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The stated pointwise boundary limits do not imply uniqueness. A counterexample to homogeneous uniqueness isIt is smooth for $x,y>0$. Writing $\eta =x/y$, differentiation givesFor fixed positive $y$, $w\to 0$ as $x\to 0$ and as $x\to \infty$; for fixed positive $x$, it also tends to zero as $y\downarrow 0$. Thus if $u_0$ is one solution, $u_0+Cw$ satisfies the same equation and every printed boundary limit for any constant $C$. Part (d) supplies such a particular solution, so this is a genuine family of distinct solutions, not merely a formal obstruction.

This is corner-source nonuniqueness for a degenerate parabolic problem. The peak $w(y,y)=1/(ey)$ diverges towards the missing corner $(0,0)$; pointwise initial limits at each fixed $x>0$ do not exclude it. The preceding maximum argument fails to force uniqueness because a supremum over an entire past strip need not be attained or finite.

A customary bounded-solution condition restores the intended result. If the difference $z$ of two solutions is bounded on every finite time strip, subtract $\delta (1+x+y)$. Its operator $x{\partial }_x^2-{\partial }_y$ equals $\delta >0$, excluding a positive past maximum; the linear barrier controls spatial infinity, and the homogeneous initial/boundary values control the parabolic boundary. Letting $\delta \downarrow 0$ gives $z\le 0$; applying the same argument to $-z$ gives $z=0$. This proves uniqueness in that additional bounded class, which the printed pointwise limits alone do not specify.