Does a Unique Solution Exist for a PDE with Specific Boundary Conditions?

[t.t.h8701]

If I have a PDE like Ux-Uy=0 and U(x,0)=f(x) when x in [0,1]. Then is there an uniqueness solution exist at point (5,1)?
How can I explain it using characteristics lines?

Thanks

 

[HallsofIvy]

If I have a PDE like Ux-Uy=0 and U(x,0)=f(x) when x in [0,1]. Then is there an uniqueness solution exist at point (5,1)?
How can I explain it using characteristics lines?

Thanks

If you only know f(x) between 0 and 1, you are going to have a problem extending the solution to x= 5!

The "characteristic lines" are of the form x+ y= C and any solution to this equation is of the form F(x+y) where F is an arbitrary function of one variable. Since you require that U(x,0)= F(x+0)= F(x)= f(x) for x between 0 and 1, you will need to take F(x) to be f(x) between 0 and 1 but that does not define it for x+ y= 5+ 1= 6. Consider any number of functions "f(x)" which are identical between 0 and 1 but differ outside that interval.