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Scattering theory for subcritical wave equation with inverse square potential

Changxing Miao, Ruipeng Shen, Tengfei Zhao

Abstract

We consider the scattering theory for the defocusing energy subcritical wave equations with an inverse square potential. By employing the energy flux method we establish energy flux estimates on the light cone. Then by the characteristic line method and radiation theorem, we show that the radial finite-energy solutions scatter to free waves outside of light cones. Using Morawetz estimates we then obtain the scattering theory for radial solutions with finite weighted energy initial data.

Scattering theory for subcritical wave equation with inverse square potential

Abstract

We consider the scattering theory for the defocusing energy subcritical wave equations with an inverse square potential. By employing the energy flux method we establish energy flux estimates on the light cone. Then by the characteristic line method and radiation theorem, we show that the radial finite-energy solutions scatter to free waves outside of light cones. Using Morawetz estimates we then obtain the scattering theory for radial solutions with finite weighted energy initial data.

Paper Structure

This paper contains 12 sections, 17 theorems, 130 equations.

Table of Contents

  1. Introduction
  2. Background and Motivations
  3. Main Results
  4. The structure of this paper
  5. Technical Lemmas and well-posedness theory
  6. Local theory
  7. Energy Flux
  8. Morawetz estimates
  9. Method of Characteristic Lines
  10. Further estimates with energy decay and the proof of Theorem \ref{['main 2']}
  11. The asymptotic behaviour of linear solution
  12. Square function estimates

Key Result

Theorem \oldthetheorem

Suppose that $3\leq d \leq 6$, $p\in[1+\frac{4}{d-1},1+\frac{4}{d-2})$, and $a>-\frac{(d-2)^2}{4}+(\frac{(d-2)p-d}{2p})^2$. Let $u$ be a radial solution to wave-La-p with initial data $(u_0,u_1) \in \dot{H}^1 \times L^2$ so that the inequality holds for a constant $\kappa \geq \kappa_0 \doteq \frac{(d+2)-(d-2)p}{p+1}$. Then the solution $u$ scatters, that is, there exists a finite-energy free wav

Theorems & Definitions (31)

  • Theorem \oldthetheorem
  • Remark \oldthetheorem
  • Proposition \oldthetheorem: Exterior scattering
  • Remark \oldthetheorem
  • Remark \oldthetheorem
  • Remark \oldthetheorem
  • Lemma \oldthetheorem: Pointwise Estimate, see Lemma 5.6 in shenenergy
  • Lemma \oldthetheorem: Equivalence of Sobolev spaces, see Theorem 1.2 in KMVZZ-2018
  • Proposition \oldthetheorem: Strichartz estimates BPST-2003 MMZ-2019
  • Remark \oldthetheorem
  • ...and 21 more