Turkish Journal of Mathematics
DOI
-
Abstract
The equation $L(u):=k(y)u_{xx}-\partial_y(\ell(y)u_y)+r(x,y)u=f(x,y,u),$ where $k(y)>0, \ell(y)>0$ for $y>0,k(0)=\ell(0)=0$ and $lim_{y\rightarrow 0}k(y)/\ell(y)$ exists, is strictly hyperbolic for $y>0$ and its order degenerates on the line $y=0$. We consider the boundary value problem $Lu=f(x,y,u)$ in $G, u\mid_{AC}=0$, where $G$ is a simply connected domain in $R^2$ with piecewise smooth boundary $\partial G=AB\cup AC\cup BC; AB=\{(x,0): 0\leq x\leq 1\}, AC : x=F(y) =\int_0^y(k(t)/\ell (t))^{1/2}dt$ and $BC: x=1-F(y)$ are characteristic curves. The existence and uniqueness of a generalized solution to this problem are proved in the linear case (where $f=f(x,y));$ the nonlinear case is treated by using the Schauder Fixed Point Theorem.
First Page
375
Last Page
388
Recommended Citation
SEMERDJIEVA, ROSSITZA (1999) "Generalized Solutions of a Class of Linear and Quasi-Linear Degenerated Hyperbolic Equations," Turkish Journal of Mathematics: Vol. 23: No. 3, Article 3. Available at: https://journals.tubitak.gov.tr/math/vol23/iss3/3
