Abstract. Let Ω be a C 2 -bounded domain of R d , 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 = f b (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 f b ∈ C 1 (R) satisfiesWe 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.

We examine the short and long-time behaviors of time-fractional diffusion equations with variable space-dependent order. More precisely, we describe the time-evolution of the solution to these equations as the time parameter goes either to zero or to infinity.Mathematics subject classification 2010: 35R11, 35B35, 35B38.