Some qualitative properties of solutions to a reaction-diffusion equation with weighted strong reaction

Abstract

We study the existence and qualitative properties of solutions to the Cauchy problem associated to the quasilinear reaction-diffusion equation $${\partial }_{t}u=\Delta u^m+(1+|x|{)}^{\sigma }{u}^p,$$ posed for $(x,t)\in {\Re }^N\times (0,\infty )$, where $m>1$, $p\in (0,1)$ and $\sigma >0$. Initial data are taken to be bounded, non-negative and compactly supported. In the range when $m+p\ge 2$, we prove \emph{local existence of solutions} together with a \emph{finite speed of propagation} of their supports for compactly supported initial conditions. We also show in this case that, for a given compactly supported initial condition, there exist \emph{infinitely many solutions} to the Cauchy problem, by prescribing the evolution of their interface. In the complementary range $m+p<2$, we establish new \emph{Aronson-Bénilan estimates} satisfied by solutions to the Cauchy problem, which are of independent interest as a priori bounds for the solutions. We apply these estimates to establish \emph{infinite speed of propagation} of the supports of solutions if $m+p<2$, that is, $u(x,t)>0$ for any $x\in {\Re }^N$, $t>0$, even in the case when the initial condition $u_0$ was compactly supported.