Convergence to self-similarity for a degenerate parabolic equation with fast-growing spatially-dependent absorption

Abstract

The large time behavior of non-negative solutions to the absorption-diffusion equation $\partial$\textsubscr{t} u = $\Delta$ u\textsuperscript{m} - |x|\textsuperscript{$\sigma$ }u\textsuperscript{m} in (0,$\infty$) x \textsuperscript{N}, with m > 1 and $\sigma$> $\sigma$\textsubscr{0}\,:= N (m-1)/(m+1) is identified. It is shown that all solutions approach a unique stationary solution in self-similar variables, which also provides a universal upper bound (friendly giant ), strongly contrasting to the standard case $\sigma$ = 0. On the one hand, the convergence proof exploits the variational structure of the equation and a suitable Caffarelli-Kohn-Nirenberg inequality, along with the B{é}nilan-Crandall homogeneity regularizing effect. On the other hand, the detailed study of the stationary problem combines elliptic estimates, Moser iteration and techniques from ordinary differential equations.