Abstract

We study the initial-boundary value problem for a nonlinear wave equation given by \[ \begin{gathered} u_{tt}- u_{xx}+ K|u|^{p-2} u+\lambda|u_t|^{q-2} u_t= F(x,t),\quad 0< x< 1, 0< t< T,\ u_x(0, t)= P(t),\quad u(1,t)= 0,\ u(x, 0)= u_0(x),\quad u_t(x,0)= u_1(x),\end{gathered}\tag{1} \] where \(p\geq 2\), \(q> 1\), \(K\), \(\lambda\) are given constants and \(u_0\), \(u_1\), \(F\) are given functions, the unknown function \(u(x, t)\) and the unknown boundary value \(P(t)\) satisfy the following nonlinear integral equation \[ P(t)= g(t)+ K_1|u(0, t)|^{\alpha-2} u(0, t)+ |u_t(0, t)|^{\beta-2} u_t(0, t)- \int^t_0 k(t- s)u(0, s) ds, \] where \(K_1\), \(\alpha\), \(\beta\) are given constants and \(g\), \(k\) are given functions. In Part 1 we prove a theorem of existence and uniqueness of a weak solution \((u, P)\) of problem (1), (2). The proof is based on the Faedo-Galerkin method associated with a priori estimates, weak convergence and compactness techniques. In Part 3 we obtain an asymptotic expansion of the solution \((u, P)\) of the problem (1), (2) up to order \(N+ 1\) in three small parameters \(K\), \(\lambda\), \(K_1\).