A boundary value problem for nonlinear hyperbolic equations with order degeneration

In this paper we study the equation L(u) := k(y)u(xx) - partial derivative(y)(l(y)u(y)) + a(x, y)u(x) + b(x, y)uy = f (x, y, u), where k(y) > 0, f(y) > 0 for y > 0, k(0) = l(0) = 0; it is strictly hyperbolic for y > 0 and its order degenerates on the line y = 0. Consider the boundary value problem Lu = f (x, y, u) in G, u\(AC) = 0, where G is a simply connected domain in R-2 with piecewise smooth boundary partial derivativeG = AB boolean OR AC boolean OR BC; AB = {(x, 0): 0 less than or equal to x less than or equal to 1}, AC: x = F(y) = integral(0)(y)(k(t)/l(t))(1/2)dt and BC: x = 1 - F(y) are characteristic curves. If f (x, y, u) = g(x, y, u) - r(x, y)u\u\(rho), rho greater than or equal to 0, we obtain existence of generalized solution by a finite element method. The uniqueness problem is considered under less restrictive assumptions on f. Namely, we prove that if f satisfies Caratheodory condition and \f (x, y, z) - f (x, y, z(2))\ less than or equal to C(\z(2)\(beta) + \z(2)\(beta))\z1 - z2\ with some constants C > 0 and beta greater than or equal to 0 then there exists at most one generalized solution. (C) 2002 Elsevier Science (USA). All rights reserved.