کد مقاله | کد نشریه | سال انتشار | مقاله انگلیسی | نسخه تمام متن |
---|---|---|---|---|
6417015 | 1338504 | 2016 | 48 صفحه PDF | دانلود رایگان |
In this paper we consider an energy critical wave equation (3â¤dâ¤5, ζ=±1)ât2uâÎu=ζÏ(x)|u|4/(dâ2)u,(x,t)âRdÃR with initial data (u,âtu)|t=0=(u0,u1)âHË1ÃL2(Rd). Here ÏâC(Rd;(0,1]) converges as |x|ââ and satisfies certain technical conditions. We generalize Kenig and Merle's results on the Cauchy problem of the equation ât2uâÎu=|u|4/(dâ2)u. Following a similar compactness-rigidity argument we prove that any solution with a finite energy must scatter in the defocusing case ζ=â1. 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 ât2vâ(ÎH3+1)v=|v|4v on the hyperbolic space H3.
Journal: Journal of Differential Equations - Volume 261, Issue 11, 5 December 2016, Pages 6437-6484