Weak parabolic maximum principle

For a classical solution of $u_t-\Delta u+cu=0$ with $c\ge 0$, the positive part of its maximum is bounded by the corresponding maximum on the initial and lateral parabolic boundary. When $c>0$, a positive maximum in space at an interior or final time gives $u_t\ge 0$, $\Delta u\le 0$, and $u_t-\Delta u+cu>0$, a contradiction. For $c=0$, a small strictly increasing exponential time perturbation gives the weak inequality by a limiting argument. Applied to both signs of the difference of two solutions, the principle proves uniqueness for fixed initial and boundary data.

On $x,t>0$, the displayed smooth nonzero function satisfies $w_t=x{w}_{xx}$. For each fixed $t>0$ it tends to zero at $x=0$ and at spatial infinity; for each fixed $x>0$ it tends to zero as $t\downarrow 0$. Thus these three pointwise limits do not determine a unique solution: a multiple of $w$ may be added to any particular solution of the same forced equation.

To verify the PDE, set $\eta =x/t$. Both $w_t$ and $x{w}_{xx}$ equal $\eta (\eta -2)e^{-\eta }/t^2$. The missing control is at the corner: $w(t,t)=1/(et)$ diverges as $t\downarrow 0$. There is no contradiction with the weak parabolic maximum principle, because a supremum on a past strip may be infinite or unattained. A bounded-solution class excludes this example. A standard uniqueness proof there applies the maximum principle to the difference minus $\delta (1+x+t)$ and lets $\delta \downarrow 0$. Conditions at infinity and at initial time must control the relevant strip, not only each fixed-variable limit.