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Decay estimates in the supremum norm for the solutions to a nonlinear evolution equation

Published online by Cambridge University Press:  16 May 2014

Petri Juutinen*
Affiliation:
Department of Mathematics and Statistics, PO Box 35, FIN-40014, University of Jyväskylä, Finland, ([email protected])

Abstract

We study the asymptotic behaviour, as t → ∞, of the solutions to the nonlinear evolution equation

where ΔpNu = Δu + (p−2) (D2u(Du/∣Du∣)) · (Du/∣Du∣) is the normalized p-Laplace equation and p ≥ 2. We show that if u(x,t) is a viscosity solution to the above equation in a cylinder Ω × (0, ∞) with time-independent lateral boundary values, then it converges to the unique stationary solution h as t → ∞. Moreover, we provide an estimate for the decay rate of maxx∈Ωu(x,t) − h(x)∣.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 2014 

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