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Dynamics of Partial Differential Equations
Volume 18 (2021)
Number 4
An invasive-invaded species dynamics with a high order diffusion operator
Pages: 257 – 278
DOI: https://dx.doi.org/10.4310/DPDE.2021.v18.n4.a1
Author
Abstract
The introduction of the Landau–Ginzburg free energy provides a framework to generalize the diffusion beyond the classical fickian approach. The analysis shows the existence and uniqueness of solutions with a priori bounds and making use of the Fixed Point Theorem to a suitable abstract evolution. Asymptotic solutions are provided with the Hamilton–Jacobi operator and a positivity condition is formulated based on an asymptotic positive kernel. Further, the positive region is characterized and a precise assessment is provided. Afterwards, the problem is analyzed in the Travelling Waves domain to show the phenomena of waves synchronization and to provide linear manifolds in the proximity of the critical points. Finally, numerical TW profiles are obtained and the amplitude of a positive region in the TW domain is provided as a function of the TW‑speed.
Keywords
high order diffusion, Fisher-KPP problem, instabilities, existence, geometric perturbation theory, asymptotic, positivity
2010 Mathematics Subject Classification
35K55, 35K91, 35K92
Received 14 April 2021
Published 2 December 2021