# 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.

## Key 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.