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So this PDE is a transport equation given as:

$u_x + xu_y = 0$ with initial condition that $u(x,0) = \exp(x)$.

My approach: Like usual I tried it to interpret this as a direction derivation, and found the curve $C_1$ (which guides me the direction to take for differenciation). Now since RHS is $0$, the value of $u$ of this curve should remain constant, but my calculation shows me the this curve intersect $x$-axis twice at different $x$ and so, the constant value should be $\exp(x_1)$ and $\exp(x_2)$ for $x_1 \neq x_2$. This appears to me as clear contradiction and hence I suspect the initial data. Could someone confirm my suspicion?

P.S If my explanation appears unclear, then try solving the above PDE by using characteristics method.

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    I didn't yet try to solve the PDE. But the problem, you describe can well happen. The solutions to a first order PDE are only defined locally (such that the characteristic lines are invertible). – 2017-01-30

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That is correct. Any solution must be constant on parabolas $x^2 - 2 y = $ constant, and thus the values at $(x,0)$ and $(-x,0)$ must be equal.

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    So that means that the PDE with the initial condition as given does not have a solution (I guess). – 2017-01-30