A Liouville property for eternal solutions to a supercritical semilinear heat equation

Abstract

We are concerned with solutions to the nonlinear heat equation $u_t=\Delta u+|u|^{p-1}u$, $x\in \mathbb{R}^N$, that are defined for all positive and negative time. If the exponent $p$ is greater or equal to the Joseph-Lundgren exponent $p_c$ and $|u|$ stays below some positive radially symmetric steady state, under a mild condition on the behaviour of $u$ as $|x|\to \infty$, we show that $u$ is independent of time. Our method of proof uses Serrin's sweeping principle, based on the strong maximum principle, applied to the linearized equation for $u_t$. Our result covers that of Pol\'{a}\v{c}ik and Yanagida [JDE (2005)] who had further assumed that the solution stays above some positive radial steady state and $p>p_c$. In contrast, they relied on the use of similarity variables and invariant manifold ideas. Remarkably, to the best of our knowledge, a corresponding Liouville property was previously missing for $p =p_c$. We emphasize that such Liouville type theorems imply the quasiconvergence of a class of solutions to the corresponding Cauchy problem. As our viewpoint originates from the study of elliptic problems, we can prove new rigidity results for the corresponding steady state problem that are motivated by the aforementioned ones for the parabolic flow.