Optimal existence, uniqueness and blow-up for a quasilinear diffusion equation with spatially inhomogeneous reaction

Abstract

Well-posedness and a number of qualitative properties for solutions to the Cauchy problem for the following nonlinear diffusion equation with a spatially inhomogeneous source $${\partial }_{t}u=\Delta u^m+|x{|}^{\sigma }{u}^p,$$ posed for $(x,t)\in {\mathbb{R}}^N\times (0,T)$, with exponents $1<p<m$ and $\sigma >0$, are established. More precisely, we identify the \emph{optimal class of initial conditions} $u_0$ for which (local in time) existence is ensured and prove \emph{non-existence of solutions} for the complementary set of data. We establish then (local in time) \emph{uniqueness and a comparison principle} for this class of data. We furthermore prove that any non-trivial solution to the Cauchy problem \emph{blows up in a finite time} $T\in (0,\infty )$ and \emph{finite speed of propagation} holds true for $t\in (0,T)$: if $u_0\in L^{\infty }({\mathbb{R}}^N)$ is an initial condition with compact support and blow-up time $T>0$, then $u(t)$ is compactly supported for $t\in (0,T)$. We also establish in this work the \emph{absence of localization at the blow-up time} $T$ for solutions stemming from compactly supported data.