 Solution

Let $\mathcal{P}u=u_t-\Delta u+a{u}_x+b{u}_y$ on the unit disk over $0<t\le 1$. Smoothness is understood to include continuity on the closed cylinder. If $u$ exceeded its nonnegative parabolic boundary maximum $A$, it would attain a positive global maximum $M>A$ at some spatial interior point at time $t_0>0$. There,At $t_0=1$ the last inequality uses the derivative from the past; at an interior time it is equality. These signs imply $\mathcal{P}u+u^3\ge M^3>0$, contradicting the given differential inequality. ThusIf $A>0$, equality at any point of $Q$ gives the same contradiction with $M=A$, so $u<A$ throughout $Q$. If $A=0$, equality is possible: $u\equiv 0$ is an example. More precisely, write the inequality as $\mathcal{P}u+cu\le 0$ with $c=u^2\ge 0$. The strong parabolic maximum principle shows that an interior zero at time $t_0>0$ forces $u\equiv 0$ on $B\times [0,t_0]$.

That conclusion need not extend into the future for a subsolution. For $t_0<1$, take a spatially constant function which is zero for $t\le t_0$ and equals $-\exp (-1/(t-t_0))$ for $t>t_0$. It is smooth, nonincreasing and nonpositive, so $\mathcal{P}u+u^3\le 0$, but it becomes negative after $t_0$.