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 ∂tu = ∆u^m + (1 + |x|)^σ u^p, posed for (x, t) ∈ R^N × (0, ∞), where m > 1, p ∈ (0, 1) and σ > 0. Initial data are taken to be bounded, non-negative and compactly supported. In the range when m + p ≥ 2, 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-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 infinite speed of propagation of the supports of solutions if m + p < 2, that is, u(x, t) > 0 for any x ∈ R^N , t > 0, even in the case when the initial condition u0 is compactly supported.