832128 |
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| author | Iagar, Razvan Gabriel Sánchez, Ariel |
| author_facet | Iagar, Razvan Gabriel Sánchez, Ariel |
| contents | Solutions in self-similar form, either global in time or presenting finite time blow-up, to the supercritical fast diffusion equation with spatially inhomogeneous source $$ \partial_tu=Δu^m+|x|^σu^p, \quad (x,t)\in\mathbb{R}^N\times(0,\infty) $$ with $$ m_c=\frac{(N-2)_+}{N}\leq m<1, \quad σ\in(\max\{-2,-N\},\infty), \quad p>\max\left\{1+\frac{σ(1-m)}{2},1\right\} $$ are considered. It is proved that global self-similar solutions with the specific tail behavior $$ u(x,t)\sim C(m)|x|^{-2/(1-m)}, \qquad {\rm as} \ |x|\to\infty $$ exist exactly for $p\in(p_F(σ),p_s(σ))$, where $$ p_F(σ)=m+\frac{σ+2}{N}, \qquad p_s(σ)=\left\{\begin{array}{ll}\frac{m(N+2σ+2)}{N-2}, & N\geq3,\\\infty, & N\in\{1,2\}, \end{array}\right. $$ are the renowned Fujita and Sobolev critical exponents. In contrast, it is shown that self-similar solutions presenting finite time blow-up exist for any $σ\in(-2,0)$ and $p$ as above, but do not exist for any $σ\geq0$ and $p\in(p_F(σ),p_s(σ))$. We stress that all these results are \emph{new also in the homogeneous case $σ=0$}. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_04714 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Global solutions versus finite time blow-up for the supercritical fast diffusion equation with inhomogeneous source Iagar, Razvan Gabriel Sánchez, Ariel Analysis of PDEs Solutions in self-similar form, either global in time or presenting finite time blow-up, to the supercritical fast diffusion equation with spatially inhomogeneous source $$ \partial_tu=Δu^m+|x|^σu^p, \quad (x,t)\in\mathbb{R}^N\times(0,\infty) $$ with $$ m_c=\frac{(N-2)_+}{N}\leq m<1, \quad σ\in(\max\{-2,-N\},\infty), \quad p>\max\left\{1+\frac{σ(1-m)}{2},1\right\} $$ are considered. It is proved that global self-similar solutions with the specific tail behavior $$ u(x,t)\sim C(m)|x|^{-2/(1-m)}, \qquad {\rm as} \ |x|\to\infty $$ exist exactly for $p\in(p_F(σ),p_s(σ))$, where $$ p_F(σ)=m+\frac{σ+2}{N}, \qquad p_s(σ)=\left\{\begin{array}{ll}\frac{m(N+2σ+2)}{N-2}, & N\geq3,\\\infty, & N\in\{1,2\}, \end{array}\right. $$ are the renowned Fujita and Sobolev critical exponents. In contrast, it is shown that self-similar solutions presenting finite time blow-up exist for any $σ\in(-2,0)$ and $p$ as above, but do not exist for any $σ\geq0$ and $p\in(p_F(σ),p_s(σ))$. We stress that all these results are \emph{new also in the homogeneous case $σ=0$}. |
| title | Global solutions versus finite time blow-up for the supercritical fast diffusion equation with inhomogeneous source |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2307.04714 |