Abstract

From the introduction: We consider the following problem: find a pair \((u,P)\) of functions satisfying \[ \begin{aligned} u_{tt}- u_{xx}+ Ku+\lambda u_t= F(x,t),\quad & 0< x< 1, 0< t< T,\ u_x(0, t)= P(t),\quad & u(1,t)= 0,\ u(x, 0)= u_0(x),\quad & u_t(x, 0)= u_1(x),\end{aligned} \] where the constants \(K\), \(\lambda\), the functions \(u_0\), \(u_1\), \(F\) are given before satisfying conditions specified later, and the unknown function \(u(x, t)\) and the unknown boundary value \(P(t)\) satisfy the following integral equation: \[ P(t)= g(t)+ K_1 |u(0, t)|^{\alpha-2} u(0, t)+ \lambda_1|u_t(0,t)|^{\beta-2}u_t(0,t)- \int^t_0 k(t- s)u(0, s) ds, \] in which the constants \(K_1\), \(\lambda_1\), \(\alpha\), \(\beta\) and the functions \(g\), \(k\) are also given before.