Nonlinear Wave Equation with Damping: Periodic Forcing and Non-Resonant Solutions to the Kuznetsov Equation

TL;DR

This paper proves the existence of non-resonant time-periodic solutions for the Kuznetsov equation with periodic forcing, using Lp estimates of a linearized damped wave equation across various domains and boundary conditions.

Contribution

It introduces a novel method employing Lp estimates of the linearized wave equation with Kelvin-Voigt damping to establish non-resonant solutions for the Kuznetsov equation.

Findings

Existence of non-resonant solutions in three-dimensional domains.

Applicability to both Dirichlet and Neumann boundary conditions.

Use of Lp estimates for the linearized wave equation with damping.

Abstract

Existence of non-resonant solutions of time-periodic type are established for the Kuznetsov equation with a periodic forcing term. The equation is considered in a three-dimensional whole-space, half-space and bounded domain, and with both non-homogeneous Dirichlet and Neumann boundary values. A method based on Lp estimates of the corresponding linearization, namely the wave equation with Kelvin-Voigt damping, is employed.