ASYMPTOTICS OF THE SOLUTION TO A PIECEWISE-SMOOTH QUASILINEAR SECOND-ORDER DIFFERENTIAL EQUATION^∗

We investigate a singularly perturbed boundary value problem for a piecewise-smooth second-order quasilinear differential equation in the case when the discontinuous curve which separates the domain is monotone. Ap-plying the boundary layer function method, the asymptotic expansion of a solution with internal layer appearing in the neighborhoods of some point on the monotone curve and the point itself is constructed. For sufficiently small parameter values, using the matching method, the existence of a smooth solution with an internal transition layer in the neighborhood of a point of the monotone curve is proved. A simple example is given to show the effectiveness of our method.