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

TL;DR

This paper investigates the long-term behavior of solutions to a nonlinear diffusion equation with spatially varying absorption, identifying various asymptotic profiles and their dependence on initial conditions and parameters.

Contribution

It classifies the large time asymptotic profiles for solutions based on critical exponents, and establishes the uniqueness of certain self-similar solutions.

Findings

Identification of multiple asymptotic profiles depending on parameters.

Uniform convergence to these profiles on expanding sets.

Proof of uniqueness for some self-similar solutions.

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.