Generalized Solutions of a Class of Linear and Quasi-Linear Degenerated Hyperbolic Equations

Authors : Rossitza SEMERDJIEVA

Pages : 375-388

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Publication Date : 0000-00-00

Article Type : Research Paper

Abstract :The equation Linsert ignore into journalissuearticles values(u);:=kinsert ignore into journalissuearticles values(y);uxx-\partialyinsert ignore into journalissuearticles values(\ellinsert ignore into journalissuearticles values(y);uy);+rinsert ignore into journalissuearticles values(x,y);u=finsert ignore into journalissuearticles values(x,y,u);, where kinsert ignore into journalissuearticles values(y);>0, \ellinsert ignore into journalissuearticles values(y);>0 for y>0,kinsert ignore into journalissuearticles values(0);=\ellinsert ignore into journalissuearticles values(0);=0 and limy\rightarrow 0kinsert ignore into journalissuearticles values(y);/\ellinsert ignore into journalissuearticles values(y); exists, is strictly hyperbolic for y>0 and its order degenerates on the line y=0. We consider the boundary value problem Lu=finsert ignore into journalissuearticles values(x,y,u); in G, u\midAC=0, where G is a simply connected domain in R2 with piecewise smooth boundary \partial G=AB\cup AC\cup BC; AB=\{insert ignore into journalissuearticles values(x,0);: 0\leq x\leq 1\}, AC : x=Finsert ignore into journalissuearticles values(y); =\int0yinsert ignore into journalissuearticles values(kinsert ignore into journalissuearticles values(t);/\ell insert ignore into journalissuearticles values(t););1/2dt and BC: x=1-Finsert ignore into journalissuearticles values(y); are characteristic curves. The existence and uniqueness of a generalized solution to this problem are proved in the linear case insert ignore into journalissuearticles values(where f=finsert ignore into journalissuearticles values(x,y););; the nonlinear case is treated by using the Schauder Fixed Point Theorem.
Keywords : Turk J Math, 23, 1999, , 375 388 Turk J Math, vol 23, iss 3