On Existence and Uniqueness of Solutions for Semilinear Fractional Wave Equations

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) satisfies f b (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 ∂ 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.
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Yavar Kian, Masahiro Yamamoto. On Existence and Uniqueness of Solutions for Semilinear Fractional Wave Equations. Fractional Calculus and Applied Analysis, De Gruyter, 2017, 20 (1), pp.117-138. ⟨10.1515/fca-2017-0006⟩. ⟨hal-01214747⟩

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