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


José Palencia (Escuela Politécnica Superior, Universidad Francisco de Vitoria, Madrid, Spain)


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.


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