Local Solutions of Semilinear Wave Equations inHs+1

Abstract

The Cauchy problem for the wave equation with power type non-linearity ±|u| σ* u and data in H s+1 (R n ) x H s (R n ) is considered, where 0 < s < (n/2)−1 and n ≥ 3. Under the growth restriction σ * ≤ 4/(n−2−2s) in many cases the existence of a local solution with u(t) ∈ H s+1 (R n ) is shown which is unique in a closely related class.