On the critical nonlinear damped wave equation with large initial data

Abstract

We study the nonlinear damped wave equation (0.1) { u t t + u t − Δ u = − | u | σ u , x ∈ R n , t > 0 , u ( 0 , x ) = u 0 ( x ) , u t ( 0 , x ) = u 1 ( x ) , x ∈ R n , in the critical case of σ = 2 n . Our aim is to prove the large time asymptotic formulas for solutions of the Cauchy problem (0.1) without any restriction on the size of the initial data and on the spatial dimension.