The Defocusing Energy-Critical Wave Equation with a Cubic Convolution

Abstract

In this paper, we study the theory of the global well-posedness and scattering for the energy-critical wave equation with a cubic convolution nonlinearity $u_{tt}-\Delta u+(|x{|}^{-4}\ast |u{|}^2)u=0$ in spatial dimension $d\ge 5$. The main difficulties are the absence of the classical finite speed of propagation (i.e. the monotonic local energy estimate on the light cone), which is a fundamental property to show the global well-posedness and then to obtain scattering for the wave equations with the local nonlinearity $u_{tt}-\Delta u+|u{|}^{\frac{4}{d-2}}u=0$. To compensate it, we resort to the extended causality and utilize the strategy derived from concentration compactness ideas. Then, the proof of the global well-posedness and scattering is reduced to show the nonexistence of the three enemies: finite time blowup; soliton-like solutions and low-to-high cascade. We will utilize the Morawetz estimate, the extended causality and the potential energy concentration to preclude the above three enemies.

19 pages, In this version, we prove the result in Proposition 4.1 in an averaged-in-time sense, and we utilize the potential energy concentration in an averaged-in-time sense and the Morawetz estimate to kill finite time blow up solutions in Section 6

References in corpus (5)

• Global well-posedness and scattering for the energy-critical, defocusing Hartree equation for radial data
• Global well-posedness and scattering for the mass-critical Hartree equation with radial data
• The defocusing energy-critical nonlinear Schrödinger equation in higher dimensions
• Global well-posedness and scattering for the defocusing cubic NLS in four dimensions
• Energy Scattering for a Klein-Gordon Equation with a Cubic Convolution

Cited by in corpus (3)

• Scattering theory for the radial ${\dot{H}}^{1/2}$-critical wave Equation with a cubic convolution
• The Defocusing Energy-critical Klein-Gordon-Hartree Equation
• Global well-posedness of the energy-critical stochastic Hartree nonlinear wave equation