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For a bounded space-time cylinder, consider the linear partial differential operatorwith bounded coefficients, a symmetric uniformly positive-definite principal matrix, and $c\ge 0$. If sufficiently smooth $u,v$ satisfy $Lu\le Lv$ inside and $u\le v$ on the parabolic boundary, the weak parabolic maximum principle applied to $u-v$ gives $u\le v$ throughout. A subsolution can therefore be compared with a supersolution. A lower bound on $c$ can replace nonnegativity after an exponential change in time.

At a first forbidden positive maximum of a strictly perturbed difference, its time derivative is nonnegative, its gradient vanishes, and its Hessian matrix is negative semidefinite. These signs contradict the strict differential inequality. The perturbation and a limit handle equality. Nonlinear comparison requires additional hypotheses rather than following automatically from this linear statement.

As a direct nonlinear example, suppose $u_t=u^2{u}_{xx}+u^3$ has a smooth periodic classical solution and $m={\min }_{x}u(x,0)>0$. The constant-in-space-and-time function $m$ is a strict subsolution, since $0-m^3<0$. A first downward contact at value $m$ would have $u_t\le 0$ and $u_{xx}\ge 0$, contradicting $u_t\ge m^3>0$. Thus $u\ge m$ for the lifetime of the smooth solution; the argument does not promise global existence.