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Well-posedness of inhomogeneous nonlinear wave equations in $\mathbb{R}^3$

Jiang Boyu Shen Jiawei, Li Kexue

Abstract

This paper is devoted to the well-posedness of the inhomogeneous nonlinear wave equations. By combining Strichartz estimates with the contraction mapping principle, we establish local and global well-posedness in the function spaces $\dot{H}^1(\mathbb{R}^3)\times L^2(\mathbb{R}^3)$ and $\dot{H}^{s+1}(\mathbb{R}^3)\times \dot{H}^{s}(\mathbb{R}^3)$. The analysis is carried out in the energy-subcritical regime. As a consequence, our results extend and improve upon previous results in the literature for general nonlinear wave equations.

Well-posedness of inhomogeneous nonlinear wave equations in $\mathbb{R}^3$

Abstract

This paper is devoted to the well-posedness of the inhomogeneous nonlinear wave equations. By combining Strichartz estimates with the contraction mapping principle, we establish local and global well-posedness in the function spaces and . The analysis is carried out in the energy-subcritical regime. As a consequence, our results extend and improve upon previous results in the literature for general nonlinear wave equations.

Paper Structure

This paper contains 5 sections, 10 theorems, 52 equations.

Table of Contents

  1. Introduction
  2. Basic Results and Preliminaries
  3. Some notations and remarks
  4. the proof of the main theory: part 1
  5. the proof of the main theory: part 2

Key Result

Theorem 1.1

Let $0 < \alpha < \frac{4-2b}{3}$ and $0 < b < 2$. Then, for any initial data $(\phi, \psi) \in \dot{H}^{1}(\mathbb{R}^3) \times L^{2}(\mathbb{R}^3)$, equation 1.1 is locally well-posed in the space $\dot{H}^1(\mathbb{R}^3) \times L^2(\mathbb{R}^3)$. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (23)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Definition 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Definition 2.4
  • Remark 2.5
  • Theorem 2.6: Strichartz estimates for the wave equation
  • ...and 13 more