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!