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

Abstract

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