具记忆和变时滞项的二阶发展方程能量衰减率的凸性估计

On Convexity for Energy Decay Rates of the Second-Order Evolution Equation with Memory and Time-Varying Delay

研究了如下一类具有记忆和变时滞项的抽象发展方程u_(tt)(t)+Au(t)-(∫_0~t g(t-s)Au(s)ds+μ_1h_1(u_t(t))+μ_2h_2(u_t(x,t-T(t)))=▽F(u(t)).通过构造合适的能量泛函和Lyapunov泛函,利用凸函数的一些性质,得到了依赖于h_1及记忆核g的能量衰减估计.此衰减估计可以应用于一些具体的模型.

In this paper, we consider the following abstract evolution equation with memory and time-varying delay of the form u_（tt）（t）＋Au（t）-（∫_0~t g（t-s）Au（s）ds＋μ_1h_1（u_t（t））＋μ_2h_2（u_t（x,t-T（t）））=▽F（u（t））.By introducing suitable energy and Lyapunov functionals, and making use of some properties of the convex functions, we establish decay estimate for the energy, which depends on the behavior of h_1 and the relaxation g. The decay estimate can be applied to various concrete models. We shall also give some applications to illustrate our result.