2021/11/30 by Gabriele Grillo, Grillo, Gabriele, Giulia Meglioli +3 · 1 citation
Computer Science · Engineering · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Nonlinear Partial Differential Equations #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.2112.00125
openalex publication_date 2021/11/30 · openalex created_date 2022/05/05 · openalex updated_date 2026/07/28
It is well known from the work of [2] that the Fujita phenomenon for reaction-diffusion evolution equations with power nonlinearities does not occur on the hyperbolic space ℍN, thus marking a striking difference with the Euclidean situation. We show that, on classes of manifolds in which the bottom Λ of the L2 spectrum of -Δ is strictly positive (the hyperbolic space being thus included), a different version of the Fujita phenomenon occurs for other kinds of nonlinearities, in which the role of the critical Fujita exponent in the Euclidean case is taken by Λ. Such nonlinearities are time-independent, in contrast to the ones studied in [2]. As a consequence of our results we show that, on a class of manifolds much larger than the case M=ℍN considered in [2], solutions to (1.1) with power nonlinearity f(u)=up, p>1, and corresponding to sufficiently small data, are global in time.