Singular limit of solutions of the equation u_t=Δ(u^m/m) as m → 0

Kin Ming Hui

Vol. 187 (1999), No. 2, 297–316

DOI: 10.2140/pjm.1999.187.297

Abstract

We will show that for n = 1, 2, as m → 0 the solution u^(m) of the fast diffusion equation ∂u∕∂t = Δ(u^m∕m), u > 0, in Rⁿ × (0,∞), u(x,0) = u₀(x) ≥ 0 in Rⁿ, where u₀ ∈ L¹(Rⁿ) ∩ L^∞(Rⁿ) will converge uniformly on every compact subset of Rⁿ × (0,T) to the maximal solution of the equation v_t = Δlog v, v(x,0) = u₀(x), where T = ∞ for n = 1 and T = ∫ _R²u₀dx∕4π for n = 2.

Milestones

Received: 4 September 1997

Published: 1 February 1999