existence and asymptotic expansion of solutions to a nonlinear wave equation with a memory condition at the boundary

We study the initial-boundary value problem for the nonlinear wave equation$$displaylines{ u_{tt} - frac{partial }{partial x} (mu ({x,t})u_x ) + K|u |^{p - 2} u + lambda |u_t |^{q - 2} u_t = f(x,t), cr u(0,t) = 0 cr - mu (1,t)u_x (1,t) = Q(t), cr u(x,0) = u_0 (x),quad u_t (x,0) = u_1 (x), cr}$$ where $pgeq 2$, $q geq 2$, $K, lambda$ are given constants and $u_0, u_1, f,mu$ are given functions. The unknown function $u(x,t)$ and the unknown boundary value $Q(t)$ satisfy the linear integral equation$$ Q(t)=K_1(t)u(1,t)+lambda_1(t)u_t(1,t)-g(t)-int_0^t {k(t-s)u(1,s)ds},$$ where $K_1, lambda_1, g, k$ are given functions satisfying some properties stated in the next section. This paper consists of two main sections. First, we prove the existence and uniqueness for the solutions in a suitable function space. Then, for the case $K_1(t)=K_1geq 0$, we find the asymptotic expansion in $K,lambda, K_1$ of the solutions, up to order $N+1$.