Polynomial bounds on the Sobolev norms of the solutions of the nonlinear wave equation with time dependent potential

2019/02/05 by Petkov, Vesselin, Tzvetkov, Nikolay
#35L15 (Secondary) #35L71 (Primary) #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.1902.01680

Abstract

We consider the Cauchy problem for the nonlinear wave equation u_tt - Δ_x u +q(t, x) u + u³ = 0 with smooth potential q(t, x) ≥ 0 having compact support with respect to x. The linear equation without the nonlinear term u³ and potential periodic in t may have solutions with exponentially increasing as t → ∞ norm H¹(\mathbb R³_x). In [2] it was established that adding the nonlinear term u³ the H¹(\mathbb R³_x) norm of the solution is polynomially bounded for every choice of q. In this paper we show that H^k(\mathbb R³_x) norm of this global solution is also polynomially bounded. To prove this we apply a different argument based on the analysis of a sequence \Y_k(nτ_k)\_{n = 0}^∞ with suitably defined energy norm Y_k(t) and 0 < τ_k <1.

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