Geodesic
Existence and asymptotic expansion of solutions to a nonlinear wave equation with a memory condition at the boundary
Electronic journal of differential equations, Tome 2007 (2007)
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

$ 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 $N+1$.
Classification : 35L20, 35L70
Keywords: nonlinear wave equation, linear integral equation, existence and uniqueness, asymptotic expansion 
@article{EJDE_2007__2007__a188,
     author = {Nguyen Thanh Long and Le Xuan Truong},
     title = {Existence and asymptotic expansion of solutions to a nonlinear wave equation with a memory condition at the boundary},
     journal = {Electronic journal of differential equations},
     year = {2007},
     volume = {2007},
     zbl = {1112.35124},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EJDE_2007__2007__a188/}
}
TY  - JOUR
AU  - Nguyen Thanh Long
AU  - Le Xuan Truong
TI  - Existence and asymptotic expansion of solutions to a nonlinear wave equation with a memory condition at the boundary
JO  - Electronic journal of differential equations
PY  - 2007
VL  - 2007
UR  - http://geodesic.mathdoc.fr/item/EJDE_2007__2007__a188/
LA  - en
ID  - EJDE_2007__2007__a188
ER  - 
%0 Journal Article
%A Nguyen Thanh Long
%A Le Xuan Truong
%T Existence and asymptotic expansion of solutions to a nonlinear wave equation with a memory condition at the boundary
%J Electronic journal of differential equations
%D 2007
%V 2007
%U http://geodesic.mathdoc.fr/item/EJDE_2007__2007__a188/
%G en
%F EJDE_2007__2007__a188
Nguyen Thanh Long; Le Xuan Truong. Existence and asymptotic expansion of solutions to a nonlinear wave equation with a memory condition at the boundary. Electronic journal of differential equations, Tome 2007 (2007). http://geodesic.mathdoc.fr/item/EJDE_2007__2007__a188/