Abstract :
In this paper we study the equation L(u) := k(y)uxx − ∂y( (y)uy ) + a(x, y)ux +
b(x, y)uy = f (x, y,u), where k(y) > 0, (y) > 0 for y >0, k(0) = (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
R2 with piecewise smooth boundary ∂G = AB ∪ AC ∪ BC; AB = {(x, 0): 0 x 1},
AC: x = F(y) =
y
0 (k(t)/ (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|ρ , ρ 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 Carathéodory condition and
|f (x, y, z1) − f (x, y, z2)| C(|z1|β + |z2|β )|z1 − z2| with some constants C > 0 and
β 0 then there exists at most one generalized solution.
2002 Elsevier Science (USA). All rights reserved.
1. Introduction
