Existence and exponential stability of solutions for transmission system with varying delay in R

Abstract

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.

A transmission problem for waves under time-varying delay and nonlinear weight

“…In the present work we improve the results in [27] where, for constant weights µ 1 (t) = µ 1 , µ 2 (t) = µ 2 and under adequate assumptions regarding the weight and time-varying delay, was proved the well posedness and singularity of solutions by using the semigroup theory. Authors also showed exponential stability by introducing an appropriate Lyapunov functional.…”

“…Under an appropriate assumption on the weights of the two feedbacks (µ 1 < µ 2 ), it was proved the well-posedness of the system and, under condition (1.7), it was established an exponential decay result. The result in [6] were improved by S. Zitouni et al [27]. There, the authors considered the problem with a time-varying delay τ (t) of the form (1.9)…”

Exponential stability of a transmission problem for wave equations with internal time‐varying delay and nonlinear degenerate weights

“…which is the same as that in Marzocchi et al [21] and similar to Datko [10] and Zitouni et al [16]. This assumption can be viewed as a restriction on the wave speeds of the two different materials.…”

“…. This is different from previous studies [16][17][18][19][20][21]29] and has not been studied by previous studies [27,28,30]. (iii) Under the assumption of 𝜉, Kato's variable norm technique can be used to obtain the well-posedness from Cauchy problem (2.7) with a non-autonomous operator  in (2.8).…”

“…It is well known that delay affects dynamical systems [10][11][12][13][14] and may be a source of instability [10,12,14,15]. For the sake of the effect of the delay in the transmission problem of wave equations, Zitouni et al considered the exponential stability of the following problem with a time-varying delay [16]:…”