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

Nguyen Thanh Long ; Le Xuan Truong

Electronic Journal of Differential Equations, Tome 2007 (2007).

Résumé

Summary: 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 $N+1$.

Détail

Classification : 35L20, 35L70
Keywords: nonlinear wave equation, linear integral equation, existence and uniqueness, asymptotic expansion