In the present paper we are going to consider in a one dimension bounded domain a transmission system with a varying delay. Under suitable assumptions on the weights of the damping and the delay terms, we prove the well-possedness and the uniqueness of solution using the semigroup theory. Also we show the exponential stability by introducing an appropriate Lyaponov functional.2000 Mathematics Subject Classification. Primary: 435B37; Secondary: 35L55.

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In the present work, we consider a one-dimensional porous-elastic system with infinite memory and a nonlinear damping term. We establish the well-posedness of the system using semigroup theory and show the general decay for the case of nonequal speeds of wave propagation. Introducing some conditions on the kernel of the infinite memory term helps estimate the nonequal speed term even if this complementary control is not strong enough to stabilize the system exponentially. Our result is an extension of many other works in this area.

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