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We are concerned with the well-posedness of solutions as well as the asymptotic behaviour of the energy related to the viscoelastic wave equation with localized memory with past history and supercritical source and damping terms, posed on a bounded domain Ω ⊂ R 3 . Avoiding any relation between the damping and source terms, we obtain the global existence of solutions and the uniform energy decay rates, introducing the notion of a potential well and defining the expanded total energy functional. We also prove that weak solutions blow-up in finite time. [ABSTRACT FROM AUTHOR]

In this work, we are concerned with the well-posedness of solutions as well as the asymptotic behavior of the energy related to the semilinear coupled heat-wave equations, posed on a bounded domain of R n. Inspired by the works (Cavalcanti and Oquendo SIAM J Control Optim. 42:1310-24, 2003) and (Nicaise and Pignotti SIAM J Control Optim. 5(45):1561-85, 2006), we consider dampings in the wave equation and the heat equation, and both dampings have a delay in time. The damping acting on the heat equation is localized and not strategically distributed. In these conditions, we prove that the solutions of this system are exponentially stable. [ABSTRACT FROM AUTHOR]