Some classes of hyperbolic equations and equations of mixed type (Q1261057)

This monograph deals with linear and quasilinear second order hyperbolic equations with two independent variables, in particular, with parabolic degeneration. It contains both classical and the author's own results. The statements are based on the unified approach of a method founded on the construction and application of so-called general integrals (solutions). In the first chapter the question of construction of general integrals is discussed. For this aim the classical characteristic method is used. Its effectiveness is shown on some examples of quasilinear equations. Further, the linear hyperbolic equations, the Laplace invariants of which are equal to zero, are considered. When the Laplace invariants or their nonlinear analogues are not equal to zero, such equations can sometimes be reduced to the previous case. As an illustration, the Liouville equation is considered and the Laplace cascade method is described. In the second chapter, the Riemann method and some ways of construction of the Riemann function are given for some linear equations for which the construction of the general integrals is not possible. The initial, characteristic and Darboux problems are also considered for hyperbolic, and parabolic degenerate hyperbolic equations of such a kind. In the third chapter, the Cauchy problem is globally investigated for a semilinear equation in the case of finite and unbounded data supports. Several nonlinear variants of Darboux and Goursat characteristic problems are also treated. Finally special attention is paid to the structures of the ranges of definitions of the solutions.