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Uniform estimates for solutions of nonlinear focusing damped wave equations

Thomas Perrin

Abstract

For a damped wave (or Klein-Gordon) equation on a bounded domain, with a focusing power-like nonlinearity satisfying some growth conditions, we prove that a global solution is bounded in the energy space, uniformly in time. Our result applies in particular to the case of a cubic equation on a bounded domain of dimension 3.

Uniform estimates for solutions of nonlinear focusing damped wave equations

Abstract

For a damped wave (or Klein-Gordon) equation on a bounded domain, with a focusing power-like nonlinearity satisfying some growth conditions, we prove that a global solution is bounded in the energy space, uniformly in time. Our result applies in particular to the case of a cubic equation on a bounded domain of dimension 3.
Paper Structure (16 sections, 6 theorems, 59 equations)

This paper contains 16 sections, 6 theorems, 59 equations.

Table of Contents

  1. Setting and main result.
  2. Examples.
  3. Connection with the existing literature.
  4. Outline of the article.
  5. Acknowledgements.
  6. Preliminaries
  7. The energy bound
  8. The $L^2(\Omega)$-estimate
  9. Step 1 : an estimate on $M^{\prime \prime}$.
  10. Step 2 : proof of (\ref{['eq_thm_L2_estimate']}).
  11. The $H^1(\Omega)$-estimates
  12. Estimate for $t \leq 1$.
  13. Estimate for $t \geq 1$.
  14. Proof of the elementary lemmas
  15. Proof of Lemma \ref{['lem_basic_explosion']}
  16. ...and 1 more sections

Key Result

Theorem 1

Assume that $f$ satisfies (hyp_f_1), with $p < \frac{d + 2}{d - 2}$, (hyp_f_2), and (hyp_f_3).

Theorems & Definitions (9)

  • Theorem 1
  • Theorem 2
  • Lemma 3
  • Lemma 4
  • Remark 5
  • Lemma 6
  • Lemma 7
  • proof
  • Remark 8