Soliton resolution for the energy critical wave equation with inverse-square potential in the radial case

TL;DR

This paper proves the soliton resolution conjecture for the energy critical wave equation with inverse-square potential in all radial cases across dimensions N≥3, advancing understanding of wave behavior with singular potentials.

Contribution

It establishes the soliton resolution for the energy critical wave equation with inverse-square potential in all radial cases, utilizing the structure of the radial linear operator and modulation analysis.

Findings

Proves soliton resolution in all radial dimensions N≥3.

Analyzes multi-solitons in specialized function spaces.

Utilizes the structure of the linear operator for energy channel analysis.

Abstract

In this paper, we establish the soliton resolution for the energy critical wave equation with inverse square potential in the radial case and in all dimensions $N\geq3$. The structure of the radial linear operator $\mathcal{L}_a :=-\Delta +\frac{a}{|x|^2}=A^*A$, is essential for the channel of energy, where $A$ is a first order differential operator and $A^*$ is its adjoint operator. Modulation and analysis of the multi-solitons are performed in the function spaces $\dot{H}^1_a(\Bbb R^N)\times L^2(\Bbb R^N)$ associated with $\mathcal{L}_a$.