Solving u_y + a(x,y) u_x = 0 for function a(x,y) that yields no solution

[lucqui]

Hi. I am taking a PDE class and struggling a little. The professor gave us this problem to practice:

Find a function a = a(x,y) continuous such that for the equation $u_y + a(x,y) u_x = 0 $ there does not exist a solution in all of $R^2$ for any non-constant initial value defined on the hyperplane ${(x,0)}$

I thought that in order to find such a function I needed to make the hyperplane characteristic at all points, but that got me nowhere. Please help me find a direction. Any help would be much appreciated!

Thanks in advance!

 

[gtoguy97]

One approach is to find a function a(x,y) such that the equation becomes inhomogeneous. That is, you need to choose a function a(x,y) such that the equation $u_y + a(x,y) u_x = f(x,y)$ has no solution in all of $R^2$ for any non-constant initial value defined on the hyperplane ${(x,0)}$. In other words, you need to find a function a(x,y) such that, for some function $f(x,y)$, the equation $u_y + a(x,y) u_x = f(x,y)$ cannot be solved for any given non-constant initial value. A possible example of such a function is $a(x,y)=e^{xy}$.