Self-similar shrinking of supports and non-extinction for a nonlinear diffusion equation with spatially inhomogeneous strong absorption
Résumé
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 $(t, x) \in (0,\infty) × \mathnn{R}^N$, 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.
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