2001 Asymptotic behaviour of solutions of ut=Δ log u$ in a bounded domain
Kin Ming Hui
Differential Integral Equations 14(2): 175-188 (2001). DOI: 10.57262/die/1356123351

Abstract

We will show that if u is the solution of ut=Δ log u, u>0, in Ω×(0,), u=c1 on Ω×(0,), u(x,0)=u0(x)0 on ΩRn where Ω is a smooth convex bounded domain, then for c1= the rescaled function w=log (u/t) will converge uniformly on every compact subset of Ω to the unique solution ψ of the equation Δψeψ=0, ψ>0, in Ω with ψ= on Ω as t. When 0<c1<, n=1,2, or 3, and u0c1 on Ω, then the function w=e(λ1/c1)tlog (u/c1) will converge uniformly on Ω¯ to Aϕ1 as t where λ1>0 and ϕ1 are the first positive eigenvalue and positive eigenfunction of the Laplace operator Δ on Ω with ϕ1L2(Ω)=1 respectively and A=limtw(,t)L2(Ω).

Citation

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Kin Ming Hui. "Asymptotic behaviour of solutions of ut=Δ log u$ in a bounded domain." Differential Integral Equations 14 (2) 175 - 188, 2001. https://doi.org/10.57262/die/1356123351

Information

Published: 2001
First available in Project Euclid: 21 December 2012

zbMATH: 1021.35055
MathSciNet: MR1797385
Digital Object Identifier: 10.57262/die/1356123351

Subjects:
Primary: 35K55
Secondary: 35B25 , 35B40 , 35K65

Rights: Copyright © 2001 Khayyam Publishing, Inc.

Vol.14 • No. 2 • 2001
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