A uniqueness theorem in linear viscoelasticity (Q1109606)

The one-dimensional homogeneous viscoelastic body governed by the constitutive equation \[ \sigma(x,t)=G(0)u_ x(x,t)+\int^{t}_{- \infty}G(t-s)u_ x(x,s)ds, \] occupying the segment \([0,\ell]\), is considered. It is proven that if \(G(0)\), \(\dot G\), \(\ddot G\), and \(\epsilon\) satisfy the conditions \(\dot G,\ddot G\in V'_{\epsilon}\), and \[ G(0)>\int^{\infty}_{0}| \dot G(\tau)| e^{-\epsilon \tau}d\tau +\frac{1}{\epsilon}(| \dot G(\quad 0)| +\int^{\infty}_{0}| \ddot G(\tau)| e^{-\epsilon \tau}d\tau) \] then the dynamic problem (a) \(\sigma_ x(x,t)+F(x,t)=\rho u_{tt}(x,t)\), and (b) \(u(0,t)=u_ 0(t)\), \(u(\ell,t)=u_ 1(t)\), \(t\in (-\infty,0)\), has a unique solution \(u\in C^ 2(\Sigma)\cap U_{\epsilon}\), \(u_ t\in U_{\epsilon}\), where \(\Sigma =[0,\ell]\times (-\infty,0)\), \(U_{\epsilon}=\{u\in C^ 2(\Sigma)| \sup | u(x,t)| e^{-\epsilon t}<\infty\}\), \(V_{\epsilon}=\{v\in C((-\infty,0))| \sup_{\infty}| v(t)| e^{-\epsilon t}<\infty \}\), and \(V'_{\epsilon}=\{v\in C((0,\infty))| \int^{\infty}_{0}| v(t)| e^{-\epsilon t}dt<\infty\}\). The solution of the problem a) and b) is not determined; we have to add supplementary conditions to the boundary conditions b). Usually, these conditions are the initial conditions. Here, the requirement regarding the behaviour of the solution as \(t_ 0\to -\infty\), described by its belonging to the set \(U_{\epsilon}\), determines the problem a) and b). The particular case of the corresponding quasi-static problem is considered and some applications of the obtained results are given.