Existence of traveling wave fronts of delayed Fisher-type equations with degenerate nonlinearities
Section snippets
Introduction and main results
In this paper, we focus on the existence of traveling wave fronts of the following two different types of degenerate degree Fisher-type equations with delays and where is a number (no need to be integer).
When , Eqs. (1.1), (1.2) are reduced to which describes some isothermal autocatalytic chemical reactions introduced in [1], [2]. In recent years, the existence
Preliminaries
In this section, we introduce two lemmas about the existence of traveling wave fronts of delayed diffusion equations with different kinds of nonlinearities. Firstly, for convenience, we let , where , and introduce the following wave equation where . Then we give some assumptions on .
(A1) , where is the constant function with value or for all ;
(A2) There exists a positive constant such
Proof of main results
In this section, we firstly apply the existence results given in Section 2 to (1.1), (1.2) to finish the proofs of Theorem 1.1, Theorem 1.2. Now, we begin to prove Theorem 1.1.
Proof of Theorem 1.1 From (1.4), . Obviously, satisfies (A1) and (A2). Then we will prove satisfies (A4). From (i) and (ii) in (A4) If and is small enough, then
Acknowledgments
The authors would like to express their sincere thanks to the referees for the valuable and helpful comments, which led the paper a significant modification. The research by YW was supported in part by the NSFC, China (11901366) and Shanxi Scholarship Council of China (2021-001). The research by MM was supported in part by NSERC, Canada Individual Discovery Grant 354724-2016.
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