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

Abstract

We investigate the asymptotic behaviour as t→∞ of the non-negative weak solution to the Cauchy problem for the equation of superslow diffusion

u_t=(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.