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Resonance effects for linear wave equations with scale invariant oscillating damping

Marina Ghisi, Massimo Gobbino

Abstract

We consider an abstract linear wave equation with a time-dependent dissipation that decays at infinity with the so-called scale invariant rate, which represents the critical case. We do not assume that the coefficient of the dissipation term is smooth, and we investigate the effect of its oscillations on the decay rate of solutions. We prove a decay estimate that holds true regardless of the oscillations. Then we show that oscillations that are too fast have no effect on the decay rate, while oscillations that are in resonance with one of the frequencies of the elastic part can alter the decay rate. In the proof we first reduce ourselves to estimating the decay of solutions to a family of ordinary differential equations, then by using polar coordinates we obtain explicit formulae for the energy decay of these solutions, so that in the end the problem is reduced to the analysis of the asymptotic behavior of suitable oscillating integrals.

Resonance effects for linear wave equations with scale invariant oscillating damping

Abstract

We consider an abstract linear wave equation with a time-dependent dissipation that decays at infinity with the so-called scale invariant rate, which represents the critical case. We do not assume that the coefficient of the dissipation term is smooth, and we investigate the effect of its oscillations on the decay rate of solutions. We prove a decay estimate that holds true regardless of the oscillations. Then we show that oscillations that are too fast have no effect on the decay rate, while oscillations that are in resonance with one of the frequencies of the elastic part can alter the decay rate. In the proof we first reduce ourselves to estimating the decay of solutions to a family of ordinary differential equations, then by using polar coordinates we obtain explicit formulae for the energy decay of these solutions, so that in the end the problem is reduced to the analysis of the asymptotic behavior of suitable oscillating integrals.
Paper Structure (20 sections, 12 theorems, 126 equations, 1 table)

This paper contains 20 sections, 12 theorems, 126 equations, 1 table.

Table of Contents

  1. Introduction
  2. Some heuristics
  3. Previous literature
  4. Our contribution
  5. Overview of the technique
  6. Resonance effects in different models
  7. Structure of the paper
  8. Statements
  9. Functional setting
  10. Main results
  11. The key tool
  12. Oscillating integrals
  13. Estimates for a family of ODEs
  14. Estimates in the "parabolic" regime
  15. Estimates in the "hyperbolic" regime
  16. ...and 5 more sections

Key Result

Theorem 2.1

Let $H$ and $A$ be as in the functional setting described at the beginning of this section. Let $t_{0}$ be a positive real number, and let $b:[t_{0},+\infty)\to\mathbb{R}$ be a measurable function. Let us assume that there exist two real numbers $M\geq m>0$ such that and let us set Then every solution to problem (eqn:main)--(eqn:data) satisfies the decay estimate for every $t\geq t_{0}$.

Theorems & Definitions (24)

  • Theorem 2.1: General oscillations
  • Remark 2.2: Better decay for coercive operators
  • Theorem 2.4: Fast oscillations
  • Theorem 2.5: Resonant oscillations
  • Remark 2.6
  • Remark 2.7: The classic model case
  • Proposition 2.8
  • Remark 2.9
  • Lemma 3.1
  • proof
  • ...and 14 more