On systems of two singularly perturbed quasilinear second-order equations

A system of two quasilinear second-order equations with a small parameter standing by the second derivatives is studied. The cases where the matrix of coefficients of the first derivatives has the following eigenvalues are considered: (a) both of them have negative real parts; (b) they are of opposite sign; (c) one of them is equal to zero. To find a solution and its asymptotics, the initial-value or boundary-value problems are posed depending on the form of these eigenvalues.