Blow-up rates and sets for a quasilinear diffusion equation with weighted source

TL;DR

This paper establishes blow-up rates and sets for solutions to a weighted quasilinear diffusion equation with exponents in the range 1<p<m, providing precise asymptotic behavior near blow-up time and characterizing blow-up sets.

Contribution

It derives explicit blow-up rates and support expansion rates for solutions, and analyzes the structure of blow-up sets under certain conditions, extending understanding of quasilinear diffusion blow-up behavior.

Findings

Blow-up rate is proportional to (T-t)^{-rac{\sigma+2}{L}}.

Support expansion rate is proportional to (T-t)^{-rac{m-p}{L}}.

Blow-up sets are either the entire space or occur only at infinity.

Abstract

Blow-up rates are established for general solutions to the quasilinear diffusion equation $$ \partial_tu=\Delta u^m+|x|^{\sigma}u^p, \quad (x,t)\in\mathbb{R}^N\times(0,T), $$ in the range of exponents $1<p<m$, $\sigma>0$. 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 $$ C_1(T-t)^{-\alpha}\leq \|u(x,t)\|_{\infty}\leq C_2(T-t)^{-\alpha}, \quad t\in(0,T), $$ for some positive constants $C_1$, $C_2$, and the upper rate of expansion of the support $$ \sup\{|x|:u(x,t)>0\}\leq C_0(T-t)^{-\beta}, \quad t\in(0,T), $$ for some constant $C_0>0$, where $$ \alpha=\frac{\sigma+2}{L}, \quad \beta=\frac{m-p}{L}, \quad L=\sigma(m-1)+2(p-1). $$ 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 $|x|\to\infty$.