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 $$ \partial_tu=\Delta u^m+(1+|x|)^{\sigma}u^p, $$ posed for \((x,t)\in\mathbb{R}^N\times(0,\infty)\), 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 \(m+p\geq2\), we prove existence of local solutions with a 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 infinitely many solutions to the Cauchy problem, by prescribing the evolution of their interface. In the complementary range \(m+p<2\), we obtain new Aronson-Benilan 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 infinite speed of propagation of the supports of solutions if \(m+p<2\), that is, \(u(x,t)>0\) for any \(x\in\mathbb{R}^N\), \(t>0\), even in the case when the initial condition \(u_0\) is compactly supported. For more information see https://ejde.math.txstate.edu/Volumes/2023/72/abstr.html