= Solution
At such an interior backward-in-time maximum, spatial differentiation gives $u_x=0$ and $u_{xx}\leq0$. Comparison with earlier times at the same spatial point gives the one-sided time derivative $u_y\geq0$; differentiability makes this the usual derivative. Consequently $xu_{xx}-u_y\leq0$, contradicting the strictly positive source $xy^{-2}e^{-x/y}$. Therefore \b[no such maximum can occur]. The temporal one-sided argument matters because the condition concerns only the past of the candidate point.
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