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

Abstract

We consider the Cauchy problem for the nonlinear wave equation uttΔxu+q(t,x)u+u3=0u_{tt} - \Delta_x u +q(t, x) u + u^3 = 0 with smooth potential q(t,x)0q(t, x) \geq 0 having compact support with respect to xx. The linear equation without the nonlinear term u3u^3 and potential periodic in tt may have solutions with exponentially increasing as t t \to \infty norm H1(Rx3)H^1({\mathbb R}^3_x). In [2] it was established that adding the nonlinear term u3u^3 the H1(Rx3)H^1({\mathbb R}^3_x) norm of the solution is polynomially bounded for every choice of qq. In this paper we show that Hk(Rx3)H^k({\mathbb R}^3_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 {Yk(nτk)}n=0\{Y_k(n\tau_k)\}_{n = 0}^{\infty} with suitably defined energy norm Yk(t)Y_k(t) and $0 < \tau_k <1.

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