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

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.