solvability, continuous dependence and asymptotic expansion of solutions in a small parameter of dirichlet problem for a nonlinear kirchhoff wave equation

We study the existence, uniqueness, continuous dependence, and asymptotic expansion of solutions of the dirichlet problem for a nonlinear kirchhoff wave equation. at first, we state and prove a theorem involving the local existence and uniqueness of a weak solution. next, we establish a sufficient condition to get an estimate of the continuous dependence of the solution with respect to the nonlinear terms. finally, an asymptotic expansion of high order in a small parameter of a weak solution is also discussed.