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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.
Wydawca
Rocznik
Tom
Strony
99--105
Opis fizyczny
Bibliogr. 13 poz.
Twórcy
autor
- Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, P.O. Box 17, Mladost 4, Sofia 1715, Bulgaria
Bibliografia
- [1] R. A. Adams, Sobolev spaces, Academic press, New York 1975.
- [2] A. V. Bitsadze, Theory of equations of mixed type, whose order degenerates along the line of change of type. In: Continuous medium mechanics and related problems of analysis, Moscow (1972) 47-52.
- [3] A. V. Bitsadze, Some classes of partial differential equations, Gordon and Breach Science Publishers, New York 1988.
- [4] R. W. Carroll, R. E. Showalter, Singular and degenerate Cauchy problems, Academic press, New York 1976.
- [5] K. Friedrichs, On the identity of weak and strong extensions of differential operators, Trans. Amer. Math. Soc., 55 (1944) 132-151.
- [6] N. Yu. Kapustin, Solvability in the class L2 of a characteristic problem for a hyperbolic equation with order degeneration, Dokl. Acad. Nauk USSR, 311 (1990) 784-788.
- [7] N. Yu. Kapustin, Solvability in the class L2 of a problem with a boundary condition on the characteristic for a hyperbolic equation with order degeneration, Differents. Uravn., 26 (1990) 66-75.
- [8] M. A. Krasnoselskiĭ, Topological methods in the theory of nonlinear integral equations, Pergamon Press, Oxford 1964.
- [9] N. A. Lar’kin, M. Schneider, Uniqueness theorems for a nonlinear Tricomi problem and the related evolution problem, Math. Meth. in the Appl. Sci., 18 (1995) 591-601.
- [10] R. I. Semerdjieva, A hyperbolic equation with degeneration of order, Serdica, 8 (1982) 57-63.
- [11] R. I. Semerdjieva, Solvability of a boundary value problem for a nonlinear degenerated hyperbolic equation, Differents. Uravn., 28 (1992) 1145-1153.
- [12] M. M. Smirnov, Degenerated hyperbolic equations, Vyshaya Skola, Minsk 1977.
- [13] S. A. Tersenov, Introduction to the theory of equations, degenerating on the boundary, Novosibirsk State University, Novosibirsk 1973.
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-article-BAT5-0001-0051