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 utt-Δu+μ1+tut=|u|pu(0,x)=u0(x),ut(0,x)=u1(x) in space dimension-n-≥-1, for-p->-1-+-2-n, where-μ is sufficiently large. We obtain estimates for the solution and its energy with the same decay rate of the linear problem. In particular, for-μ-≥-2-+-n, the damping term is effective with respect to the L1-L2 low-frequency estimates for the solution and its energy. In this case, we may prove global existence in any space dimension-n-≥-3, by assuming smallness of the initial data in some weighted energy space. In space dimension-n-=-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 to-μ-≥-5-3 in space dimension-n-=-1 and to-μ-≥-3 in space dimension-n-=-2.