Thursday, 14 January 2010

Damped wave equation in the subcritical case

aDepartment of Mathematics, Graduate School of Science, Osaka University, Osaka, Toyonaka 560-0043, JapanbDepartamento de Ciencias Básicas, Instituto Tecnológico de Morelia, Morelia CP 58120, Michoacán, Mexico
cInstituto de Matemáticas, Unam Campus Morelia, AP 61-3 (Xangari), Morelia CP 58089, Michoacán, Mexico

Received 22 July 2003; 
revised 31 March 2004. 
Available online 12 August 2004.

Abstract

We study large time asymptotics of small solutions to the Cauchy problem for the one dimensional nonlinear damped wave equation

(1)
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in the sub critical case σset membership, variant(2-var epsilon3,2). We assume that the initial data v0,(1+∂x)-1v1set membership, variantLL1,a, aset membership, variant(0,1) where

View the MathML source
Also we suppose that the mean value of initial data

Then there exists a positive value var epsilon such that the Cauchy problem (1) has a unique global solution v(t,x)set membership, variantC([0,∞);LL1,a), satisfying the following time decay estimate:

View the MathML source
for large t>0, here 2-var epsilon3<σ<2.
Keywords: Damped wave equation; Subcritical nonlinearity; Asymptotic expansion; Large time behavior

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