Mathematical model for a shock problem involving a linear viscoelastic bar

The following problem, finding a pair (u,P) of functions satisfying utt-uxx+F(u,ut) = f(x,t), 0<x<1, 0<t<T; ux(0,t) = P(t); ux(1,t)+K1u(1,t)+λ1ut(1,t) = 0; u(x,0) = u0(x), ut(x,0) = u1(x), and F(u,ut) = Ku+λut, is considered, where K,λ,K1,λ1 are given nonnegative constants and u0, u1, f are given functions satisfying conditions specified later, and the unknown function u(x,t) and the unknown boundary value P(t) satisfy the following Cauchy problem for ordinary differential equation P″(t)+ω+2$/P(t) = hutt(0,t), 0<t<T; P(0) = P0, P′(0) = P1, where ω>0, h≥0, P0, P1 are given constants. The theorem of global existence and uniqueness of a weak solution to the problems was proved, based on a Galerkin decomposition method.