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

In this paper, we study some new connections between parabolic Liouvilletype 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 ut−∆u = up on a domain Ω, possibly unbounded and not necessarily convex, we obtain initial and final blow-up rate estimates of the form u(x, t) ≤ C(Ω, 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 ∗Supported in part by NSF Grant DMS-0400702 †Supported in part by VEGA Grant 1/3021/06