Existence,General Decay And Blow Up In The Finite Time To Viscoelastic Wave Equations With Dynamic Boundary Conditions

Hybrid power system with dynamic boundary control has become one of the hot spots in the domestic and foreign scholars. As one of the branches, the equation of the dynamic boundary condition has been paid more and more attentions by scholars. In this work, we investigate viscoelastic wave equations with dynamic boundary condi-tions, several results on the solution existence, general decay and blow up in the finite time are given. The thesis is organized as follows:In the first Chapter, we review the background and some development of the related problems and summarize the main work of the present dissertation.In Chapter 2, we consider a viscoelastic Kirchhoff equation with Balakrishnan-Taylor damping, dynamic boundary conditions and a time-varying delay term acting on the boundary. By using the Faedo-Galerkin approximation method, we prove of the well-posedness of the solutions and by introducing suitable energy and perturbed Lya-punov functionals, we prove the general decay results, from which the usual exponential and polynomial decay rates are only special cases.In Chapter 3, we consider a semilinear damped wave equation with a viscoelastic term and dynamical boundary conditions. Under certain assumptions, we give the upper and lower bounds for the blow-up time according to the exponent number m and p of the nonlinear boundary damping term and the source term. For the case 2≤m<p, we extend the earlier exponentially growth result in Gerbi and Said-Houari (Adv. Nonlinear Analysis 2 (2):163-193,2013) to a blow-up in finite time result with positive initial energy and get the upper bound for the blow-up time. For the case m= 2, by using the concavity method, we prove a finite time blow-up result and get the upper bound for the blow-up time, which is a supplement to Gerbi and Said-Houari (Adv. Nonlinear Analysis 2 (2):163-193,2013). Moreover, for the case m≥2, under certain conditions on the data, we give a lower bound for the blow-up time when blow-up occurs.