= Strong parabolic maximum principle
Let $\Omega$ be a connected domain and $u$ a classical subsolution of
$$
u_t-\sum_{i,j}a_{ij}(x,t)u_{x_ix_j}+\sum_i b_i(x,t)u_{x_i}+c(x,t)u\leq0,
$$
with uniformly positive definite diffusion matrix, continuous bounded coefficients, and $c\geq0$. If $u\leq0$ on $\Omega\times(0,T]$ and $u(x_0,t_0)=0$ at a spatial interior point with $t_0>0$, then $u=0$ on $\Omega\times(0,t_0]$. Continuity supplies the same conclusion at initial and lateral points where the solution extends continuously. There is generally no conclusion about times after $t_0$ for a subsolution.
One proof applies the strict positivity principle to $w=-u\geq0$, a supersolution: any positive value at an earlier interior point propagates to positive values at every later interior point along a chain of overlapping interior balls, contradicting $w(x_0,t_0)=0$. Nonnegative zeroth-order terms can be bounded on each compact time cylinder and removed from this positivity argument by an exponential multiplier. For the inequality $u_t-\Delta u+b\cdot\nabla u+u^3\leq0$, use $c=u^2\geq0$ after the weak <maximum principle> has established $u\leq0$.
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