On the nonlinear wave equation with time periodic potential

TL;DR

This paper demonstrates that adding a nonlinear defocusing term to a wave equation with time periodic potential prevents exponential energy growth, ensuring global existence and polynomial energy bounds, and analyzes the instability of the zero solution.

Contribution

It introduces a nonlinear defocusing interaction that stabilizes solutions of wave equations with time periodic potentials, preventing exponential energy growth.

Findings

Solutions exist globally with polynomial energy bounds.

Zero solution is unstable under certain linear conditions.

Nonlinear interaction stabilizes the wave equation behavior.

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.