Damped wave equation in the subcritical case

Abstract We study large time asymptotics of small solutions to the Cauchy problem for the one dimensional nonlinear damped wave equation (1) v tt + v t - v xx + v 1 + σ = 0 , x ∈ R , t > 0 , v 0 , x = ɛ v 0 x , v t 0 , x = ɛ v 1 x in the sub critical case σ ∈ 2 - ɛ 3 , 2 . We assume that the initial data v 0 , 1 + ∂ x - 1 v 1 ∈ L ∞ ∩ L 1 , a , a ∈ 0 , 1 where L 1 , a = φ ∈ L 1 ; φ L 1 , a = · a φ L 1 ∞ , x = 1 + x 2 . Also we suppose that the mean value of initial data ∫ R v 0 x + v 1 x dx > 0 . Then there exists a positive value ɛ 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: v t L ∞ ⩽ C ɛ t - 1 σ for large t > 0 , here 2 - ɛ 3 σ 2 .

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