2013/07/31 by Franz Achleitner, Christian Kuehn
Biochemistry, Genetics and Molecular Biology · Engineering · Mathematics · Medicine · Physics and Astronomy · #Bounded function #Domain (mathematical analysis) #Invariant (physics) #Mathematical Biology Tumor Growth #Mathematical analysis #Mathematical and Theoretical Epidemiology and Ecology Models #Mathematical physics #Mathematics #Nonlinear system #Perturbation (astronomy) #Physics #Pure mathematics #Quantum #Quantum entanglement #Quantum mechanics #Quantum nonlocality #Sobolev space #Stability and Controllability of Differential Equations #math.AP #math.DS #nlin.PS #q-bio.QM
paper · pdf · doi:10.1016/j.na.2014.09.004
published as Nonlinear Analysis A: Theory, Methods & Applications, Vol. 112, pp. 15-29, 2015 · 24 pages, 1 figure; revised version
arxiv created 2014/08/09 · openalex publication_date 2014/09/21 · arxiv updated 2015/03/06 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
We study the existence of stationary solutions for a nonlocal version of the Fisher-Kolmogorov-Petrovskii-Piscounov (FKPP) equation. The main motivation is a recent study by Berestycki et al. [Nonlinearity 22 (2009), pp.~2813--2844] where the nonlocal FKPP equation has been studied and it was shown for the spatial domain ℝ andsufficiently small nonlocality that there are only two bounded non-negative stationary solutions. Here we provide a similar result for ℝd using a completely different approach. In particular, an abstract perturbation argument is used in suitable weighted Sobolev spaces. One aim of the alternative strategy is that it can eventually be generalized to obtain persistence results for hyperbolic invariant sets for other nonlocal evolution equations on unbounded domains with small nonlocality, i.e., to improve our understanding in applications when a small nonlocal influence alters the dynamics and when it does not.