Stability, uniqueness and recurrence of generalized traveling waves in time heterogeneous media of ignition type

Shen, Wenxian; Shen, Zhongwei

doi:10.1090/tran/6726

Stability, uniqueness and recurrence of generalized traveling waves in time heterogeneous media of ignition type
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by Wenxian Shen and Zhongwei Shen PDF

Trans. Amer. Math. Soc. 369 (2017), 2573-2613 Request permission

Abstract:

The present paper is devoted to the study of stability, uniqueness and recurrence of generalized traveling waves of reaction-diffusion equations in time heterogeneous media of ignition type, whose existence has been proven by the authors of the present paper in a previous work. It is first shown that generalized traveling waves exponentially attract wave-like initial data. Next, properties of generalized traveling waves, such as space monotonicity and exponential decay ahead of interface, are obtained. Uniqueness up to space translations of generalized traveling waves is then proven. Finally, it is shown that the wave profile and the front propagation velocity of the unique generalized traveling wave are of the same recurrence as the media. In particular, if the media is time almost periodic, then so are the wave profile and the front propagation velocity of the unique generalized traveling wave.

References

Nicholas D. Alikakos, Peter W. Bates, and Xinfu Chen, Periodic traveling waves and locating oscillating patterns in multidimensional domains, Trans. Amer. Math. Soc. 351 (1999), no. 7, 2777–2805. MR 1467460, DOI 10.1090/S0002-9947-99-02134-0
Sigurd Angenent, The zero set of a solution of a parabolic equation, J. Reine Angew. Math. 390 (1988), 79–96. MR 953678, DOI 10.1515/crll.1988.390.79
D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, Partial differential equations and related topics (Program, Tulane Univ., New Orleans, La., 1974) Lecture Notes in Math., Vol. 446, Springer, Berlin, 1975, pp. 5–49. MR 0427837
D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. in Math. 30 (1978), no. 1, 33–76. MR 511740, DOI 10.1016/0001-8708(78)90130-5
Henri Berestycki and François Hamel, Front propagation in periodic excitable media, Comm. Pure Appl. Math. 55 (2002), no. 8, 949–1032. MR 1900178, DOI 10.1002/cpa.3022
Henri Berestycki and François Hamel, Generalized travelling waves for reaction-diffusion equations, Perspectives in nonlinear partial differential equations, Contemp. Math., vol. 446, Amer. Math. Soc., Providence, RI, 2007, pp. 101–123. MR 2373726, DOI 10.1090/conm/446/08627
Henri Berestycki and François Hamel, Generalized transition waves and their properties, Comm. Pure Appl. Math. 65 (2012), no. 5, 592–648. MR 2898886, DOI 10.1002/cpa.21389
H. Berestycki, B. Larrouturou, and P.-L. Lions, Multi-dimensional travelling-wave solutions of a flame propagation model, Arch. Rational Mech. Anal. 111 (1990), no. 1, 33–49. MR 1051478, DOI 10.1007/BF00375699
H. Berestycki and L. Nirenberg, On the method of moving planes and the sliding method, Bol. Soc. Brasil. Mat. (N.S.) 22 (1991), no. 1, 1–37. MR 1159383, DOI 10.1007/BF01244896
Henri Berestycki, Basil Nicolaenko, and Bruno Scheurer, Traveling wave solutions to combustion models and their singular limits, SIAM J. Math. Anal. 16 (1985), no. 6, 1207–1242. MR 807905, DOI 10.1137/0516088
Peter W. Bates and Adam Chmaj, A discrete convolution model for phase transitions, Arch. Ration. Mech. Anal. 150 (1999), no. 4, 281–305. MR 1741258, DOI 10.1007/s002050050189
Fengxin Chen, Stability and uniqueness of traveling waves for system of nonlocal evolution equations with bistable nonlinearity, Discrete Contin. Dyn. Syst. 24 (2009), no. 3, 659–673. MR 2505676, DOI 10.3934/dcds.2009.24.659
Xinfu Chen, Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations, Adv. Differential Equations 2 (1997), no. 1, 125–160. MR 1424765
Xinfu Chen and Jong-Shenq Guo, Existence and asymptotic stability of traveling waves of discrete quasilinear monostable equations, J. Differential Equations 184 (2002), no. 2, 549–569. MR 1929888, DOI 10.1006/jdeq.2001.4153
Xinfu Chen, Jong-Shenq Guo, and Chin-Chin Wu, Traveling waves in discrete periodic media for bistable dynamics, Arch. Ration. Mech. Anal. 189 (2008), no. 2, 189–236. MR 2413095, DOI 10.1007/s00205-007-0103-3
W. Ding, F. Hamel, and X.-Q. Zhao, Bistable pulsating fronts for reaction-diffusion equations in a periodic habitat. arXiv:1408.0723.
Paul C. Fife and J. B. McLeod, The approach of solutions of nonlinear diffusion equations to travelling front solutions, Arch. Rational Mech. Anal. 65 (1977), no. 4, 335–361. MR 442480, DOI 10.1007/BF00250432
Paul C. Fife and J. B. McLeod, A phase plane discussion of convergence to travelling fronts for nonlinear diffusion, Arch. Rational Mech. Anal. 75 (1980/81), no. 4, 281–314. MR 607901, DOI 10.1007/BF00256381
A. M. Fink, Almost periodic differential equations, Lecture Notes in Mathematics, Vol. 377, Springer-Verlag, Berlin-New York, 1974. MR 0460799
Yoshinori Kametaka, On the nonlinear diffusion equation of Kolmogorov-Petrovskii-Piskunov type, Osaka Math. J. 13 (1976), no. 1, 11–66. MR 422875
Ja. I. Kanel′, The behavior of solutions of the Cauchy problem when the time tends to infinity, in the case of quasilinear equations arising in the theory of combustion, Soviet Math. Dokl. 1 (1960), 533–536. MR 0126627
Ja. I. Kanel′, Certain problems on equations in the theory of burning, Soviet Math. Dokl. 2 (1961), 48–51. MR 0117429
Ja. I. Kanel′, Stabilization of solutions of the Cauchy problem for equations encountered in combustion theory, Mat. Sb. (N.S.) 59 (101) (1962), no. suppl., 245–288 (Russian). MR 0157130
Ja. I. Kanel′, Stabilization of the solutions of the equations of combustion theory with finite initial functions, Mat. Sb. (N.S.) 65 (107) (1964), 398–413 (Russian). MR 0177209
Timothy J. Lewis and James P. Keener, Wave-block in excitable media due to regions of depressed excitability, SIAM J. Appl. Math. 61 (2000), no. 1, 293–316. MR 1776397, DOI 10.1137/S0036139998349298
Antoine Mellet, Jean-Michel Roquejoffre, and Yannick Sire, Generalized fronts for one-dimensional reaction-diffusion equations, Discrete Contin. Dyn. Syst. 26 (2010), no. 1, 303–312. MR 2552789, DOI 10.3934/dcds.2010.26.303
Antoine Mellet, James Nolen, Jean-Michel Roquejoffre, and Lenya Ryzhik, Stability of generalized transition fronts, Comm. Partial Differential Equations 34 (2009), no. 4-6, 521–552. MR 2530708, DOI 10.1080/03605300902768677
Shiwang Ma and Jianhong Wu, Existence, uniqueness and asymptotic stability of traveling wavefronts in a non-local delayed diffusion equation, J. Dynam. Differential Equations 19 (2007), no. 2, 391–436. MR 2333414, DOI 10.1007/s10884-006-9065-7
Grégoire Nadin, Critical travelling waves for general heterogeneous one-dimensional reaction-diffusion equations, Ann. Inst. H. Poincaré C Anal. Non Linéaire 32 (2015), no. 4, 841–873. MR 3390087, DOI 10.1016/j.anihpc.2014.03.007
Grégoire Nadin and Luca Rossi, Propagation phenomena for time heterogeneous KPP reaction-diffusion equations, J. Math. Pures Appl. (9) 98 (2012), no. 6, 633–653 (English, with English and French summaries). MR 2994696, DOI 10.1016/j.matpur.2012.05.005
James Nolen and Lenya Ryzhik, Traveling waves in a one-dimensional heterogeneous medium, Ann. Inst. H. Poincaré C Anal. Non Linéaire 26 (2009), no. 3, 1021–1047. MR 2526414, DOI 10.1016/j.anihpc.2009.02.003
James Nolen, Jean-Michel Roquejoffre, Lenya Ryzhik, and Andrej Zlatoš, Existence and non-existence of Fisher-KPP transition fronts, Arch. Ration. Mech. Anal. 203 (2012), no. 1, 217–246. MR 2864411, DOI 10.1007/s00205-011-0449-4
A. Kolmogorov, I. Petrowsky, and N. Piscunov, Study of the diffusion equation with growth of the quantity of matter and its application to a biology problem, Bjul. Moskovskogo Gos. Univ. 1 (1937), 1–26.
Jean-Michel Roquejoffre, Convergence to travelling waves for solutions of a class of semilinear parabolic equations, J. Differential Equations 108 (1994), no. 2, 262–295. MR 1270581, DOI 10.1006/jdeq.1994.1035
D. H. Sattinger, On the stability of waves of nonlinear parabolic systems, Advances in Math. 22 (1976), no. 3, 312–355. MR 435602, DOI 10.1016/0001-8708(76)90098-0
Wenxian Shen, Travelling waves in time almost periodic structures governed by bistable nonlinearities. I. Stability and uniqueness, J. Differential Equations 159 (1999), no. 1, 1–54. MR 1726918, DOI 10.1006/jdeq.1999.3651
Wenxian Shen, Travelling waves in time almost periodic structures governed by bistable nonlinearities. II. Existence, J. Differential Equations 159 (1999), no. 1, 55–101. MR 1726919, DOI 10.1006/jdeq.1999.3652
Wenxian Shen, Traveling waves in diffusive random media, J. Dynam. Differential Equations 16 (2004), no. 4, 1011–1060. MR 2110054, DOI 10.1007/s10884-004-7832-x
Wenxian Shen, Traveling waves in time dependent bistable equations, Differential Integral Equations 19 (2006), no. 3, 241–278. MR 2215558
Wenxian Shen, Existence, uniqueness, and stability of generalized traveling waves in time dependent monostable equations, J. Dynam. Differential Equations 23 (2011), no. 1, 1–44. MR 2772198, DOI 10.1007/s10884-010-9200-3
Wenxian Shen, Existence of generalized traveling waves in time recurrent and space periodic monostable equations, J. Appl. Anal. Comput. 1 (2011), no. 1, 69–93. MR 2881848
W. Shen and Z. Shen, Transition fronts in time heterogeneous and random media of ignition type. arXiv:1407.7579.
Hal L. Smith and Xiao-Qiang Zhao, Global asymptotic stability of traveling waves in delayed reaction-diffusion equations, SIAM J. Math. Anal. 31 (2000), no. 3, 514–534. MR 1740724, DOI 10.1137/S0036141098346785
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
W. A. Veech, Almost automorphic functions on groups, Amer. J. Math. 87 (1965), 719–751. MR 187014, DOI 10.2307/2373071
Hans F. Weinberger, On spreading speeds and traveling waves for growth and migration models in a periodic habitat, J. Math. Biol. 45 (2002), no. 6, 511–548. MR 1943224, DOI 10.1007/s00285-002-0169-3
Xue Xin, Existence and uniqueness of travelling waves in a reaction-diffusion equation with combustion nonlinearity, Indiana Univ. Math. J. 40 (1991), no. 3, 985–1008. MR 1129338, DOI 10.1512/iumj.1991.40.40044
Jack X. Xin, Existence of planar flame fronts in convective-diffusive periodic media, Arch. Rational Mech. Anal. 121 (1992), no. 3, 205–233. MR 1188981, DOI 10.1007/BF00410613
Jack X. Xin, Existence and nonexistence of traveling waves and reaction-diffusion front propagation in periodic media, J. Statist. Phys. 73 (1993), no. 5-6, 893–926. MR 1251222, DOI 10.1007/BF01052815
Wenxian Shen and Yingfei Yi, Almost automorphic and almost periodic dynamics in skew-product semiflows, Mem. Amer. Math. Soc. 136 (1998), no. 647, x+93. MR 1445493, DOI 10.1090/memo/0647
Andrej Zlatoš, Transition fronts in inhomogeneous Fisher-KPP reaction-diffusion equations, J. Math. Pures Appl. (9) 98 (2012), no. 1, 89–102. MR 2935371, DOI 10.1016/j.matpur.2011.11.007
Andrej Zlatoš, Generalized traveling waves in disordered media: existence, uniqueness, and stability, Arch. Ration. Mech. Anal. 208 (2013), no. 2, 447–480. MR 3035984, DOI 10.1007/s00205-012-0600-x