Electronic Journal of Differential Equations, Vol. 2023 (2023), No. 72, pp. 1-21.

Title: Qualitative properties of solutions to a reaction-diffusion equation  with weighted strong reaction
       
Authors: Razvan Gabriel Iagar (Univ. Rey Juan Carlos, Madrid, Spain)
         Ana I. Munoz  (Univ. Rey Juan Carlos, Madrid, Spain)
         Ariel Sanchez (Univ. Rey Juan Carlos, Madrid, Spain)
         
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.

Submitted June 13, 2023. Published October 23, 2023.
Math Subject Classifications: 35B44, 35B45, 35K57, 35K59.
Key Words: Reaction-diffusion equations; weighted reaction; strong reaction; Aronson-Benilan estimates.