Optimal Liouville theorems for superlinear parabolic problems

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 related initial initial-boundary value problems. In case heat ut??u=upinRn×R,p>1, nonexistence classical in subcritical range p(n?2)<n+2 has been conjectured a long time, but all known results require either more restrictive assumption on p deal with special class (time-independent radially symmetric satisfying suitable decay conditions). We solve this open problem and—by using same arguments—we also prove superlinear systems. equation, straightforward applications our theorem several long-standing For example, they an ancient solutions, global corresponding Cauchy problem, blowup rate estimate nonconvex domains, The proof main result is based refined energy suitably rescaled solutions.