A porous medium equation with spatially inhomogeneous absorption. Part II: Large time behavior

Abstract

We study the large time behavior of solutions to the Cauchy problem for the quasilinear absorption-diffusion equation $$ \partial_tu=\Delta u^m-|x|^{\sigma}u^p, \quad (x,t)\in\real^N\times(0,\infty), $$ with exponents $p>m>1$ and $\sigma>0$ and with initial conditions either satisfying $$ u_0\in L^{\infty}(\real^N)\cap C(\real^N), \quad \lim\limits_{|x|\to\infty}|x|^{\theta}u_0(x)=A\in(0,\infty) $$ for some $\theta\geq0$. A number of different asymptotic profiles are identified, and uniform convergence on time-expanding sets towards them is established, according to the position of both $p$ and $\theta$ with respect to the following critical exponents $$ p_F(\sigma)=m+\frac{\sigma+2}{N}, \quad \theta_*=\frac{\sigma+2}{p-m}, \quad \theta^*=N. $$ More precisely, solutions in radially symmetric self-similar form decaying as $|x|\to\infty$ with the rates $$ u(x,t)\sim A|x|^{-\theta_*}, \quad {\rm or} \quad u(x,t)\sim \left(\frac{1}{p-1}\right)^{1/(p-1)}|x|^{-\sigma/(p-1)}, $$ are obtained as asymptotic profiles in some of these cases, while asymptotic simplifications or logarithmic corrections in the time scales also appear in other cases. The uniqueness of some of these self-similar solutions, left aside in the first part of this work, is also established.