Singularity and decay estimates in superlinear problems via Liouville-type theorems. Part II: Parabolic equations Peter PolacikPavol QuittnerPhilippe Souplet 35K5535K1535K2035B4535B40semilinear parabolic equationssingularity and decay estimatesLiouville theoremsblow-up ratedecay ratedoubling lemma 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. Indiana University Mathematics Journal 2007 text pdf 10.1512/iumj.2007.56.2911 10.1512/iumj.2007.56.2911 en Indiana Univ. Math. J. 56 (2007) 879 - 908 state-of-the-art mathematics http://iumj.org/access/