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We consider in this paper the problem of asymptotic behavior of solutions for two viscoelastic   wave equations with infinite memory. We show that the stability of the system holds for a much larger class of kernels   and get better decay rate than the ones known in the literature. More precisely, we consider infinite memory kernels    satisfying, where and are given functions. Under this very general assumption on the behavior of g at infinity and for   each viscoelastic wave equation, we provide a relation between the decay rate of the solutions and the growth of g at   infinity, which improves the decay rates obtained in [15, 16, 17, 19, 40]. Moreover, we drop the boundedness assumptions   on the history data considered in [15, 16, 17, 40].

NEW GENERAL DECAY RATES OF SOLUTIONS FOR TWO VISCOELASTIC WAVE EQUATIONS WITH INFINITE MEMORY