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 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 - Δu = u^p 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 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