The threshold of effective damping for semilinear wave equations

In this paper, we obtain the global existence of small data solutions to the Cauchy problem U-tt - Delta u + mu/1+t u(t) = vertical bar u vertical bar(p) u(o,x) = u(0)(x), u(t)(o,x) = u(1)(x) in space dimensionn1, forp>1+2/n, where is sufficiently large. We obtain estimates for the solution and its energy with the same decay rate of the linear problem. In particular, for2+n, the damping term is effective with respect to the L-1-L-2 low-frequency estimates for the solution and its energy. In this case, we may prove global existence in any space dimensionn3, by assuming smallness of the initial data in some weighted energy space. In space dimensionn=1,2, we only assume smallness of the data in some Sobolev spaces, and we introduce an approach based on fractional Sobolev embedding to improve the threshold for global existence to5/3 in space dimensionn=1 and to3 in space dimensionn=2. Copyright (c) 2014 John Wiley & Sons, Ltd.