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June  2016, 36(6): 3317-3338. doi: 10.3934/dcds.2016.36.3317

Diffusive KPP equations with free boundaries in time almost periodic environments: I. Spreading and vanishing dichotomy

Fang Li 1, , Xing Liang 1, and  Wenxian Shen 2,

1. 

School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui, 230026, China, China
2. 

Department of Mathematics and Statistics, Auburn University, Auburn University, AL 36849-5310

Received  May 2015 Revised  October 2015 Published  December 2015

In this series of papers, we investigate the spreading and vanishing dynamics of time almost periodic diffusive KPP equations with free boundaries. Such equations are used to characterize the spreading of a new species in time almost periodic environments with free boundaries representing the spreading fronts. In this first part, we show that a spreading-vanishing dichotomy occurs for such free boundary problems, that is, the species either successfully spreads to all the new environment and stabilizes at a time almost periodic positive solution, or it fails to establish and dies out eventually. The results of this part extend the existing results on spreading-vanishing dichotomy for time and space independent, or time periodic and space independent, or time independent and space periodic diffusive KPP equations with free boundaries. The extension is nontrivial and is ever done for the first time.
Keywords: principal Lyapunov exponent, time almost periodic environment, free boundary, spreading-vanishing dichotomy, time almost periodic positive solution., part metric, Diffusive KPP equation.
Mathematics Subject Classification: 35K20, 35R35, 35K57, 35B15, 92B0.
Citation: Fang Li, Xing Liang, Wenxian Shen. Diffusive KPP equations with free boundaries in time almost periodic environments: I. Spreading and vanishing dichotomy. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 3317-3338. doi: 10.3934/dcds.2016.36.3317
References:
[1]

H. Berestycki and F. Hamel, Generalized transition waves and their properties,, Comm. Pure Appl. Math., 65 (2012), 592. doi: 10.1002/cpa.21389. Google Scholar

[2]

H. Berestycki, F. Hamel and G. Nadin, Asymptotic spreading in heterogeneous diffusive excitable media,, J. Funct. Anal., 255 (2008), 2146. doi: 10.1016/j.jfa.2008.06.030. Google Scholar

[3]

H. Berestycki and G. Nadin, Spreading speeds for one-dimensional monostable reaction-diffusion equations,, J. Math. Phys., 53 (2012). doi: 10.1063/1.4764932. Google Scholar

[4]

Y. Du and Z. Guo, Spreading-vanishing dichotomy in a diffusive logistic model with a free boundary, II,, J. Differential Equations, 250 (2011), 4336. doi: 10.1016/j.jde.2011.02.011. Google Scholar

[5]

Y. Du, Z. Guo and R. Peng, A diffusive logistic model with a free boundary in time-periodic environment,, J. Functional Analysis, 265 (2013), 2089. doi: 10.1016/j.jfa.2013.07.016. Google Scholar

[6]

Y. Du and Z. Lin, Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary,, SIAM J. Math. Anal., 42 (2010), 377. doi: 10.1137/090771089. Google Scholar

[7]

Y. Du and X. Liang, Pulsating semi-waves in periodic media and spreading speed determined by a free boundary model,, Ann. I. H. Poincaré-AN, 32 (2015), 279. doi: 10.1016/j.anihpc.2013.11.004. Google Scholar

[8]

Y. Du and B. Lou, Spreading and vanishing in nonlinear diffusion problems with free boundaries,, J. Eur. Math. Soc., 17 (2015), 2673. doi: 10.4171/JEMS/568. Google Scholar

[9]

A. M. Fink, Almost Periodic Differential Equations,, Lectures Notes in Mathematics, (1974). Google Scholar

[10]

R. Fisher, The wave of advance of advantageous genes,, Ann. of Eugenics, 7 (1937), 355. doi: 10.1111/j.1469-1809.1937.tb02153.x. Google Scholar

[11]

J. Huang and W. Shen, Speeds of spread and propagation of KPP models in time almost and space periodic media,, SIAM J. Appl. Dyn. Syst., 8 (2009), 790. doi: 10.1137/080723259. Google Scholar

[12]

A. Kolmogorov, I. Petrowsky and N. Piscunov, A study of the equation of diffusion with increase in the quantity of matter, and its application to a biological problem,, Bjul. Moskovskogo Gos. Univ., 1 (1937), 1. Google Scholar

[13]

L. Kong and W. Shen, Liouville type property and spreading speeds of KPP equations in periodic media with localized spatial inhomogeneity,, J. Dynam. Differential Equations, 26 (2014), 181. doi: 10.1007/s10884-014-9351-8. Google Scholar

[14]

X. Liang and X.-Q. Zhao, Spreading speeds and traveling waves for abstract monostable evolution systems,, J. Funct. Anal., 259 (2010), 857. doi: 10.1016/j.jfa.2010.04.018. Google Scholar

[15]

J. L. Lockwood, M. F. Hoppes and M. P. Marchetti, Invasion Ecology,, Blackwell Publishing, (2007). Google Scholar

[16]

J. Mierczynski and W. Shen, Exponential separation and principal Lyapunov exponent/spectrum for random/nonautonomous parabolic equations,, J. Differential Equations, 191 (2003), 175. doi: 10.1016/S0022-0396(03)00016-0. Google Scholar

[17]

J. Mierczynski and W. Shen, Lyapunov exponents and asymptotic dynamics in random kolmogorov models,, J. Evolution Equations, 4 (2004), 371. doi: 10.1007/s00028-004-0160-0. Google Scholar

[18]

J. Mierczynski and W. Shen, Spectral Theory for Random and Nonautonomous Parabolic Equations and Applications,, Chapman & Hall/CRC Monogr. Surv. Pure Appl. Math., (2008). doi: 10.1201/9781584888963. Google Scholar

[19]

G. Nadin, Traveling fronts in space-time periodic media,, J. Math. Pures Appl., 92 (2009), 232. doi: 10.1016/j.matpur.2009.04.002. Google Scholar

[20]

G. Nadin and L. Rossi, Propagation phenomena for time heterogeneous KPP reaction-diffusion equations,, J. Math. Pures Appl., 98 (2012), 633. doi: 10.1016/j.matpur.2012.05.005. Google Scholar

[21]

J. Nolen, J.-M. Roquejoffre, L. Ryzhik and A. Zlatoš, Existence and non-existence of Fisher-KPP transition fronts,, Arch. Ration. Mech. Anal., 203 (2012), 217. doi: 10.1007/s00205-011-0449-4. Google Scholar

[22]

J. Nolen and J. Xin, Existence of KPP type fronts in space-time periodic shear flows and a study of minimal speeds based on variational principle,, Discrete Contin. Dyn. Syst., 13 (2005), 1217. doi: 10.3934/dcds.2005.13.1217. Google Scholar

[23]

J. Nolen and J. Xin, A variational principle based study of KPP minimal front speeds in random shears,, Nonlinearity, 18 (2005), 1655. doi: 10.1088/0951-7715/18/4/013. Google Scholar

[24]

A. Pazy, Semigroups of Linear Operators and Application to Partial Differential Equations,, Springer-Verlag, (1983). doi: 10.1007/978-1-4612-5561-1. Google Scholar

[25]

R. Peng and X.-Q. Zhao, The diffusive logistic model with a free boundary and seasonal succession,, Discrete and Continuous Dynamical Systems, 33 (2013), 2007. doi: 10.3934/dcds.2013.33.2007. Google Scholar

[26]

W. Shen, Variational principle for spreading speeds and generalized propagating speeds in time almost periodic and space periodic KPP models,, Trans. Amer. Math. Soc., 362 (2010), 5125. doi: 10.1090/S0002-9947-10-04950-0. Google Scholar

[27]

W. Shen, Existence, uniqueness, and stability of generalized traveling waves in time dependent monostable equations,, J. Dynam. Differential Equations, 23 (2011), 1. doi: 10.1007/s10884-010-9200-3. Google Scholar

[28]

W. Shen, Existence of generalized traveling waves in time recurrent and space periodic monostable equations,, J. Appl. Anal. Comput., 1 (2011), 69. Google Scholar

[29]

W. Shen and Y. Yi, Almost Automorphic and Almost Periodic Dynamics in Skew-Product Semiflow,, Memoirs of the American Mathmatical Society, (1998). doi: 10.1090/memo/0647. Google Scholar

[30]

N. Shigesada and K. Kawasaki, Biological Invasions: Theory and Practice,, Oxford Series in Ecology and Evolution, (1997). Google Scholar

[31]

P. Takáč, Convergence in the part metric for discrete dynamical systems in ordered topological cones,, Nonlinear Anal., 26 (1996), 1753. doi: 10.1016/0362-546X(95)00015-N. Google Scholar

[32]

T. Tao, B. Zhu and A. Zlatoš, Transition fronts for inhomogeneous monostable reaction-diffusion equations via linearization at zero,, Nonlinearity, 27 (2014), 2409. doi: 10.1088/0951-7715/27/9/2409. Google Scholar

[33]

A. C. Thompson, On certain contraction mappings in a partially ordered vector space,, in Proceedings of the American Mathematical Society, (1963), 438. Google Scholar

[34]

H. Weinberger, On spreading speed and travelling waves for growth and migration models in a periodic habitat,, J. Math. Biol., 45 (2002), 511. doi: 10.1007/s00285-002-0169-3. Google Scholar

[35]

A. Zlatoš, Transition fronts in inhomogeneous Fisher-KPP reaction-diffusion equations,, J. Math. Pures Appl., 98 (2012), 89. doi: 10.1016/j.matpur.2011.11.007. Google Scholar

[36]

X.-Q. Zhao, Global attractivity in monotone and subhomogeneous almost periodic systems,, J. Differential Equations, 187 (2003), 494. doi: 10.1016/S0022-0396(02)00054-2. Google Scholar

show all references

References:
[1]

H. Berestycki and F. Hamel, Generalized transition waves and their properties,, Comm. Pure Appl. Math., 65 (2012), 592. doi: 10.1002/cpa.21389. Google Scholar

[2]

H. Berestycki, F. Hamel and G. Nadin, Asymptotic spreading in heterogeneous diffusive excitable media,, J. Funct. Anal., 255 (2008), 2146. doi: 10.1016/j.jfa.2008.06.030. Google Scholar

[3]

H. Berestycki and G. Nadin, Spreading speeds for one-dimensional monostable reaction-diffusion equations,, J. Math. Phys., 53 (2012). doi: 10.1063/1.4764932. Google Scholar

[4]

Y. Du and Z. Guo, Spreading-vanishing dichotomy in a diffusive logistic model with a free boundary, II,, J. Differential Equations, 250 (2011), 4336. doi: 10.1016/j.jde.2011.02.011. Google Scholar

[5]

Y. Du, Z. Guo and R. Peng, A diffusive logistic model with a free boundary in time-periodic environment,, J. Functional Analysis, 265 (2013), 2089. doi: 10.1016/j.jfa.2013.07.016. Google Scholar

[6]

Y. Du and Z. Lin, Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary,, SIAM J. Math. Anal., 42 (2010), 377. doi: 10.1137/090771089. Google Scholar

[7]

Y. Du and X. Liang, Pulsating semi-waves in periodic media and spreading speed determined by a free boundary model,, Ann. I. H. Poincaré-AN, 32 (2015), 279. doi: 10.1016/j.anihpc.2013.11.004. Google Scholar

[8]

Y. Du and B. Lou, Spreading and vanishing in nonlinear diffusion problems with free boundaries,, J. Eur. Math. Soc., 17 (2015), 2673. doi: 10.4171/JEMS/568. Google Scholar

[9]

A. M. Fink, Almost Periodic Differential Equations,, Lectures Notes in Mathematics, (1974). Google Scholar

[10]

R. Fisher, The wave of advance of advantageous genes,, Ann. of Eugenics, 7 (1937), 355. doi: 10.1111/j.1469-1809.1937.tb02153.x. Google Scholar

[11]

J. Huang and W. Shen, Speeds of spread and propagation of KPP models in time almost and space periodic media,, SIAM J. Appl. Dyn. Syst., 8 (2009), 790. doi: 10.1137/080723259. Google Scholar

[12]

A. Kolmogorov, I. Petrowsky and N. Piscunov, A study of the equation of diffusion with increase in the quantity of matter, and its application to a biological problem,, Bjul. Moskovskogo Gos. Univ., 1 (1937), 1. Google Scholar

[13]

L. Kong and W. Shen, Liouville type property and spreading speeds of KPP equations in periodic media with localized spatial inhomogeneity,, J. Dynam. Differential Equations, 26 (2014), 181. doi: 10.1007/s10884-014-9351-8. Google Scholar

[14]

X. Liang and X.-Q. Zhao, Spreading speeds and traveling waves for abstract monostable evolution systems,, J. Funct. Anal., 259 (2010), 857. doi: 10.1016/j.jfa.2010.04.018. Google Scholar

[15]

J. L. Lockwood, M. F. Hoppes and M. P. Marchetti, Invasion Ecology,, Blackwell Publishing, (2007). Google Scholar

[16]

J. Mierczynski and W. Shen, Exponential separation and principal Lyapunov exponent/spectrum for random/nonautonomous parabolic equations,, J. Differential Equations, 191 (2003), 175. doi: 10.1016/S0022-0396(03)00016-0. Google Scholar

[17]

J. Mierczynski and W. Shen, Lyapunov exponents and asymptotic dynamics in random kolmogorov models,, J. Evolution Equations, 4 (2004), 371. doi: 10.1007/s00028-004-0160-0. Google Scholar

[18]

J. Mierczynski and W. Shen, Spectral Theory for Random and Nonautonomous Parabolic Equations and Applications,, Chapman & Hall/CRC Monogr. Surv. Pure Appl. Math., (2008). doi: 10.1201/9781584888963. Google Scholar

[19]

G. Nadin, Traveling fronts in space-time periodic media,, J. Math. Pures Appl., 92 (2009), 232. doi: 10.1016/j.matpur.2009.04.002. Google Scholar

[20]

G. Nadin and L. Rossi, Propagation phenomena for time heterogeneous KPP reaction-diffusion equations,, J. Math. Pures Appl., 98 (2012), 633. doi: 10.1016/j.matpur.2012.05.005. Google Scholar

[21]

J. Nolen, J.-M. Roquejoffre, L. Ryzhik and A. Zlatoš, Existence and non-existence of Fisher-KPP transition fronts,, Arch. Ration. Mech. Anal., 203 (2012), 217. doi: 10.1007/s00205-011-0449-4. Google Scholar

[22]

J. Nolen and J. Xin, Existence of KPP type fronts in space-time periodic shear flows and a study of minimal speeds based on variational principle,, Discrete Contin. Dyn. Syst., 13 (2005), 1217. doi: 10.3934/dcds.2005.13.1217. Google Scholar

[23]

J. Nolen and J. Xin, A variational principle based study of KPP minimal front speeds in random shears,, Nonlinearity, 18 (2005), 1655. doi: 10.1088/0951-7715/18/4/013. Google Scholar

[24]

A. Pazy, Semigroups of Linear Operators and Application to Partial Differential Equations,, Springer-Verlag, (1983). doi: 10.1007/978-1-4612-5561-1. Google Scholar

[25]

R. Peng and X.-Q. Zhao, The diffusive logistic model with a free boundary and seasonal succession,, Discrete and Continuous Dynamical Systems, 33 (2013), 2007. doi: 10.3934/dcds.2013.33.2007. Google Scholar

[26]

W. Shen, Variational principle for spreading speeds and generalized propagating speeds in time almost periodic and space periodic KPP models,, Trans. Amer. Math. Soc., 362 (2010), 5125. doi: 10.1090/S0002-9947-10-04950-0. Google Scholar

[27]

W. Shen, Existence, uniqueness, and stability of generalized traveling waves in time dependent monostable equations,, J. Dynam. Differential Equations, 23 (2011), 1. doi: 10.1007/s10884-010-9200-3. Google Scholar

[28]

W. Shen, Existence of generalized traveling waves in time recurrent and space periodic monostable equations,, J. Appl. Anal. Comput., 1 (2011), 69. Google Scholar

[29]

W. Shen and Y. Yi, Almost Automorphic and Almost Periodic Dynamics in Skew-Product Semiflow,, Memoirs of the American Mathmatical Society, (1998). doi: 10.1090/memo/0647. Google Scholar

[30]

N. Shigesada and K. Kawasaki, Biological Invasions: Theory and Practice,, Oxford Series in Ecology and Evolution, (1997). Google Scholar

[31]

P. Takáč, Convergence in the part metric for discrete dynamical systems in ordered topological cones,, Nonlinear Anal., 26 (1996), 1753. doi: 10.1016/0362-546X(95)00015-N. Google Scholar

[32]

T. Tao, B. Zhu and A. Zlatoš, Transition fronts for inhomogeneous monostable reaction-diffusion equations via linearization at zero,, Nonlinearity, 27 (2014), 2409. doi: 10.1088/0951-7715/27/9/2409. Google Scholar

[33]

A. C. Thompson, On certain contraction mappings in a partially ordered vector space,, in Proceedings of the American Mathematical Society, (1963), 438. Google Scholar

[34]

H. Weinberger, On spreading speed and travelling waves for growth and migration models in a periodic habitat,, J. Math. Biol., 45 (2002), 511. doi: 10.1007/s00285-002-0169-3. Google Scholar

[35]

A. Zlatoš, Transition fronts in inhomogeneous Fisher-KPP reaction-diffusion equations,, J. Math. Pures Appl., 98 (2012), 89. doi: 10.1016/j.matpur.2011.11.007. Google Scholar

[36]

X.-Q. Zhao, Global attractivity in monotone and subhomogeneous almost periodic systems,, J. Differential Equations, 187 (2003), 494. doi: 10.1016/S0022-0396(02)00054-2. Google Scholar

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