Liouville theorems for superlinear parabolic problems with gradient structure

Abstract

We improve one of the methods for obtaining Liouville theorems for superlinear parabolic problems. In particular, if we consider a positively homogeneous gradient nonlinearity $$F:{{\mathbb {R}}}^m\rightarrow {{\mathbb {R}}}^m$$ of degree $$p>1$$, where $$n>2$$, $$p<n/(n-2)$$ and F satisfies some additional hypotheses, then the method introduced in [Math. Ann. 364 (2016), 269–292] guarantees the nonexistence of positive solutions of the problem $$\begin{aligned} U_t-\varDelta U=F(U), \qquad x\in {{\mathbb {R}}}^n,\ t\in {{\mathbb {R}}}. \end{aligned}$$We show that the condition $$p<n/(n-2)$$ can be weakened to $$p\le n/(n-2)$$: This covers the important case of cubic or quadratic nonlinearities if $$n=3$$ or $$n=4$$, respectively. Our improvement also applies to problems with nonlinear boundary conditions and other problems.