Singular convergence for semilinear wave equations with steep potential well

TL;DR

This paper studies the behavior of solutions to a semilinear wave equation with a deep potential well, showing they converge to solutions of a boundary wave problem as the well deepens.

Contribution

It establishes the convergence of solutions from a semilinear wave equation with a steep potential well to a boundary wave equation as the potential well depth increases.

Findings

Solutions converge to boundary wave solutions as well depth increases

The limit problem involves a wave equation with Dirichlet boundary conditions

Provides rigorous proof of convergence in the deep well limit

Abstract

We consider a semilinear wave equation in the whole space with a deep potential well. We prove that as the depth of the well tends to infinity, the solutions of the equation converge to the solutions of a wave equation defined on the bottom of the well, with Dirichlet condition on the boundary.