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Uniqueness theorem for nonlinear hyperbolic equations with order degeneration

Semerdjieva R.

Bulletin of the Polish Academy of Sciences. Mathematics

2003

Vol. 51, no 1

99-105

Let k(y) > O, l(y) > O for y > O, k(0) = l(0) = 0; then the equation L(u) := k(y)u(xx) - (delta)y(l(y)uy) +a(x,y)ux = f (x,y,u) is strictly hyperbolic for y > O and its order degenerates on the line y = 0. We 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 [delta]G = 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 x = 1 - F(y) are characteristic curves. It is proved that if f satisfies the Caratheodory condition and \f{x,y,z(1)}-f{x,y,z(2))\ less than or equal to C(\z(1)\(beta) + \z(2)\(beta))\z1-z2\ with some constants C > O and beta is greater than or equal to O then there exists at most one generalized solution.