Abstract
In this paper, we consider a transmission problem in the presence of history and delay terms.Under appropriate assumptions, we prove well-posedness by using the semigroup theory. Our stability estimate proves that the unique dissipation given by the history term is strong enough to stabilize exponentially the system in presence of delay by introducing a suitable Lyaponov functional.
Highlights
IntroductionIn this paper we study the following transmission system with a past history and a delay term
In this paper we study the following transmission system with a past history and a delay term utt(x, t) − auxx(x, t) + g(s)uxx(x, t − s)ds
In [7] the authors examined a system of wave equations with a linear boundary damping term with a delay:
Summary
In this paper we study the following transmission system with a past history and a delay term. In [7] the authors examined a system of wave equations with a linear boundary damping term with a delay:. Vtt(x, t) − bvxx(x, t) = 0, (x, t) ∈ (L1, L2) × (0, +∞), and under the assumption μ2 ≤ μ1 They proved that the solution is exponentially stable. In [11], authors considered the equation utt(x, t) − ∆xu(x, t) − μ1∆xut(x, t) − μ2∆xut(x, t − τ ) = 0, and under the assumption. They proved the well-posedness and the exponential decay of energy.
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