Existence and asymptotic expansion for a viscoelastic problem with a mixed nonhomogeneous condition (Q881617)

The following mixed problem is considered: \[ u_{tt}-u_{xx}+\int_0^t k(t-s)u_{xx}(s)\,ds+K| u|^{p-2}u+\lambda | u_t|^{q-2}u_t =f(x,t),\quad 0<x<1, \] \[ u_x(0,t)=u(0,t),\quad u_x(1,t)+\eta u(1,t)=g(t), \] \[ u(x,0)=\tilde{u}_0(x),\quad u_t(x,0)=\widetilde{u}_1(x), \] where \(\eta\geq 0\), \(p\geq 2\), \(q \geq 2\); \(K,\lambda\) are given constants and \(\widetilde{u}_0, \widetilde{u}_1,\,f,g,k\) are given functions. The authors claim the existence and uniqueness of a generalized solution and give an asymptotic expansion of a solution with respect to \(\eta\). In the reviewer's opinion, the existence part of Theorem 2.1 is questionable since regularity of solutions claimed, see (2.6), needs compatibility conditions, like (1.2), for the initial data. To prove the existence of second derivative, (1.6), it is necessary to estimate the trace \(u^N_{tt}(x,0)\) of Galerkin approximations which is impossible even for \(k(s)=K=\lambda=0\) while using an arbitrary basis in \(H^2(0,1)\) because basis functions, generally speaking, do not satisfy (1.2). These difficulties are well-known for specialists in hyperbolic equations and can be overcome by various ways.