We first consider positive classical solutions to a semilinear heat equation and discuss the associated Liouville-type theorems and their consequences on a priori estimates of solutions (in particular, blow-up rate estimates). Then we discuss the same issues for several related problems: Sign-changing solutions of the semilinear heat equation, parabolic systems with superlinear gradient nonlinearities (not necessarily homogeneous) and the heat equation with nonlinear boundary conditions.
