Universal Bounds at the Blow-Up Time for Nonlinear Parabolic Equations

We prove a priori supremum bounds for solutions to doubly degenerate nonlinear parabolic equations, with a forcing term f(x)u^p where u is the solution, p> 1, f is strongly dependent on the space variable x, as t approaches the time when u becomes unbounded. Such bounds are universal in the sense that they do not depend on u. Here f may become unbounded, or vanish, as x goes to 0. When f =1, we also prove a bound below, as well as uniform localization of the support, for subsolutions to the corresponding Cauchy problem.