On the Nonlinear Wave Equation with Time-periodic Potential

Abstract

Abstract It is known that for some time-periodic potentials $q(t, x) \geq 0$ having compact support with respect to $x$ some solutions of the Cauchy problem for the wave equation $\partial _t^2 u - \Delta _x u + q(t,x)u = 0$ have exponentially increasing energy as $t \to \infty $. We show that if one adds a nonlinear defocusing interaction $|u|^ru, 2\leq r < 4,$ then the solution of the nonlinear wave equation exists for all $t \in{\mathbb R}$, and its energy is polynomially bounded as $t \to \infty $ for every choice of $q$. Moreover, we prove that the zero solution of the nonlinear wave equation is instable if the corresponding linear equation has the property mentioned above.