Asymptotic behaviour for an equation of superslow diffusion. The Cauchy problem

Article type: Research Article

Authors: Galaktionov, Victor A. | Vazquez, Juan L.

Affiliations: Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, Miusskaya Pl. 4, 125047 Moscow, Russia | Departamento de Matemáticas, Univ. Autónoma de Madrid, 28049 Madrid, Spain

Abstract: We investigate the asymptotic behaviour as t→∞ of the non-negative weak solution to the Cauchy problem for the equation of superslow diffusion ut=(e−1/u)xx for x∈R, t>0, with non-negative initial function u0∈L∞(R)∩L1(R), u0$\not\equiv$0. We prove that asymptotic separation of variables takes place if we make the change of variables v=e−1/u and η=x/log t. The precise result says that as t→∞ tv(η log t,t)→½(a2−η2)+, and the convergence is uniform in η∈R. The constant a>0 is exactly one half of the initial energy: a=½∫u0(x)dx>0. This implies that u evolves for large t towards a mesa-like profile of height 1/(log t) and width =2a log t.

DOI: 10.3233/ASY-1994-8202

Journal: Asymptotic Analysis, vol. 8, no. 2, pp. 145-159, 1994