Existence and multiplicity of blow-up profiles for a quasilinear diffusion equation with source

Abstract

We classify radially symmetric self-similar profiles presenting finite time blow-up to the quasilinear diffusion equation with weighted source $$ u_t=\Delta u^m+|x|^{\sigma}u^p, $$ posed for $(x,t)\in\real^N\times(0,T)$, $T>0$, in dimension $N\geq1$ and in the range of exponents $-2<\sigma<\infty$, $1<m<p<p_s(\sigma)$, where $$ p_s(\sigma)=\left\{\begin{array}{ll}\frac{m(N+2\sigma+2)}{N-2}, & N\geq3,\\ +\infty, & N\in\{1,2\},\end{array}\right. $$ is the renowned Sobolev critical exponent. The most interesting result is the \emph{multiplicity of two different types} of self-similar profiles for $p$ sufficiently close to $m$ and $\sigma$ sufficiently close to zero in dimension $N\geq2$, including \emph{dead-core profiles}. For $\sigma=0$, this answers in dimension $N\geq2$ a question still left open in \cite[Section IV.1.4, pp. 195-196]{S4}, where only multiplicity in dimension $N=1$ had been established. Besides this result, we also prove that, for any $\sigma\in(-2,0)$, $N\geq1$ and $m<p<p_s(\sigma)$ \emph{existence} of at least a self-similar blow-up profile is granted. In strong contrast with the previous results, given any $N\geq1$, $\sigma\geq\sigma^*=(mN+2)/(m-1)$ and $p\in(m,p_s(\sigma))$, \emph{non-existence} of any radially symmetric self-similar profile is proved.