Existence of Generalized Traveling Waves in Time Recurrent and Space Periodic Monostable Equations

By Wenxian Shen. 2011

61 citations recorded by the source corpus.

This paper is concerned with the extension of the concepts and theories of traveling wave solutions of time and space periodic monostable equations to time recurrent and space periodic ones. It first introduces the concept of generalized traveling wave solutions of time recurrent and space periodic monostable equations, which extends the concept of periodic traveling wave solutions of time and space periodic monostable equations to time recurrent and space periodic ones. It then proves that in the direction of any unit vector $\xi$, there is $c^*(\xi)$ such that for any $c>c^*(\xi)$, a generalized traveling wave solution in the direction of $\xi$ with averaged propagation speed $c$ exists. It also proves that if the time recurrent and space periodic monostable equation is indeed time periodic, then $c^*(\xi)$ is the minimal wave speed in the direction of $\xi$ and the generalized traveling wave solution in the direction of $\xi$ with averaged speed $c>c^*(\xi)$ is a periodic traveling wave solution with speed $c$, which recovers the existing results on the existence of periodic traveling wave solutions in the direction of $\xi$ with speed greater than the minimal speed in that direction.

Subfields: Applied Mathematics, Modeling and Simulation, Public Health, Environmental and Occupational Health

View Paper: https://www.semanticscholar.org/paper/a0a87e2d0dc255799253ee496896e4a9401f564e

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In Applied Mathematics

In Modeling and Simulation

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Citations in the corpus

Traveling Wave Solutions of Spatially Periodic Nonlocal Monostable Equations (2012, background): “he evolutionary take-over of a habitat by a ﬁtter genotype. See, for example, [1], [3], [4], [5], [16], [18], [23], [27], [28], [33], [37], [38], [39], [40], [42], [43], [44], [47], [48], [49], [51], [52], [56], [57], [58], and references therein, for the study of the spatial spread and front propag”
Spreading Speeds and Traveling Waves of Nonlocal Monostable Equations in Time and Space Periodic Habitats (2013, background): “The reader is referred to [25] and [26] for spreading properties and existence of traveling waves in homogeneous and periodic systems, respectively, and to [21], [28], [33], [34] for the extensions of the above results to the cases that f(t, x, u) is almost periodic in t and periodic in x and that f”
Transition waves for Fisher-KPP equations with general time-heterogeneous and space-periodic coefficients (2015, methodology): “When the coeﬃcients only depend on t in a general way, the existence of transition waves was ﬁrst proved by Shen for bistable nonlinearities [25] (that is, nonlinearities vanishing at u = 0 and u = 1 but negative near these two equilibria) and for monostable equations with time uniquely ergodic coeﬃ”
Global existence and asymptotic behavior of classical solutions to a parabolic-elliptic chemotaxis system with logistic source on $\mathbb{R}^{N}$ (2016, unknown): “24) where f(t, x, u) < 0 for u ≫ 1, ∂uf(t, x, u) < 0 for u ≥ 0 (see [3, 4, 5, 6, 10, 11, 25, 26, 27, 29, 30, 33, 34, 41, 42, 53], etc.”