In this paper, we consider the viscoelastic wave equation with a delay term in internal feedbacks; namely, we investigate the following problemtogether with initial conditions and boundary conditions of Dirichlet type. Here (x, t) ∈ Ω × (0, ∞), g is a positive real valued decreasing function and μ 1 , μ 2 are positive constants. Under an hypothesis between the weight of the delay term in the feedback and the weight of the term without delay, using the Faedo-Galerkin approximations together with some energy estimates, we prove the global existence of the solutions. Under the same assumptions, general decay results of the energy are established via suitable Lyapunov functionals.
Mathematics Subject Classification (2000). 35L05 · 35L15 · 35L70 · 93D15.
We consider the inverse problem of finding the temperature distribution and the heat source whenever the temperatures at the initial time and the final time are given. The problem considered is one dimensional and the unknown heat source is supposed to be space dependent only. The existence and uniqueness results are proved.
Two inverse problems for the wave equation with involution are considered. Results on existence and uniqueness of solutions of these problems are presented.
In this paper, we consider one-dimensional linear Bresse systems in a bounded open domain under Dirichlet-Neumann-Neumann boundary conditions with two infinite memories acting only on two equations. First, we establish the well-posedness in the sense of semigroup theory. Then, we prove two (uniform and weak) decay estimates depending on the speeds of wave propagations, the smoothness of initial data and the arbitrarily growth at infinity of the two relaxation functions.
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