On the nonlinear wave equation with the mixed nonhomogeneousconditions: Linear approximation and asymptotic expansion of solutions

摘要

In this paper, we consider the following nonlinear wave equationwhere ilo, ill, it, f, g are given functions. To problem (1), we associate a linear recursivescheme for which the existence of a local and unique weak solution is proved by applyingthe Faedo-Galerkin method and the weak compact method. In the case of eft E CN+2(R), Pi E CN+1(R), li(z) > tro > 0, /([ (z) > 0, for all z E R, andg E C3(R+),f E CN+1([0, 1 l xR+ x R3), fl E CN([0, 11 x 14 x R3), a weak solution u,,,e2(x, t) having an asymptotic expansion of order N + 1 in two small parameters e,, e2 is established for the followingequation associated to (1)2,3: utt — (111(u) + eliil (u)lux) = f (x, t, u, ux, ut) +(X, t, u, ux, ut).(2)