Asymptotic Self-Similarity in Diffusion Equations with Nonconstant Radial Limits at Infinity
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Asymptotic Self-Similarity in Diffusion Equations with Nonconstant Radial Limits at Infinity Thierry Gallay1
· Romain Joly1 · Geneviève Raugel2
Received: 29 May 2020 / Accepted: 5 September 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We study the long-time behavior of localized solutions to linear or semilinear parabolic equations in the whole space Rn , where n ≥ 2, assuming that the diffusion matrix depends on the space variable x and has a finite limit along any ray as |x| → ∞. 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. Keywords Diffusion equations · Inhomogeneous media · Long-time asymptotics · Self-similar solutions
1 Introduction We consider semilinear parabolic equations of the form ∂t u(x, t) = div A(x)∇u(x, t) + N (u(x, t)),
x ∈ Rn , t > 0,
(1.1)
Geneviève Raugel: († May 10, 2019).
B
Thierry Gallay [email protected] Romain Joly [email protected]
1
Institut Fourier, Université Grenoble Alpes et CNRS, 100 rue des Maths, 38610 Gières, France
2
Département de Mathématiques, CNRS et Université Paris-Saclay, 91405 Orsay, France
123
Journal of Dynamics and Differential Equations
which describe the evolution of a scalar quantity u(x, t) ∈ R under the action of inhomogeneous diffusion and nonlinear self-interaction. We assume that the diffusion matrix A(x) in (1.1) is symmetric, Lipschitz continuous as a function of x ∈ Rn , and satisfies the following uniform ellipticity condition: there exist positive constants λ1 , λ2 such that for all x ∈ Rn and all ξ ∈ Rn , (1.2) λ1 |ξ |2 ≤ A(x)ξ, ξ ) ≤ λ2 |ξ |2 , where (·, ·) denotes the Euclidean scalar product in Rn . As for the nonlinearity, we suppose that N is globally Lipschitz, that N (0) = 0, and that N (u) = O(|u|σ ) as u → 0 for some σ > 1+2/n. Our goal is to investigate the long-time behavior of all solutions of (1.1) starting from sufficiently small and localized initial data. Even in the linear case where N = 0, it is necessary to make further assumptions on the diffusion matrix A(x) to obtain accurate results on the long-time behavior of solutions of (1.1). In fact, two classical situations are well understoo
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