JOURNAL OF INTEGRAL EQUATIONS AND APPLICATIONS Vol. , No. , YEAR https://doi.org/jie.YEAR..PAGE GENERAL DECAY RATE FOR AN ABSTRACT WEAKLY DISSIPATIVE MOORE-GIBSON-THOMPSON EQUATION JAMILU HASHIM HASSAN AND SALIM A. MESSAOUDI ABSTRACT. In this paper we study an abstract class of weakly dissipative Moore-Gibson-Thompson equation with finite memory. We establish a general decay rate for the solution of the system under some appropriate conditions on the relaxation function. 1. Introduction Let H be a real separable Hilbert space whose associated inner product and norm are respectively denoted by (·, ·) and k·k. The main objective of this work is the study of a class of third-order abstract integro-differential equations of the form (1)    u ttt + α u tt + β Au t + γ Au - Z t 0 g(t - s)A σ u(s)ds = 0, t > 0, u(0)= u 0 , u t (0)= u 1 , u tt (0)= u 2 , where α , β , γ , σ are positive constants with σ ∈ (0, 1), A : D (A) ⊂ H -→ H is a positive definite self-adjoint operator on H such that the injection D (A r ) ,→ D (A s ) is compact for any r > s ≥ 0. System (1) is an abstract version of interpolating cases of Moore-Gibson-Thompson (MGT) equation that appears in nonlinear acoustics and arises in modeling the dynamics of high-frequency ultrasound propagation taking into consideration both thermal flux and molecular relaxation times. In fact, in some materials, the “viscoelastic” operator A σ is weaker than the principal operator A. For instance, in a viscoelastic plate, A = Δ 2 and the “viscoelatic” operator is -Δ (σ = 1/2). This was considered earlier by Rivera and Naso [11] and Rivera et al. [12]. There are may works in the literature that dealt with the well-posedness and the asymptotic stability of MGT equation, see [1, 3, 5, 6, 10] and the references therein. Memory-type MGT equation had been studied by many researchers and different types of asymptotic stability results of the system had been established depending on the values of parameters in the equation and the decay rate of the relaxation function. In [8], Lasiecka and Wang considered the following memory-type MGT equation (2) τ u ttt + α u tt + c 2 Au + bAu t - Z t 0 g(t - s)Aw(s)ds = 0, 2020 Mathematics Subject Classification. 34G10; 35B35; 35B40; 35L90; 45D05. Key words and phrases. Moore-Gibson-Thompson equation, weakly dissipative system, viscoelastic, general decay. Submitted to Journal of Integral Equations and Applications - NOT THE PUBLISHED VERSION 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 1