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We study the existence and qualitative properties of solutions to the Cauchyproblem associated to the quasilinear reaction-diffusion equation $$\partial_tu=\Delta u^m+(1+|x|)^{\sigma}u^p, $$ posed for$(x,t)\in\real^N\times(0,\infty)$, where $m>1$, $p\in(0,1)$ and $\sigma>0$.Initial data are taken to be bounded, non-negative and compactly supported. Inthe range when $m+p\geq2$, we prove \emph{local existence of solutions}together with a \emph{finite speed of propagation} of their supports forcompactly supported initial conditions. We also show in this case that, for agiven compactly supported initial condition, there exist \emph{infinitely manysolutions} to the Cauchy problem, by prescribing the evolution of theirinterface. In the complementary range $m+p<2$, we establish new\emph{Aronson-B\'enilan estimates} satisfied by solutions to the Cauchyproblem, which are of independent interest as a priori bounds for thesolutions. We apply these estimates to establish \emph{infinite speed ofpropagation} of the supports of solutions if $m+p<2$, that is, $u(x,t)>0$ forany $x\in\real^N$, $t>0$, even in the case when the initial condition $u_0$ wascompactly supported.

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