Local Solutions of Semilinear Wave Equations in Hs+1

The Cauchy problem for the wave equation with power type non-linearity ±∣u∣u and data in H^s+1(ℝⁿ)×H^s(ℝⁿ) 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(ℝⁿ) is shown which is unique in a closely related class.