2019/09/01 by Christos Sourdis, Sourdis, Christos
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.1909.00498
openalex publication_date 2019/09/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We are concerned with solutions to the nonlinear heat equation ut=\Δν+|u|p-1u, x\∈ \ℝN, that are defined for all positive and\nnegative time. If the exponent p is greater or equal to the Joseph-Lundgren\nexponent pc and |u| stays below some positive radially symmetric steady\nstate, under a mild condition on the behaviour of u as |x|\→ \∞, we\nshow that u is independent of time. Our method of proof uses Serrin's\nsweeping principle, based on the strong maximum principle, applied to the\nlinearized equation for ut. Our result covers that of Pol 'a vcik and\nYanagida [JDE (2005)] who had further assumed that the solution stays above\nsome positive radial steady state and p>pc. In contrast, they relied on the\nuse of similarity variables and invariant manifold ideas. Remarkably, to the\nbest of our knowledge, a corresponding Liouville property was previously\nmissing for p =pc. We emphasize that such Liouville type theorems imply the\nquasiconvergence of a class of solutions to the corresponding Cauchy problem.\nAs our viewpoint originates from the study of elliptic problems, we can prove\nnew rigidity results for the corresponding steady state problem that are\nmotivated by the aforementioned ones for the parabolic flow.\n