A semi-linear energy critical wave equation with an application

In this paper we consider an energy critical wave equation (3 <= d <= 5, zeta = +/- 1) partial derivative(2)(t)u - Delta u = zeta phi(x)vertical bar u vertical bar(4/(d-2))u, (x, t) is an element of R-d x R with initial data (u, partial derivative(t)u)vertical bar(t=0) = (u(0), u(1)) is an element of (H) over dot(1) x L-2(R-d). Here phi is an element of C(R-d; (0, 1]) converges as vertical bar x vertical bar <- infinity and satisfies certain technical conditions. We generalize Kenig and Merle's results on the Cauchy problem of the equation partial derivative(2)(t)u - Delta u = vertical bar u vertical bar(4/(d-2))u. Following a similar compactness-rigidity argument we prove that any solution with a finite energy must scatter in the defocusing case zeta = -1. While in the focusing case zeta = 1 we give a criterion for global behaviour of the solutions, either scattering or finite-time blow-up when the energy is smaller than a certain threshold. As an application we give a similar criterion on the global behaviour of radial solutions to the focusing, energy critical shifted wave equation partial derivative(2)(t)v (Delta(H3) +1)v = vertical bar v vertical bar(4)v on the hyperbolic space H-3. (C) 2016 Elsevier Inc. All rights reserved.