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Note on the existence of classical solutions of derivative semilinear models for one dimensional wave equation

Yuki Haruyama, Takiko Sasaki, Hiroyuki Takamura

TL;DR

The work analyzes the lifespan of classical solutions to the 1D derivative semilinear wave equation with product-type nonlinearity $|u_t|^p|u_x|^q$ under small data. It employs a fixed-point iteration on the integral representation involving the operators $L$, $L'$, and $\overline{L'}$ to derive a lower bound on the lifespan, yielding $T(\varepsilon)\ge c\varepsilon^{-(p+q-1)}$ for $p>1$, $q>1$. A complementary blow-up analysis for the special model $u_{tt}-u_{xx}=|u_t\pm u_x|^{p-1}(u_t\pm u_x)$ shows an upper bound $T(\varepsilon)\le C\varepsilon^{-(p-1)}$, with sharpness in certain regimes, illustrating the optimal exponent behavior. Overall, the results extend known single-component cases to multi-term derivative nonlinearities in one dimension and inform the structure of the blow-up boundary in relation to Takamura’s work on nonlinear wave equations.

Abstract

This note is a supplement with a new result to the review paper by Takamura [13] on nonlinear wave equations in one space dimension. We are focusing here to the long-time existence of classical solutions of semilinear wave equations in one space dimension, especially with derivative nonlinear terms of product-type. Our result is an extension of the single component case, but it is meaningful to provide models as possible as many to cover the optimality of the general theory. The proof is based on the classical iteration argument of the point-wise estimate of the solution.

Note on the existence of classical solutions of derivative semilinear models for one dimensional wave equation

TL;DR

The work analyzes the lifespan of classical solutions to the 1D derivative semilinear wave equation with product-type nonlinearity under small data. It employs a fixed-point iteration on the integral representation involving the operators , , and to derive a lower bound on the lifespan, yielding for , . A complementary blow-up analysis for the special model shows an upper bound , with sharpness in certain regimes, illustrating the optimal exponent behavior. Overall, the results extend known single-component cases to multi-term derivative nonlinearities in one dimension and inform the structure of the blow-up boundary in relation to Takamura’s work on nonlinear wave equations.

Abstract

This note is a supplement with a new result to the review paper by Takamura [13] on nonlinear wave equations in one space dimension. We are focusing here to the long-time existence of classical solutions of semilinear wave equations in one space dimension, especially with derivative nonlinear terms of product-type. Our result is an extension of the single component case, but it is meaningful to provide models as possible as many to cover the optimality of the general theory. The proof is based on the classical iteration argument of the point-wise estimate of the solution.
Paper Structure (4 sections, 4 theorems, 47 equations)

This paper contains 4 sections, 4 theorems, 47 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and main result
  3. Proof of Theorem \ref{['thm:lower-bound']}
  4. Blow-up example including product terms

Key Result

Proposition 2.1

Assume that $(f,g)\in C^2({\bf R})\times C^1({\bf R})$. Let $(v,w)$ be a $C^1$ solution of a system of integral equations; with some $T>0$. Then, $u$ defined by is a classical solution of (IVP) in ${\bf R}\times[0,T]$, and $v\equiv u_t, w\equiv u_x$ in ${\bf R}\times[0,T]$ hold.

Theorems & Definitions (10)

  • Proposition 2.1
  • Theorem 2.1
  • Remark 2.1
  • Proposition 3.1
  • Remark 3.1
  • Remark 3.2
  • Theorem 4.1
  • Remark 4.1
  • Remark 4.2
  • Remark 4.3