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Interpolation scattering for wave equations with singular potentials and singular data

Pham Truong Xuan

Abstract

In this paper we investigate a construction of scattering for wave-type equations with singular potentials on the whole space $\mathbb{R}^n$ in a framework of weak-$L^p$ spaces. First, we use an Yamazaki-type estimate for wave groups on Lorentz spaces and fixed point arguments to prove the global well-posedness for wave-type equations on weak-$L^p$ spaces. Then, we provide a corresponding scattering results in such singular framework. Finally, we use also the dispersive estimates to establish the polynomial stability and improve the decay of scattering in weak-$L^p$ spaces.

Interpolation scattering for wave equations with singular potentials and singular data

Abstract

In this paper we investigate a construction of scattering for wave-type equations with singular potentials on the whole space in a framework of weak- spaces. First, we use an Yamazaki-type estimate for wave groups on Lorentz spaces and fixed point arguments to prove the global well-posedness for wave-type equations on weak- spaces. Then, we provide a corresponding scattering results in such singular framework. Finally, we use also the dispersive estimates to establish the polynomial stability and improve the decay of scattering in weak- spaces.
Paper Structure (7 sections, 3 theorems, 55 equations)

This paper contains 7 sections, 3 theorems, 55 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Wave-type equations and dispersive estimates
  4. Lorentz spaces and Yamazaki-type estimates for wave groups
  5. Well-posedness, interpolation scattering and stability on weak-$L^p$ spaces
  6. Global well-posedness and interpolation scattering
  7. Polynomial stability

Key Result

Proposition 2.1

(Yamazaki-type estimate). Let $f$ be radially symmetric and $n\geqslant 3$ odd. If $1<d_1,d_2 < \dfrac{2(n-1)}{n-1}\,\, (\infty \text{ if } n=3)$ with $\left( \dfrac{1}{d_1},\dfrac{1}{d_2} \right) \in \Delta_{P_2P_4P_5}$, then $|t|^{n\left( \frac{1}{d_1}-\frac{1}{d_2}\right)-2}W(t)f \in L^1(\math for all $f\in L^{(d_1,1)}_{rad}(\mathbb{R}^n)$.

Theorems & Definitions (8)

  • Proposition 2.1
  • proof
  • Theorem 3.1
  • proof
  • Remark 3.2
  • Theorem 3.3
  • proof
  • Remark 3.4