Asymptotic self-similarity in diffusion equations with nonconstant radial limits at infinity

TL;DR

This paper investigates the long-time behavior of solutions to diffusion equations with space-dependent diffusion matrices approaching a limit at infinity, demonstrating convergence to self-similar Gaussian-like profiles under certain conditions.

Contribution

It introduces a novel energy estimate method involving an antiderivative to analyze convergence to self-similar solutions with nonconstant asymptotic diffusion matrices.

Findings

Solutions converge to self-similar Gaussian-like profiles.

The convergence rate can be arbitrarily slow depending on matrix properties.

The method applies to both linear and semilinear diffusion equations.

Abstract

We study the long-time behavior of localized solutions to linear or semilinear parabolic equations in the whole space $\mathbb{R}^n$, where $n \ge 2$, assuming that the diffusion matrix depends on the space variable $x$ and has a finite limit along any ray as $|x| \to \infty$. Under suitable smallness conditions in the nonlinear case, we prove convergence to a self-similar solution whose profile is entirely determined by the asymptotic diffusion matrix. Examples are given which show that the profile can be a rather general Gaussian-like function, and that the approach to the self-similar solution can be arbitrarily slow depending on the continuity and coercivity properties of the asymptotic matrix. The proof of our results relies on appropriate energy estimates for the diffusion equation in self-similar variables. The new ingredient consists in estimating not only the difference $w$ between the solution and the self-similar profile, but also an antiderivative $W$ obtained by solving a linear elliptic problem which involves $w$ as a source term. Hence, a good part of our analysis is devoted to the study of linear elliptic equations whose coefficients are homogeneous of degree zero.