We consider the wave equation with a weak internal damping with non-constant delay and nonlinear weights given by utt(x, t) − uxx(x, t) + µ 1 (t)ut(x, t) + µ 2 (t)ut(x, t − τ (t)) = 0 in a bounded domain. Under proper conditions on nonlinear weights µ 1 (t), µ 2 (t) and non-constant delay τ (t), we prove global existence and estimative the decay rate for the energy.

In this paper, we study the well-posedness and asymptotic stability to a thermoelastic laminated beam with nonlinear weights and time-varying delay. To the best of our knowledge, there are no results on the system and related Timoshenko systems with nonlinear weights. On suitable premises about the time delay and the hypothesis of equal-speed wave propagation, existence and uniqueness of solution is obtained by combining semigroup theory with Kato variable norm technique. The exponential stability is proved by energy method in two cases, with and without the structural damping, by using suitably sophisticated estimates for multipliers to construct an appropriated Lyapunov functional.

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In this article we study the well-posedness and exponential stability to the one-dimensional system in the linear isothermal theory of swelling porous elastic soils subject with time-varying weights and time-varying delay. We prove existence of global solution for the problems combining semigroup theory with the Kato's variable norm technique. To prove exponential stability, we apply the energy method without the equal wave speeds assumption.

This paper is concerned with the well-posedness of global solution and exponential stability to the Timoshenko system subject with time-varying weights and time-varying delay. We consider two problems: full and partially damped systems. We prove existence of global solution for both problems combining semigroup theory with the Kato's variable norm technique. To prove exponential stability, we apply the Energy Method. For partially damped system the exponential stability is proved under assumption of equal-speed wave propagation in the transversal and angular directions. For full damped system the exponential stability is obtained without the hypothesis of equal-speed wave propagation.