Electron. J. Diff. Eqns., Vol. 2007(2007), No. 48, pp. 1-19.

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

Nguyen Thanh Long, Le Xuan Truong

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.

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Nguyen Thanh Long
Department of Mathematics
Hochiminh City National University
227 Nguyen Van Cu, Q5, HoChiMinh City, Vietnam
email: longnt@hcmc.netnam.vn, longnt2@gmail.com
Le Xuan Truong
Department of Mathematics, Faculty of General Science
University of Technical Education in HoChiMinh City
01 Vo Van Ngan Str., Thu Duc Dist., HoChiMinh City, Vietnam
email: lxuantruong@gmail.com

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