ON EXISTENCE AND UNIQUENESS OF SOLUTIONS FOR SEMILINEAR FRACTIONAL WAVE EQUATIONS

Let O be a C-2-bounded domain of R-d, d = 2, 3, and fix Q = (0, T) x Omega with T is an element of (0,+infinity]. In the present paper we consider a Dirichlet initial-boundary value problem associated to the semilinear fractional wave equation. a partial derivative(alpha)(t)u + Au = f(b)(u) in Q where 1 < alpha < 2, partial derivative(alpha)(t) corresponds to the Caputo fractional derivative of order alpha, A is an elliptic operator and the nonlinearity f(b) is an element of C-1(R) satisfies f(b)(0) = 0 and vertical bar f(b)' (u)vertical bar <= C vertical bar u vertical bar(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 partial derivative(alpha)(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.