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Singular convergence for semilinear wave equations with steep potential well

Martino Prizzi

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.

Singular convergence for semilinear wave equations with steep potential well

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.
Paper Structure (6 sections, 16 theorems, 156 equations)

This paper contains 6 sections, 16 theorems, 156 equations.

Table of Contents

  1. Introduction
  2. Resolvent convergence
  3. Spectral convergence
  4. The linear hyperbolic equation and its semigroup
  5. A singular Trotter-Kato theorem
  6. The nonlinear problem

Key Result

Theorem 2.2

Let $(\beta_n)_{n\in\mathbb N}$ be a sequence of positive numbers, $\beta_n\to+\infty$ as $n\to\infty$. Let $(f_n)_{n\in\mathbb N}$ be a sequence in $L^2(\mathbb R^N)$, and assume that $f_n\to f$ strongly in $L^2(\mathbb R^N)$ as $n\to\infty$, where $f\in L^2_\Omega(\mathbb R^N)$. Let $\lambda>-1$ b Moreover, let $u_\Omega\in H^1_\Omega(\mathbb R^N)$ be the weak solution of the problem Then, up t

Theorems & Definitions (32)

  • Theorem 2.2
  • proof
  • Corollary 2.3
  • Theorem 3.1
  • Theorem 3.2
  • proof
  • Remark 4.1
  • Theorem 4.2
  • Lemma 4.3
  • proof
  • ...and 22 more