Damped wave equation in the subcritical case

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Abstract

We study large time asymptotics of small solutions to the Cauchy problem for the one dimensional nonlinear damped wave equation (1) {vtt + vt - vxx + v1+σ = 0, x ∈ R, t > 0, v (0, x) = ε v0 (x), vt (0, x) = ε v1 (x) in the sub critical case σ ∈ (2 - ε3, 2). We assume that the initial data v0, (1 + ∂x)-1 v1 ∈ L∞ ∩ L1,a, a ∈ (0, 1) where L1,a = { ∈ L1; ∥φ∥ L1a, = ∥〈·〉a φ∥ L1 < ∞}, 〈x〉 = 1 + x2. Also we suppose that the mean value of initial data ∫R (v0 (x) + v1 (x)) dx > 0. Then there exists a positive value ε such that the Cauchy problem (1) has a unique global solution v (t, x) ∈ C ([0, ∞); L∞ ∩ L1,a), satisfying the following time decay estimate: ∥v (t)∥ L∞ ≤ C ε 〈t〉 -1/σ for large t > 0, here 2 - ε3 < σ < 2. © 2004 Elsevier Inc. All rights reserved.

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Hayashi, N., Kaikina, E. I., & Naumkin, P. I. (2004). Damped wave equation in the subcritical case. Journal of Differential Equations, 207(1), 161–194. https://doi.org/10.1016/j.jde.2004.06.018

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