RESEARCH PAPER FINITE-TIME ATTRACTIVITY FOR SEMILINEAR TEMPERED FRACTIONAL WAVE EQUATIONS Tran Dinh Ke 1 , Nguyen Nhu Quan 2 Abstract We prove the existence and finite-time attractivity of solutions to semi- linear tempered fractional wave equations with sectorial operator and su- perlinear nonlinearity. Our analysis is based on the α-resolvent theory, the fixed point theory for condensing maps and the local estimates of solutions. An application to a class of partial differential equations will be given. MSC 2010 : Primary 35B35; Secondary 37C75, 47H08, 47H10 Key Words and Phrases: tempered fractional wave equation; finite- time attractivity; measure of non-compactness; condensing map; sectorial operator 1. Introduction Let (X, ‖·‖) be a Banach space. We consider the following problem D α,σ 0 u(t)= Au(t)+ f (t, u t ),t ∈ [0,T ], (1.1) u(s)= ϕ(s),s ∈ [-h, 0], (1.2) u ′ (0) = y, (1.3) where α ∈ (1, 2),σ> 0, D α,σ 0 represents the tempered fractional derivative of order α in the Caputo sense (see Definition 2.4), the state function u takes values in X with the history state u t ∈ C ([-h, 0]; X) defined by u t (s)= u(t + s),s ∈ [-h, 0], A is a closed linear operator on X such that -A is sectorial, and the nonlinearity function f is defined on [0,T ] ×C ([-h, 0]; X). c  2018 Diogenes Co., Sofia pp. 1471–1492 , DOI: 10.1515/fca-2018-0077 Brought to you by | University of Sussex Library Authenticated Download Date | 2/25/19 10:13 AM