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We study a temporally third order (Moore-Gibson-Thompson) equation with amemory term. Previously it is known that, in non-critical regime, the globalsolutions exist and the energy functionals decay to zero. More precisely, it isknown that the energy has exponential decay if the memory kernel decaysexponentially. The current work is a generalization of the previous one (PartI) in that it allows the memory kernel to be more general and shows that theenergy decays the same way as the memory kernel does, exponentially or not.

Moore-Gibson-Thompson equation with memory, part II: general decay of energy