DISCRETE AND CONTINUOUS doi:10.3934/dcdsb.2019255 DYNAMICAL SYSTEMS SERIES B Volume 25, Number 2, February 2020 pp. 569–597 WELL-POSEDNESS RESULTS FOR FRACTIONAL SEMI-LINEAR WAVE EQUATIONS Jean-Daniel Djida Departamento de An´ alise Matem´ atica Universidade de Santiago de Compostela 15782 Santiago de Compostela, Spain Arran Fernandez ∗ Department of Mathematics, Eastern Mediterranean University Famagusta, Northern Cyprus, via Mersin-10, Turkey Iv´ an Area Departamento de Matem´ atica Aplicada II E.E. Aeron´autica e do Espazo, Universidade de Vigo 32004-Ourense, Spain Dedicated to Prof. Juan J. Nieto on the occasion of his 60th birthday Abstract. This work is concerned with well-posedness results for nonlocal semi-linear wave equations involving the fractional Laplacian and fractional derivative operator taken in the sense of Caputo. Representations for solutions, existence of classical solutions, and some L p -estimates are derived, by consid- ering a quasi-stationary elliptic problem that comes from the realisation of the fractional Laplacian as the Dirichlet-to-Neumann map for a non-uniformly el- liptic problem posed on a semi-infinite cylinder. We derive some properties such as existence of global weak solutions of the extended semi-linear integro- differential equations. 1. Introduction. In this paper we study well-posedness results for nonlocal semi- linear integro-differential wave equations which involve both the fractional Laplacian (in space) and the Caputo fractional derivative operator (in time), as follows:    D α t u +(−Δ) s u = f (u),u = u(x,t), (x,t) ∈ R d × (0, ∞), u(0) = ϕ, ϕ = ϕ(x) x ∈ R d , u t (0) = ψ, ψ = ψ(x) x ∈ R d , (1) where s ∈ (0, 1), α ∈ (1, 2], the integer d> 2s, and f ∈C 1 (R). We fix the variables t, x to be in the spaces t> 0, x ∈ R d , and we consider nonnegative solutions u of (1). The integro-partial differential equation (1) interpolates between the fractional heat equation (α = 1) and the fractional wave equation (α = 2). The problem (1) has been intensively studied by several authors (see e.g., [28, 29, 42, 3] and 2010 Mathematics Subject Classification. Primary: 26A33, 74G20, 74G25; Secondary: 35R11, 35G31, 35B65. Key words and phrases. Fractional partial differential equations, fractional semi-linear wave equations, local and global existence of solutions, regularity estimates. ∗ Corresponding author: Arran Fernandez. 569