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 ∂α t u + Au = fb(u) in Q where 1 < α < 2, ∂α t 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)∣∣ 6 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 ∂α t u+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.