AI Summary & Key Takeaways
AI InsightThe 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.
📌 3 Key Takeaways
Abstract
The equation L(u):=k(y)uxx-\partialy(\ell(y)uy)+r(x,y)u=f(x,y,u), where k(y)>0, \ell(y)>0 for y>0,k(0)=\ell(0)=0 and limy\rightarrow 0k(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\midAC=0, where G is a simply connected domain in R2 with piecewise smooth boundary \partial G=AB\cup AC\cup BC; AB=\{(x,0): 0\leq x\leq 1\}, AC : x=F(y) =\int0y(k(t)/\ell (t))1/2dt 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.
Özet
The equation L(u):=k(y)uxx-\partialy(\ell(y)uy)+r(x,y)u=f(x,y,u), where k(y)>0, \ell(y)>0 for y>0,k(0)=\ell(0)=0 and limy\rightarrow 0k(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\midAC=0, where G is a simply connected domain in R2 with piecewise smooth boundary \partial G=AB\cup AC\cup BC; AB=\{(x,0): 0\leq x\leq 1\}, AC : x=F(y) =\int0y(k(t)/\ell (t))1/2dt 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.