The purpose of this work is to investigate the stabilization of a system of weakly coupled wave equations with one or two locally internal Kelvin-Voigt damping and non-smooth coefficient at the interface. The main novelty in this paper is that the considered system is a coupled system and that the geometrical situations covered (see Remarks 5.6, 5.12) are richer than all previous results, even for simple wave equation with Kelvin-Voigt damping. Firstly, using a unique continuation result, we prove that the system is strongly stable. Secondly, we show that the system is not always exponentially stable, instead, we establish some polynomial energy decay estimates. Further, we prove that a polynomial energy decay rate of order t −1/2 is optimal in some sense.
The existence, uniqueness, strong and exponential stability of a generalized telegraph equation set on one dimensional star shaped networks are established. It is assumed that a dissipative boundary condition is applied at all the external vertices and an improved Kirchhoff law at the common internal vertex is considered. First, using a general criteria of Arendt-Batty (see Arendt and Batty in Trans. Am. Math. Soc. 306(2):837-852, 1988), combined with a new uniqueness result, we prove that our system is strongly stable. Next, using a frequency domain approach, combined with a multiplier technique and the construction of a new multiplier satisfying some ordinary differential inequalities, we show that the energy of the system decays exponentially to zero.
This work is devoted to the study of a compressible viscoelastic fluids satisfying the Oldroyd-B model in a regular bounded domain. We prove the local existence of solutions and uniqueness of flows by a classical fixed point argument.
In this work, we are interested in a mathematical problem arising from the dynamics of dislocation densities in crystals. The model, originally developed by Groma, Czikor, and Zaiser in [], is a coupled singular parabolic system that describes the motion of dislocations in a bounded crystal, taking into account the short‐range interactions and the effect of exterior stresses. We show the derivation and the short time existence and uniqueness of a regular solution in a Hölder space using a fixed point argument and a particular comparison principle.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.