Title: Singularity and decay estimates in superlinear problems via Liouville-type theorems. Part II: Parabolic equations

Authors: Peter Polacik, Pavol Quittner and Philippe Souplet

Issue: Volume 56 (2007), Issue 2, 879-908

Abstract: In this paper, we study some new connections between parabolic Liouville-type theorems and local and global properties of nonnegative classical solutions to superlinear parabolic problems, with or without boundary conditions. Namely, we develop a general method for derivation of universal, pointwise a~priori estimates of solutions from Liouville-type theorems, which unifies and improves many results concerning a priori bounds, decay estimates and initial and final blow-up rates. For example, for the equation $u_t-\Delta u=u^p$ on a domain $\Omega$, possibly unbounded and not necessarily convex, we obtain initial and final blow-up rate estimates of the form $u(x,t)\leq C(\Omega,p)(1+t^{-1/(p-1)}+(T-t)^{-1/(p-1)})$. Our method is based on rescaling arguments combined with a key "doubling" property, and it is facilitated by parabolic Liouville-type theorems for the whole space or the half-space. As an application of our universal estimates, we prove a nonuniqueness result for an initial boundary value problem.