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.