Construction of type II blow-up solutions for the energy-critical wave equation with an inverse-square potential in dimension 3

TL;DR

This paper constructs finite-time type-II blow-up solutions for a 3D energy-critical wave equation with an inverse-square potential, analyzing the influence of the potential using spectral methods and extending previous frameworks.

Contribution

It introduces a novel construction of blow-up solutions for the wave equation with an inverse-square potential, incorporating spectral analysis and the distorted Hankel transform.

Findings

Successful construction of finite-time blow-up solutions.

Analysis of the potential's influence on blow-up behavior.

Extension of existing frameworks to include inverse-square potentials.

Abstract

In this paper, we construct finite-time type-II blow-up solutions for the focusing energy-critical wave equation with an inverse-square potential $$\partial_t^2 u-\Delta u+\frac{\alpha}{|x|^2}u = u^5,$$ with discussions of the influence of the potential being made. The key ingredients include an approxiamte construction of solutions and spectral analysis of the linearized operator, especially the distorted Hankel transform. The construction is based on the framework established by Krieger-Schlag-Tataru in 2009.