A semi-linear energy critical wave equation with an application.

In this paper we consider an energy critical wave equation ( 3 ≤ d ≤ 5 , ζ = ± 1 ) ∂ t 2 u − Δ u = ζ ϕ ( x ) | u | 4 / ( d − 2 ) u , ( x , t ) ∈ R d × R with initial data ( u , ∂ t u ) | t = 0 = ( u 0 , u 1 ) ∈ H ˙ 1 × L 2 ( R d ) . Here ϕ ∈ C ( R d ; ( 0 , 1 ] ) converges as | x | → ∞ and satisfies certain technical conditions. We generalize Kenig and Merle's results on the Cauchy problem of the equation ∂ t 2 u − Δ u = | u | 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 ζ = − 1 . While in the focusing case ζ = 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 ∂ t 2 v − ( Δ H 3 + 1 ) v = | v | 4 v on the hyperbolic space H 3 . [ABSTRACT FROM AUTHOR]