Persistence and asymptotic analysis of solutions of nonlinear wave equations

TL;DR

This paper investigates the persistence, unique continuation, and asymptotic behavior of solutions to a broad class of nonlinear wave equations, including models in elastic and shallow water dynamics, establishing key properties and non-existence results.

Contribution

It provides a unified analysis of persistence and asymptotic profiles for various nonlinear wave equations, extending understanding of their solution structures.

Findings

Proves unique continuation for the class of equations.

Establishes asymptotic profiles of solutions.

Shows non-existence of non-trivial compactly supported solutions.

Abstract

We consider persistence properties of solutions for a generalised wave equation including vibration in elastic rods and shallow water models, such as the BBM, the Dai's, the Camassa-Holm, and the Dullin-Gottwald-Holm equations, as well as some recent shallow water equations with the Coriolis effect. We establish unique continuation results and exhibit asymptotic profiles for the solutions of the general class considered. From these results we prove the non-existence of non-trivial spatially compactly supported solutions for the equation. As an aftermath, we study the equations earlier mentioned in light of our results for the general class.