Optimal Liouville theorems for superlinear parabolic problems

Abstract

Liouville theorems for scaling invariant nonlinear parabolic equations and systems (saying that the equation or system does not possess positive entire solutions) guarantee optimal universal estimates of solutions of related initial and initial-boundary value problems. In the case of the nonlinear heat equation

$${\mathit{u}}_{\mathit{t}}-\mathrm{\Delta }\mathit{u}={\mathit{u}}^{\mathit{p}}\text{in}{\mathbb{R}}^{\mathit{n}}\times \mathbb{R},\mathit{p}\mathrm{>}1,$$

the nonexistence of positive classical solutions in the subcritical range $\mathit{p}(\mathit{n}-2)\mathrm{<}\mathit{n}+2$ has been conjectured for a long time, but all known results require either a more restrictive assumption on p or deal with a special class of solutions (time-independent or radially symmetric or satisfying suitable decay conditions). We solve this open problem and—by using the same arguments—we also prove optimal Liouville theorems for a class of superlinear parabolic systems. In the case of the nonlinear heat equation, straightforward applications of our Liouville theorem solve several related long-standing problems. For example, they guarantee an optimal Liouville theorem for ancient solutions, optimal decay estimates for global solutions of the corresponding Cauchy problem, optimal blowup rate estimate for solutions in nonconvex domains, and optimal universal estimates for solutions of the corresponding initial-boundary value problems. The proof of our main result is based on refined energy estimates for suitably rescaled solutions.