具边界反馈时滞的粘弹方程的能量衰减(英文)

General Decay of the Energy for a Viscoelastic Wave Equation with a Delay Term in the Boundary Feedback

考虑如下具边界反馈时滞的粘弹方程ut(x,t)-Δu(x,t)+∫0tg(t-s)Δu(x,s)ds=0,x∈Ω,t>0,u(x,t)=0,x∈Γ0,t>0,?u /?v=∫0tg(t-s)/vu(s)ds-μ1ut(x,t)-μ2ut(x,t-τ),x∈Γ1,t>0,u(x,0)=u0(x),ut(x,0)=u1(x),x∈Ω,ut(x,t-τ)=f0(x,t-τ),x∈Ω,0<t<τ,其中Ω∈Rn(n≥1)是具C2类边界Ω的有界域.此外,g是所谓的"记忆核",μ1,μ2是两个实数,τ为时滞.在假设|μ2|<μ1下,通过构造合适的Lyapunov函数,证明上述问题能量的一般衰减性,使得指数型衰减和多项式衰减仅仅是其特殊情况.

We consider a viscoelastic wave equation with a delay term in the boundary feedback; namely, we study the following problem ut（x,t）-Δu（x,t）＋∫0tg（t-s）Δu（x,s）ds=0,x∈Ω,t〉0,u（x,t）=0,x∈Γ0,t〉0,？u /？v=∫0tg（t-s）/vu（s）ds-μ1ut（x,t）-μ2ut（x,t-τ）,x∈Γ1,t〉0,u（x,0）=u0（x）,ut（x,0）=u1（x）,x∈Ω,ut（x,t-τ）=f0（x,t-τ）,x∈Ω,0〈t〈τ,whereΩ∈Rn（n≥1） be a regular and bounded domain with a boundary ЭΩ of class C2. Moreover,g is so-called ＂memory kernel＂,μ1,μ2 are two real coefficients which are not necessarily positive, and r represents the time delay. Under the restriction ｜μ2｜〈μ1 , general decay results of the energy of the concerned problem are obtained via an appropriate Lyapunov functional. And the exponential and polynomial types of decay are only special cases.