Existence and Multiplicity of Blow-Up Profiles for a Quasilinear Diffusion Equation with Source

We classify radially symmetric self-similar solutions presenting finite time blow-up to the quasilinear diffusion equation with weighted source \(\begin{aligned} u_t=\Delta u^m+|x|^{\sigma }u^p, \end{aligned}\) $$\begin{array}{c}\hfill u_t=\mathrm{\Delta }{u}^m+{|x|}^{\sigma }{u}^p,\end{array}$$ posed for \((x,t)\in \mathbb {R}^N\times (0,T)\) $(x,t)\in {\mathbb{R}}^N\times (0,T)$ , \(T>0\) $T>0$ , in dimension \(N\ge 1\) $N\ge 1$ and in the range of exponents \(-2<\sigma <\infty \) $-2<\sigma <\infty$ , \(1<m<p<p_s(\sigma )\) $1<m<p<p_s(\sigma )$ , where \(\begin{aligned} p_s(\sigma )=\left\{ \begin{array}{ll}\frac{m(N+2\sigma +2)}{N-2}, & N\ge 3,\\ +\infty , & N\in \{1,2\},\end{array}\right. \end{aligned}\) $$\begin{array}{c}\hfill p_s(\sigma )=\left\{\begin{array}{cc}\frac{m(N+2\sigma +2)}{N-2},\hfill & N\ge 3,\hfill \\ +\infty ,\hfill & N\in \{1,2\},\hfill \end{array}\right)\end{array}$$ is the renowned Sobolev critical exponent. The most interesting result is the multiplicity of two different types of self-similar solutions for p sufficiently close to m and \(\sigma \) $\sigma$ sufficiently close to zero in dimension \(N\ge 2\) $N\ge 2$ , including solutions with dead-core profiles. For \(\sigma =0\) $\sigma =0$ , this answers in dimension \(N\ge 2\) $N\ge 2$ a question still left open in Samarskii et al. (Blow-up in quasilinear parabolic problems. de Gruyter expositions in mathematics, W. de Gruyter, Berlin, 1995, Section IV.1.4, pp. 195–196), where only multiplicity in dimension \(N=1\) $N=1$ had been established. Besides this result, we also prove that, for any \(\sigma \in (-2,0)\) $\sigma \in (-2,0)$ , \(N\ge 1\) $N\ge 1$ and \(m<p<p_s(\sigma )\) $m<p<p_s(\sigma )$ existence of at least a self-similar blow-up solution is granted. In strong contrast with the previous results, given any \(N\ge 1\) $N\ge 1$ , \(\sigma \ge \sigma ^*=(mN+2)/(m-1)\) $\sigma \ge {\sigma }^{\ast }=(mN+2)/(m-1)$ and \(p\in (m,p_s(\sigma ))\) $p\in (m,p_s(\sigma ))$ , non-existence of any radially symmetric self-similar solution is proved.