On Existence and Uniqueness of Solutions for Semilinear Fractional Wave Equations

Let Ω be a 𝒞2-bounded domain of ℝd, d = 2,3, and fix Q = (0,T)× Ω with T ∈ (0,+∞). In the present paper we consider a Dirichlet initial-boundary value problem associated to the semilinear fractional wave equation ∂αt+Au=fb(u) in Q where 1<α<2, ∂αt corresponds to the Caputo fractional derivative of order α, 𝒜 is an elliptic operator and the nonlinearity fb ∈ 𝒞1 (ℝ) satisfies fb(0) = 0 and |f″b(u)| ≤ C |u|b-1 for some b > 1. We first provide a definition of local weak solutions of this problem by applying some properties of the associated linear equation ∂αt +Au=f(t,x) in Q. Then, we prove existence of local solutions of the semilinear fractional wave equation for some suitable values of b > 1. Moreover, we obtain an explicit dependence of the time of existence of solutions with respect to the initial data that allows longer time of existence for small initial data.