Stability of non-monotone critical traveling waves for reaction–diffusion equations with time-delay

Journal of Differential Equations - Tập 259 - Trang 1503-1541 - 2015
I-Liang Chern1,2, Ming Mei3,4, Xiongfeng Yang5, Qifeng Zhang6
1Department of Mathematics, National Central University, Jhongli, Taoyuan, 32001, Taiwan, ROC
2Institute of Applied Mathematical Science, National Taiwan University, Taipei, Taiwan, ROC
3Department of Mathematics, Champlain College Saint-Lambert Quebec, J4P 3P2, Canada
4Department of Mathematics and Statistics, McGill University, Montreal, Quebec, H3A 2K6, Canada
5Department of Mathematics, and MOE-LSC, Shanghai Jiao Tong University, Shanghai, 200240, China
6School of Science, Zhejiang Sci-Tech University, Hangzhou, Zhejiang, 310018, China

Tài liệu tham khảo

Aguerrea, 2012, On uniqueness of semi-wavefronts, Math. Ann., 354, 73, 10.1007/s00208-011-0722-8 Boese, 1987, Some stability charts and stability conditions for a class of difference-differential equations, ZAMM Z. Angew. Math. Mech., 67, 56 Fang, 2010, Existence and uniqueness of traveling waves for non-monotone integral equations with in applications, J. Differential Equations, 248, 2199, 10.1016/j.jde.2010.01.009 Faria, 2006, Traveling waves for delayed reaction–diffusion equations with global response, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 462, 229, 10.1098/rspa.2005.1554 Faria, 2006, Nonmonotone traveling waves in single species reaction–diffusion equation with delay, J. Differential Equations, 228, 357, 10.1016/j.jde.2006.05.006 Faria, 2007, Positive heteroclinics and traveling waves for scalar population models with a single delay, Appl. Math. Comput., 185, 594 Gomez, 2014, Global continuation of monotone wavefronts, J. Lond. Math. Soc., 89, 47, 10.1112/jlms/jdt050 Gourley, 2003, Non-locality of reaction–diffusion equations induced by delay: biological modeling and nonlinear dynamics, 84 Gourley, 2006, Delayed nonlocal diffusive system in biological invasion and disease spread, vol. 48, 137 Huang, 2012, Planar traveling waves for nonlocal dispersal equation with monostable nonlinearity, Discrete Contin. Dyn. Syst. Ser. A, 32, 3621, 10.3934/dcds.2012.32.3621 R. Huang, M. Mei, K.-J. Zhang, Q.-F. Zhang, Stability of oscillating wavefronts for time-delayed nonlocal dispersal equations, preprint, 2014. Liang, 2003, Traveling waves and numerical approximations in a reaction–diffusion equation with nonlocal delayed effect, J. Nonlinear Sci., 13, 289, 10.1007/s00332-003-0524-6 Lin, 2014, Exponential stability of non-monotone traveling waves for Nicholson's blowflies equation, SIAM J. Math. Anal., 46, 1053, 10.1137/120904391 Ma, 2007, Traveling waves for non-local delayed diffusion equations via auxiliary equations, J. Differential Equations, 237, 259, 10.1016/j.jde.2007.03.014 Matsumura, 1997, Nonlinear stability of viscous shock profile for a non-convex system of viscoelasticity, Osaka J. Math., 34, 589 Mei, 2009, Traveling wavefronts for time-delayed reaction–diffusion equation: (I) local nonlinearity, J. Differential Equations, 247, 495, 10.1016/j.jde.2008.12.026 Mei, 2009, Traveling wavefronts for time-delayed reaction–diffusion equation: (II) nonlocal nonlinearity, J. Differential Equations, 247, 511, 10.1016/j.jde.2008.12.020 Mei, 2010, Global stability of monostable traveling waves for nonlocal time-delayed reaction–diffusion equations, SIAM J. Math. Anal., 42, 2762, 10.1137/090776342 Mei, 2008, Stability of strong traveling waves for a nonlocal time-delayed reaction–diffusion equation, Proc. Roy. Soc. Edinburgh Sect. A, 138, 551, 10.1017/S0308210506000333 Mei, 2004, Asymptotic stability of traveling waves for the Nicholson's blowflies equation with diffusion, Proc. Roy. Soc. Edinburgh Sect. A, 134, 579, 10.1017/S0308210500003358 Mei, 2011, Remark on stability of traveling waves for nonlocal Fisher–KPP equations, Int. J. Numer. Anal. Model. Ser. B, 2, 379 Moet, 1979, A note on asymptotic behavior of solutions of the KPP equation, SIAM J. Math. Anal., 10, 728, 10.1137/0510067 Nishihara, 2000, Lp-convergence rate to nonlinear diffusion waves for p-system with damping, J. Differential Equations, 161, 191, 10.1006/jdeq.1999.3703 Radu, 2010, Decay estimates for wave equations with variable coefficients, Trans. Amer. Math. Soc., 362, 2279, 10.1090/S0002-9947-09-04742-4 Rothe, 1978, Convergence to travelling fronts in semilinear parabolic equations, Proc. Roy. Soc. Edinburgh Sect. A, 80, 213, 10.1017/S0308210500010258 Sherratt, 1998, On the transition from initial data to travelling waves in the Fisher–KPP equation, Dyn. Stab. Syst., 13, 167, 10.1080/02681119808806258 Smith, 2011, An Introduction to Delay Differential Equations with Applications to the Life Sciences, vol. 57 So, 1998, Dirichlet problem for the diffusive Nicholson's blowflies equation, J. Differential Equations, 150, 317, 10.1006/jdeq.1998.3489 So, 2001, Traveling waves for the diffusive Nicholson's blowflies equation, Appl. Math. Comput., 122, 385 Tang, 1998, Oscillation of delay differential equations with variable coefficients, J. Math. Anal. Appl., 217, 32, 10.1006/jmaa.1997.5693 Todorova, 2007, Nonlinear dissipative wave equations with potential, vol. 426, 317 Trofimchuk, 2008, Admissible wavefront speeds for a single species reaction–diffusion equation with delay, Discrete Contin. Dyn. Syst. Ser. A, 20, 407, 10.3934/dcds.2008.20.407 Trofimchuk, 2008, Slowly oscillating wave solutions of a single species reaction–diffusion equation with delay, J. Differential Equations, 245, 2307, 10.1016/j.jde.2008.06.023 van Saarloos, 2003, Front propagation into unstable states, Phys. Rep., 386, 29, 10.1016/j.physrep.2003.08.001 Wang, 2007, Existence and stability of planar diffusion waves for 2-D Euler equations with damping, J. Differential Equations, 242, 40, 10.1016/j.jde.2007.07.002 Zhao, 2003