Traveling waves for non-local delayed diffusion equations via auxiliary equations☆

By Shiwang Ma. 2007

171 citations recorded by the source corpus.

In this paper, we study the existence of traveling wave solutions for a class of delayed non-local reaction– diffusion equations without quasi-monotonicity. The approach is based on the construction of two associated auxiliary reaction–diffusion equations with quasi-monotonicity and a profile set in a suitable Banach space by using the traveling wavefronts of the auxiliary equations. Under monostable assumption, by using the Schauder’s fixed point theorem, we then show that there exists a constant c∗ > 0 such that for each c>c ∗, the equation under consideration admits a traveling wavefront solution with speed c, which is not necessary to be monotonic.

Subfields: Applied Mathematics, Numerical Analysis, Public Health, Environmental and Occupational Health

View Paper: https://www.semanticscholar.org/paper/1289266a3c4e790f1de58c8a87f87c5569ab7ea8

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In Applied Mathematics

In Numerical Analysis

In Public Health, Environmental and Occupational Health

Citations in the corpus

Stability of non-monotone critical traveling waves for reaction-diffusion equations with time-delay (2015, background): “For details, we refer to [3,5,7,14,32,33], see also the summary in [13].”
On the existence of traveling waves for delayed reaction–diffusion equations☆ (2009, methodology): “Similar results for different reaction terms can be found in [10,11, 19] and others.”
Traveling Wave Solutions for Delayed Reaction–Diffusion Systems and Applications to Diffusive Lotka–Volterra Competition Models with Distributed Delays (2014, unknown): “[17], Ma [18,19], Ma and Wu [20], Mei [22], Ou and Wu [23], Schaaf [26], Smith and Zhao [29], Thieme and Zhao [31], Wang [33], Wang et al.”
Traveling Fronts in Monostable Equations with Nonlocal Delayed Effects (2008, methodology): “For the case b0 > e, that is, ve−v is nonmonotone on v ∈ [0, ln b0], we refer to [16,17,29].”
Existence and uniqueness of traveling waves for non-monotone integral equations with applications (2010, methodology): “In [15], Ma employed Schauder’s fixed point theorem to prove the existence of traveling waves with speed c > c∗ for the following time-delayed nonlocal reaction–diffusion equation ut(t, z) = Duxx(t, z) − g ( u(t, z) ) + h(u(t, z)) ∫ R f ( u(t − r, y)) J (z − y)dy, t 0, z ∈ R.”
Spreading speeds and travelling waves for non-monotone time-delayed lattice equations (2010, methodology): “Ma (2007) also applied Schauder’s fixed-point theorem to study travelling waves for a time-delayed and non-local reaction–diffusion equation.”