= Solution
The <parabolic boundary> values are zero. An exact solution satisfies the preceding <differential inequality>, so its maximum estimate with $A=0$ gives $u\leq0$. The function $v=-u$ satisfies the same equation because both the linear differential operator and the cubic term change sign. Applying the same <maximum principle> to $v$ gives $-u\leq0$. Hence the complete solution set is
$$
\boxed{\{u\equiv0\}.}
$$
Substitution verifies existence, and the two inequalities prove uniqueness among the assumed smooth solutions for every continuous choice of the drift coefficients.
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