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

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 $|x|\to\infty$.