Parabolic comparison principle

A first-contact argument compares a smooth solution with a subsolution or supersolution. At a first spatial minimum of their difference, the Hessian is nonnegative; a strictly favorable time derivative then contradicts contact. For $u_t=u^2{u}_{xx}+u^3$, a positive decreasing constant-in-space barrier proves that positive periodic initial data stay positive for the lifetime of the smooth solution.