 Solution

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