Wavefront’s stability with asymptotic phase in the delayed monostable equations

Solar, Abraham; Trofimchuk, Sergei

doi:10.1090/proc/15988

Wavefront’s stability with asymptotic phase in the delayed monostable equations
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by Abraham Solar and Sergei Trofimchuk PDF

Proc. Amer. Math. Soc. 150 (2022), 4349-4358 Request permission

Abstract:

We extend the class of initial conditions for scalar delayed reaction-diffusion equations $u_t (t,x)=u_{xx}(t,x)+f(u(t, x), u(t-h, x))$ which evolve in solutions converging to monostable traveling waves. Our approach allows to compute, in the moving reference frame, the phase distortion $\alpha$ of the limiting travelling wave with respect to the position of solution at the initial moment $t=0$. In general, $\alpha \not =0$ for the Mackey-Glass type diffusive equation. Nevertheless, $\alpha =0$ for the KPP-Fisher delayed equation: the related theorem also improves existing stability conditions for this model.

References

Richard Bellman and Kenneth L. Cooke, Differential-difference equations, Academic Press, New York-London, 1963. MR 0147745
Rafael D. Benguria and Abraham Solar, An iterative estimation for disturbances of semi-wavefronts to the delayed Fisher-KPP equation, Proc. Amer. Math. Soc. 147 (2019), no. 6, 2495–2501. MR 3951427, DOI 10.1090/proc/14381
I-Liang Chern, Ming Mei, Xiongfeng Yang, and Qifeng Zhang, Stability of non-monotone critical traveling waves for reaction-diffusion equations with time-delay, J. Differential Equations 259 (2015), no. 4, 1503–1541. MR 3345859, DOI 10.1016/j.jde.2015.03.003
Arnaud Ducrot and Grégoire Nadin, Asymptotic behaviour of travelling waves for the delayed Fisher-KPP equation, J. Differential Equations 256 (2014), no. 9, 3115–3140. MR 3171769, DOI 10.1016/j.jde.2014.01.033
Jian Fang and Xiao-Qiang Zhao, Monotone wavefronts of the nonlocal Fisher-KPP equation, Nonlinearity 24 (2011), no. 11, 3043–3054. MR 2844826, DOI 10.1088/0951-7715/24/11/002
Teresa Faria, Wenzhang Huang, and Jianhong Wu, Travelling waves for delayed reaction-diffusion equations with global response, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 462 (2006), no. 2065, 229–261. MR 2189262, DOI 10.1098/rspa.2005.1554
Avner Friedman, Partial differential equations of parabolic type, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. MR 0181836
Karel Hasík, Jana Kopfová, Petra Nábělková, and Sergei Trofimchuk, On pushed wavefronts of monostable equation with unimodal delayed reaction, Discrete Contin. Dyn. Syst. 41 (2021), no. 12, 5979–6000. MR 4342144, DOI 10.3934/dcds.2021103
Karel Hasik and Sergei Trofimchuk, Slowly oscillating wavefronts of the KPP-Fisher delayed equation, Discrete Contin. Dyn. Syst. 34 (2014), no. 9, 3511–3533. MR 3190991, DOI 10.3934/dcds.2014.34.3511
Rui Huang, Chunhua Jin, Ming Mei, and Jingxue Yin, Existence and stability of traveling waves for degenerate reaction-diffusion equation with time delay, J. Nonlinear Sci. 28 (2018), no. 3, 1011–1042. MR 3800254, DOI 10.1007/s00332-017-9439-5
A. Kolmogorov, I. Petrovskii, and N. Piskunov, Study of a diffusion equation that is related to the growth of a quality of matter and its application to a biological problem, Byul. Mosk. Gos. Univ. Ser. A Mat. Mekh. 1 (1937), 1–26.
Guangying Lv and Mingxin Wang, Nonlinear stability of travelling wave fronts for delayed reaction diffusion equations, Nonlinearity 23 (2010), no. 4, 845–873. MR 2602017, DOI 10.1088/0951-7715/23/4/005
Abraham Solar and Sergei Trofimchuk, Speed selection and stability of wavefronts for delayed monostable reaction-diffusion equations, J. Dynam. Differential Equations 28 (2016), no. 3-4, 1265–1292. MR 3537371, DOI 10.1007/s10884-015-9482-6
Abraham Solar and Sergei Trofimchuk, Asymptotic convergence to pushed wavefronts in a monostable equation with delayed reaction, Nonlinearity 28 (2015), no. 7, 2027–2052. MR 3366635, DOI 10.1088/0951-7715/28/7/2027
K\B{o}hei Uchiyama, The behavior of solutions of some nonlinear diffusion equations for large time, J. Math. Kyoto Univ. 18 (1978), no. 3, 453–508. MR 509494, DOI 10.1215/kjm/1250522506
Zhi-Cheng Wang, Wan-Tong Li, and Shigui Ruan, Traveling fronts in monostable equations with nonlocal delayed effects, J. Dynam. Differential Equations 20 (2008), no. 3, 573–607. MR 2429437, DOI 10.1007/s10884-008-9103-8