Self-similar shrinking of supports and non-extinction for a nonlinear diffusion equation with spatially inhomogeneous strong absorption

Abstract

We study the dynamics of the following porous medium equation with strong absorption $${\partial }_{t}u=\Delta u^m-|x{|}^{\sigma }{u}^q,$$ posed for $(t,x)\in (0,\infty )\times {\mathbb{R}}^N$, with $m>1$, $q\in (0,1)$ and $\sigma >2(1-q)/(m-1)$. Considering the Cauchy problem with non-negative initial condition $u_0\in L^{\infty }({\mathbb{R}}^N)$ instantaneous shrinking and localization of supports for the solution $u(t)$ at any $t>0$ are established. With the help of this property, existence and uniqueness of a nonnegative compactly supported and radially symmetric forward self-similar solution with algebraic decay in time are proven. Finally, it is shown that finite time extinction does not occur for a wide class of initial conditions and this unique self-similar solution is the pattern for large time behavior of these general solutions.