Blow-up rates and sets for a quasilinear diffusion equation with weighted source
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866908941211926528 |
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| author | Ferreira, Raúl Iagar, Razvan Gabriel Sánchez, Ariel |
| author_facet | Ferreira, Raúl Iagar, Razvan Gabriel Sánchez, Ariel |
| contents | Blow-up rates are established for general solutions to the quasilinear diffusion equation $$ \partial_tu=Δu^m+|x|^σu^p, \quad (x,t)\in\mathbb{R}^N\times(0,T), $$ in the range of exponents $1<p<m$, $σ>0$. More precisely, if we consider a compactly supported solution $u(x,t)$ with blow-up time $T=T(u)\in(0,\infty)$, we derive the blow-up rate $$ C_1(T-t)^{-α}\leq \|u(x,t)\|_{\infty}\leq C_2(T-t)^{-α}, \quad t\in(0,T), $$ for some positive constants $C_1$, $C_2$, and the upper rate of expansion of the support $$ \sup\{|x|:u(x,t)>0\}\leq C_0(T-t)^{-β}, \quad t\in(0,T), $$ for some constant $C_0>0$, where $$ α=\frac{σ+2}{L}, \quad β=\frac{m-p}{L}, \quad L=σ(m-1)+2(p-1). $$ We also analyze the blow-up sets of solutions $u$, showing, under a suitable condition, that either $B(u)=\mathbb{R}^N$ or blow-up takes place only as $|x|\to\infty$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_05101 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Blow-up rates and sets for a quasilinear diffusion equation with weighted source Ferreira, Raúl Iagar, Razvan Gabriel Sánchez, Ariel Analysis of PDEs Blow-up rates are established for general solutions to the quasilinear diffusion equation $$ \partial_tu=Δu^m+|x|^σu^p, \quad (x,t)\in\mathbb{R}^N\times(0,T), $$ in the range of exponents $1<p<m$, $σ>0$. More precisely, if we consider a compactly supported solution $u(x,t)$ with blow-up time $T=T(u)\in(0,\infty)$, we derive the blow-up rate $$ C_1(T-t)^{-α}\leq \|u(x,t)\|_{\infty}\leq C_2(T-t)^{-α}, \quad t\in(0,T), $$ for some positive constants $C_1$, $C_2$, and the upper rate of expansion of the support $$ \sup\{|x|:u(x,t)>0\}\leq C_0(T-t)^{-β}, \quad t\in(0,T), $$ for some constant $C_0>0$, where $$ α=\frac{σ+2}{L}, \quad β=\frac{m-p}{L}, \quad L=σ(m-1)+2(p-1). $$ We also analyze the blow-up sets of solutions $u$, showing, under a suitable condition, that either $B(u)=\mathbb{R}^N$ or blow-up takes place only as $|x|\to\infty$. |
| title | Blow-up rates and sets for a quasilinear diffusion equation with weighted source |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2604.05101 |