scholarly journals Existence and Asymptotic Behavior of Traveling Wave Fronts for a Time-Delayed Degenerate Diffusion Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-20
Author(s):  
Weifang Yan ◽  
Rui Liu
Keyword(s):  
Asymptotic Behavior ◽  
Time Delay ◽  
Diffusion Equation ◽  
Traveling Wave ◽  
Sufficient Conditions ◽  
Degenerate Diffusion ◽  
Wave Fronts ◽  
Traveling Wave Fronts ◽  
Necessary And Sufficient ◽  
Degenerate Diffusion Equation

This paper is concerned with traveling wave fronts for a degenerate diffusion equation with time delay. We first establish the necessary and sufficient conditions to the existence of monotone increasing and decreasing traveling wave fronts, respectively. Moreover, special attention is paid to the asymptotic behavior of traveling wave fronts connecting two uniform steady states. Some previous results are extended.

scholarly journals Some Qualitative Properties of Traveling Wave Fronts of Nonlocal Diffusive Competition-Cooperation Systems of Three Species with Delays

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-19
Author(s):  
Meiping Yao ◽  
Pengzhi Qiao ◽  
Yang Wang
Keyword(s):  
Asymptotic Behavior ◽  
Traveling Wave ◽  
Nonlocal Diffusion ◽  
Strict Monotonicity ◽  
Qualitative Properties ◽  
Wave Fronts ◽  
Traveling Wave Fronts ◽  
Sliding Method ◽  
Diffusion Systems ◽  
Stable Manifold Theorem

This paper is concerned with nonlocal diffusion systems of three species with delays. By modified version of Ikehara’s theorem, we prove that the traveling wave fronts of such system decay exponentially at negative infinity, and one component of such solutions also decays exponentially at positive infinity. In order to obtain more information of the asymptotic behavior of such solutions at positive infinity, for the special kernels, we discuss the asymptotic behavior of such solutions of such system without delays, via the stable manifold theorem. In addition, by using the sliding method, the strict monotonicity and uniqueness of traveling wave fronts are also obtained.

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