Linear and nonlinear speed selection for mono-stable wave propagations. (English) Zbl 1407.35114
Authors’ abstract: We study the selection mechanism of the minimal wave speed for traveling waves to an abstract monotone semi-flow. A necessary and sufficient condition for the nonlinear selection of the minimal wave speed is established. Based on this result, we then derive conditions under which the linear or nonlinear selection is realized by way of a comparison principle. Our results on nonlinear selection are new and novel, and they can be viewed as breakthroughs in this topic; and for the linear selection, we successfully improve previous conventional results that always require that the monotone semi-flow is dominated by its linear map. The applications to various biological models are also successful. We establish a series of new results to reaction-diffusion models with delay interactions, a lattice system, a scalar integro-difference equation, and a cooperative system, which completely solve some open problems and conjectures in the related references.
Reviewer: E. Ahmed (Mansoura)
MSC:
| 35K57 | Reaction-diffusion equations |
| 35B20 | Perturbations in context of PDEs |
| 35C07 | Traveling wave solutions |
| 35K15 | Initial value problems for second-order parabolic equations |
| 35K91 | Semilinear parabolic equations with Laplacian, bi-Laplacian or poly-Laplacian |
| 92D25 | Population dynamics (general) |
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