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I'm trying to figure out this problem:

Find a smooth function $a(x, y)$ in $\mathbb{R}^2$ such that, for the equation of the form $$u_y + a(x, y)u_x = 0,$$ there does not exist any solution in the entire $\mathbb{R}^2$ for any nonconstant initial value prescribed on $\{y = 0\}$.

Solving the PDE I think I should get $$x_0 = x - \int{a(x(s),s)ds}$$ and $$u(x,y) = u_0(x_0)$$ but I'm not seeing how to use this to find such a function.

TrivialCase
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    I'm not convinced the statement is true... since $a$ is smooth, characteristic curves cannot intersect by the uniqueness for ODEs. They can certainly run off into infinity in finite time, but this is more of an issue for uniqueness than existence. Consider the initial values $u_0$ of compact support. The characteristic curves emanating from the points of $\operatorname{supp} u_0$ sweep out some part of the plane and determine what $u$ should be there. In the rest of the plane, set $u=0$. Why wouldn't this be an entire solution? –  Jan 07 '17 at 05:40
  • Does "any nonconstant initial value prescribed on ${y=0}$" mean that $u(x,0)$ is necessarily a nonconstant function of $x$? – dafinguzman Jan 28 '17 at 14:44

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