Long time behavior for the inhomogeneous PME in a medium with slowly decaying density

Abstract

We study the long-time behavior of non-negative solutions to theCauchy problem(P) $\qquad \rho(x) \partial_t u= \Delta u^m\qquad$in $Q$:$=\mathbb R^n\times\mathbb R_+$$u(x, 0)=u_0$in dimensions $n\ge 3$. We assume that $m> 1$ (slowdiffusion) and $\rho(x)$ is positive, bounded and behaveslike $\rho(x)$~$|x|^{-\gamma}$ as $|x|\to\infty$, with$0\le \gammaOur asymptotic analysis leads to the associatedsingular equation $|x|^{-\gamma}u_t= \Delta u^m,$ whichadmits a one-parameter family of selfsimilar solutions $U_E(x,t)=t^{-\alpha}F_E(xt^{-\beta})$, $E>0$, which aresource-type in the sense that $|x|^{-\gamma}u(x,0)=E\delta(x)$. Weshow that these solutions provide the first term in the asymptoticexpansion of generic solutions to problem (P) for largetimes, both in the weighted $L^1$ sense$u(t)=U_E(t)+o(1)\qquad$ in $L^1_\rho$and in the uniform sense $u(t)=U_E(t)+o(t^{-\alpha})$ in $L^\infty $ as $t\to \infty$ for the explicit rate$\alpha=\alpha(m,n,\gamma)>0$ which is precisely the time-decayrate of $U_E$. For a given solution, the proper choice of theparameter is $E=\int \rho(x)u_0 dx$.