Existence of ground states for fourth-order wave equations

TL;DR

This paper proves the existence of ground state solutions for a class of fourth-order wave equations with various nonlinearities, valid for a range of wave speeds, advancing understanding of such equations in mathematical physics.

Contribution

It establishes the existence of ground states for fourth-order wave equations across a broad range of speeds and nonlinearities, which was previously unconfirmed.

Findings

Ground state solutions exist for certain wave speeds.

Existence results hold for a variety of nonlinear functions.

The work extends known results to higher-order wave equations.

Abstract

Focusing on the fourth-order wave equation $u_{tt} + \Delta^2 u + f(u)= 0$, we prove the existence of ground state solutions $u=u(x+ct)$ for an optimal range of speeds $c\in\mathbb{R}^n$ and a variety of nonlinearities $f$.