A priori growth estimates for nonnegative supertemperatures and solutions of semilinear heat equations in a Lipschitz domain

In a bounded Lipschitz domain, we give a priori growth estimates near the parabolic boundary for a certain class of nonnegative supertemperatures which includes nonnegative continuous solutions of semilinear heat equations of the form ∂ t u ( x , t ) − Δ u ( x , t ) = V ( x , t ) u p ( x , t ) , where V ( x, t ) and p ( x, t ) are nonnegative Borel measurable functions satisfying weak conditions. A growth rate and the range of p depend on the shape of a domain. Our estimates make improvements to a priori estimates given by Bidaut-Véron (1998), Poláčik–Quittner–Souplet (2007) and Taliaferro (2007, 2011). Also, the C 1 -regularity with respect to the spatial variables and a pointwise gradient estimate are shown.