Damped wave equation in the subcritical case

^aDepartment of Mathematics, Graduate School of Science, Osaka University, Osaka, Toyonaka 560-0043, Japan^bDepartamento 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)

in the sub critical case σ(2-³,2). We assume that the initial data v₀,(1+∂_x)^-1v₁L^∞∩L^1,a, a(0,1) where

Also we suppose that the mean value of initial data

∫_R(v₀(x)+v₁(x))dx>0.

Then there exists a positive value such that the Cauchy problem (1) has a unique global solution v(t,x)C([0,∞);L^∞∩L^1,a), satisfying the following time decay estimate:

for large t>0, here 2-³<σ<2.

Keywords: Damped wave equation; Subcritical nonlinearity; Asymptotic expansion; Large time behavior