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Singularity and decay estimates in superlinear problems via Liouville-type theorems. II: Parabolic equations. (English) Zbl 1122.35051

Summary: 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 \[ u(x,t)\leq C(\Omega,p)(1 + t^{-1/(p-i)} + (T-t)^{-1/(p-1)}). \] 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 halfspace. As an application of our universal estimates, we prove a nonuniqueness result for an initial boundary value problem.
[For part I, see Duke Math. J. 139, No. 3, 555–579 (2007; Zbl 1146.35038).]

MSC:

35K55 Nonlinear parabolic equations
35K15 Initial value problems for second-order parabolic equations
35K20 Initial-boundary value problems for second-order parabolic equations
35B45 A priori estimates in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs

Citations:

Zbl 1146.35038
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