Existence of traveling wave fronts of delayed Fisher-type equations with degenerate nonlinearities

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Abstract

In this paper, two different kinds of degenerate n-degree Fisher-type equations with delays are considered. Due to the difference of the reaction terms, the existence of traveling front are proved by different methods. More precisely, when the reaction term satisfies the weak quasimonotonicity condition, for c>2, the existence result is given by the super-sub solution method and the fixed point theorem. Then for c<c2, where c is the minimal speed of degenerate p-degree Fisher-type equations without delays, the existence result is proved by the perturbation method and the implicit function theory. For the other type reaction term, we apply the monotone iteration method and the super-sub solution method to obtain the existence conclusion.

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 pdegree Fisher-type equations with delays u(x,t)t=2u(x,t)x2+up(x,t)(1u(x,tτ)),and u(x,t)t=2u(x,t)x2+up(x,tτ)(1u(x,t)),where p>1 is a number (no need to be integer).

When τ=0, Eqs. (1.1), (1.2) are reduced to u(x,t)t=2u(x,t)x2+up(x,t)(1u(x,t)),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 ϕ(s)ϕ(ξ)(s)=ϕ(ξ+s), where s[cτ,0], and introduce the following wave equation ϕ(ξ)cϕ(ξ)+f(ϕ(ξ)(0),ϕ(ξ)(s))=0,where s[cτ,0]. Then we give some assumptions on f.

(A1) f(0,0̃)=f(1,1̃)=0, where ϕ̃:[cτ,0]R is the constant function with value 0 or 1 for all ξR;

(A2) There exists a positive constant L 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), f(ϕ(0),ϕ(s))=ϕp(0)(1ϕ(cτ)). Obviously, f satisfies (A1) and (A2). Then we will prove f satisfies (A4). From (i) and (ii) in (A4) f(ϕ(0),ϕ(s))f(ψ(0),ψ(s))=ϕp(0)(1ϕ(cτ))ψp(0)(1ψ(cτ))=(1ϕ(cτ))(ϕp(0)ψp(0))+ψp(0)(ψ(cτ)ϕ(cτ))eβcτ(ϕ(0)ψ(0)). If β>1 and τ is small enough, then f(ϕ(0),ϕ(s))f(ψ(0),ψ(s))

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.

References (24)

  • W.J. Bo, G. Lin, Y.W. Qi, The role of delay and degeneracy on propagation dynamics in diffusion equations. J. Dyn....
  • BuZ.H. et al.

    Multidimensional stability of traveling fronts in combustion and non-KPP monostable equations

    Z. Angew. Math. Phys.

    (2018)
  • Cited by (0)

    View full text