Strong parabolic maximum principle

Let $\Omega$ be a connected domain and $u$ a classical subsolution ofwith uniformly positive definite diffusion matrix, continuous bounded coefficients, and $c\ge 0$. If $u\le 0$ 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 $w=-u\ge 0$, 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 $u_t-\Delta u+b\cdot \nabla u+u^3\le 0$, use $c=u^2\ge 0$ after the weak maximum principle has established $u\le 0$.