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Recent developments on the lifespan estimate for classical solutions of nonlinear wave equations in one space dimension

Hiroyuki Takamura

Abstract

In this paper, we overview the recent progresses on the lifespan estimates of classical solutions of the initial value problems for nonlinear wave equations in one space dimension. There are mainly two directions of the developments on the model equations which ensure the optimality of the general theory. One is on the so-called "combined effect" of two kinds of the different nonlinear terms, which shows the possibility to improve the general theory. Another is on the extension to the non-autonomous nonlinear terms which includes the application to nonlinear damped wave equations with the time-dependent critical case.

Recent developments on the lifespan estimate for classical solutions of nonlinear wave equations in one space dimension

Abstract

In this paper, we overview the recent progresses on the lifespan estimates of classical solutions of the initial value problems for nonlinear wave equations in one space dimension. There are mainly two directions of the developments on the model equations which ensure the optimality of the general theory. One is on the so-called "combined effect" of two kinds of the different nonlinear terms, which shows the possibility to improve the general theory. Another is on the extension to the non-autonomous nonlinear terms which includes the application to nonlinear damped wave equations with the time-dependent critical case.
Paper Structure (11 sections, 7 theorems, 70 equations)

This paper contains 11 sections, 7 theorems, 70 equations.

Table of Contents

  1. Introduction.
  2. Case of constant coefficients and the combined effect.
  3. The generalized combined effect.
  4. Comparison with the general theory.
  5. An improvement of the general theory.
  6. Strategy of the proof of Theorem \ref{['thm:gcombined']}.
  7. The case of variable coefficients
  8. Motivation of the problem
  9. Spatially weighted nonlinear terms.
  10. Weighted nonlinear terms in the characteristic directions.
  11. Other models.

Key Result

Theorem 1

The conjecture (lifespan_conjecture) is true except for the case where In this case, we have that which is strictly shorter than the second case in (lifespan_conjecture).

Theorems & Definitions (9)

  • Theorem 1: Morisawa, Sasaki and Takamura MSTMST_erratum, Kido, Sasaki, Takamatsu and Takamura KSTT
  • Theorem 2: Takamatsu Takamatsu
  • Theorem 3: Kitamura, Morisawa and Takamura KMT22
  • Remark 1
  • Theorem 4: Zhou Zhou01, Kitamura, Morisawa and Takamura KMT23
  • Theorem 5: Kitamura, Wakasa and Takamura KTW23
  • Remark 2
  • Theorem 6: Kitamura Kitamura
  • Theorem 7: Sasaki, Takamatsu and Takamura STT23