This study examines a nonlinear wave equation with boundary fractional damping and time-varying delay. After proving the existence of a global solution, we establish exponential stability by constructing an appropriate Lyapunov functional. Finally, we prove the blow-up phenomenon using the potential-well method.

In this manuscript, we study a thin and narrow plate equation that models the deck of a suspension bridge that is subject to a Balakrishnan-Taylor damping and a strong damping. First, by using the Faedo Galerkin method, we prove the existence of both global weak and regular solutions. Second, we prove the exponential stability of the energy for regular solutions by combining the multiplier method and a well-known result of Komornik.

In this article, we consider the Kirchhoff plate equation with delay terms on the boundary control. We give instability examples of systems for some choices of delays. Finally, we prove its well-posedness, strong stability, and exponential stability under a multiplier geometric control condition. Foro more information see https://ejde.math.txstate.edu/Volumes/2023/68/abstr.html

In this manuscript we consider a coupled, by second order terms, system of two wave equations with a past history acting on the first equation as a stabilizer. We show that the solution of this system decays exponentially by constructing an appropriate Lyapunov function.

In the present paper, we examine a suspension bridges model subject to frictional damping, a fractional delay term, and a source term. First, we prove the existence of global solutions of the problem. Second, for small initial data, we establish the exponential stability of the system by using the energy method. Additionally, we show that if the initial energy assumes a negative value, the solution blows up in finite time.

In this manuscript, we study the asymptotic stability of solutions of two coupled quasi-linear viscoelastic Kirchhoff plate equations involving free boundary conditions, and accounting for rotational forces \begin{document}$ \begin{eqnarray*} &&\vert y_t\vert^{\rho}y_{tt}-\Delta y_{tt}+\Delta^{2}y- \int_0^t h_1(t-s)\Delta^2 y(s)\;ds+f_1(y, z) = 0,\\\\ &&\vert z_t\vert^{\rho}z_{tt}-\Delta z_{tt}+\Delta^{2}z- \int_0^t h_2(t-s)\Delta^2 z(s)\;ds+f_2(y, z) = 0. \end{eqnarray*} $\end{document} The system under study in this contribution could be seen as a model for two stacked plates. This work is motivated by previous works about coupled quasi-linear wave equations or concerning single quasi-linear Kirchhoff plate. The existence of local weak solutions is established by the Faedo-Galerkin approach. By using the perturbed energy method, we prove a general decay rate of the energy for a wide class of relaxation functions.