Electronic Journal of Differential Equations,
Vol. 2007(2007), No. 48, pp. 1-19.

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

Authors: Nguyen Thanh Long (Hochiminh City National Univ., Vietnam)
         Le Xuan Truong (Univ. of Technical Education in HoChiMinh City Vietnam)

 
Abstract:
 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 $p\geq 2$, $q \geq 2$, $K, \lambda$ 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
 $K_1(t)=K_1\geq 0$, we find the asymptotic expansion in
 $K,\lambda, K_1$ of the solutions, up to order $N+1$.
  
Submitted October 27, 2006. Published March 20, 2007.
Math Subject Classifications: 35L20, 35L70.
Key Words: Nonlinear wave equation; linear integral equation; 
           existence and uniqueness; asymptotic expansion.