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Sharp criteria of Liouville type for some nonlinear systems
Diffusive KPP equations with free boundaries in time almost periodic environments: I. Spreading and vanishing dichotomy
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 |
References:
[1] |
H. Berestycki and F. Hamel, Generalized transition waves and their properties, Comm. Pure Appl. Math., 65 (2012), 592-648.
doi: 10.1002/cpa.21389. |
[2] |
H. Berestycki, F. Hamel and G. Nadin, Asymptotic spreading in heterogeneous diffusive excitable media, J. Funct. Anal., 255 (2008), 2146-2189.
doi: 10.1016/j.jfa.2008.06.030. |
[3] |
H. Berestycki and G. Nadin, Spreading speeds for one-dimensional monostable reaction-diffusion equations, J. Math. Phys., 53 (2012), 115619, 23pp.
doi: 10.1063/1.4764932. |
[4] |
Y. Du and Z. Guo, Spreading-vanishing dichotomy in a diffusive logistic model with a free boundary, II, J. Differential Equations, 250 (2011), 4336-4366.
doi: 10.1016/j.jde.2011.02.011. |
[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), 2089-2142.
doi: 10.1016/j.jfa.2013.07.016. |
[6] |
Y. Du and Z. Lin, Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary, SIAM J. Math. Anal., 42 (2010), 377-405.
doi: 10.1137/090771089. |
[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), 279-305.
doi: 10.1016/j.anihpc.2013.11.004. |
[8] |
Y. Du and B. Lou, Spreading and vanishing in nonlinear diffusion problems with free boundaries, J. Eur. Math. Soc., 17 (2015), 2673-2724.
doi: 10.4171/JEMS/568. |
[9] |
A. M. Fink, Almost Periodic Differential Equations, Lectures Notes in Mathematics, Springer-Verlag, Berlin-New York, 1974. |
[10] |
R. Fisher, The wave of advance of advantageous genes, Ann. of Eugenics, 7 (1937), 355-369.
doi: 10.1111/j.1469-1809.1937.tb02153.x. |
[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), 790-821.
doi: 10.1137/080723259. |
[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), 1-26. |
[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), 181-215.
doi: 10.1007/s10884-014-9351-8. |
[14] |
X. Liang and X.-Q. Zhao, Spreading speeds and traveling waves for abstract monostable evolution systems, J. Funct. Anal., 259 (2010), 857-903.
doi: 10.1016/j.jfa.2010.04.018. |
[15] |
J. L. Lockwood, M. F. Hoppes and M. P. Marchetti, Invasion Ecology, Blackwell Publishing, 2007. |
[16] |
J. Mierczynski and W. Shen, Exponential separation and principal Lyapunov exponent/spectrum for random/nonautonomous parabolic equations, J. Differential Equations, 191 (2003), 175-205.
doi: 10.1016/S0022-0396(03)00016-0. |
[17] |
J. Mierczynski and W. Shen, Lyapunov exponents and asymptotic dynamics in random kolmogorov models, J. Evolution Equations, 4 (2004), 371-390.
doi: 10.1007/s00028-004-0160-0. |
[18] |
J. Mierczynski and W. Shen, Spectral Theory for Random and Nonautonomous Parabolic Equations and Applications, Chapman & Hall/CRC Monogr. Surv. Pure Appl. Math., Chapman & Hall/CRC, Boca Raton, FL, 2008.
doi: 10.1201/9781584888963. |
[19] |
G. Nadin, Traveling fronts in space-time periodic media, J. Math. Pures Appl., 92 (2009), 232-262.
doi: 10.1016/j.matpur.2009.04.002. |
[20] |
G. Nadin and L. Rossi, Propagation phenomena for time heterogeneous KPP reaction-diffusion equations, J. Math. Pures Appl., 98 (2012), 633-653.
doi: 10.1016/j.matpur.2012.05.005. |
[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), 217-246.
doi: 10.1007/s00205-011-0449-4. |
[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), 1217-1234.
doi: 10.3934/dcds.2005.13.1217. |
[23] |
J. Nolen and J. Xin, A variational principle based study of KPP minimal front speeds in random shears, Nonlinearity, 18 (2005), 1655-1675.
doi: 10.1088/0951-7715/18/4/013. |
[24] |
A. Pazy, Semigroups of Linear Operators and Application to Partial Differential Equations, Springer-Verlag, New York, 1983.
doi: 10.1007/978-1-4612-5561-1. |
[25] |
R. Peng and X.-Q. Zhao, The diffusive logistic model with a free boundary and seasonal succession, Discrete and Continuous Dynamical Systems, Ser. A, 33 (2013), 2007-2031.
doi: 10.3934/dcds.2013.33.2007. |
[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), 5125-5168.
doi: 10.1090/S0002-9947-10-04950-0. |
[27] |
W. Shen, Existence, uniqueness, and stability of generalized traveling waves in time dependent monostable equations, J. Dynam. Differential Equations, 23 (2011), 1-44.
doi: 10.1007/s10884-010-9200-3. |
[28] |
W. Shen, Existence of generalized traveling waves in time recurrent and space periodic monostable equations, J. Appl. Anal. Comput., 1 (2011), 69-93. |
[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. |
[30] |
N. Shigesada and K. Kawasaki, Biological Invasions: Theory and Practice, Oxford Series in Ecology and Evolution, Oxford, Oxford University Press, 1997. |
[31] |
P. Takáč, Convergence in the part metric for discrete dynamical systems in ordered topological cones, Nonlinear Anal., 26 (1996), 1753-1777.
doi: 10.1016/0362-546X(95)00015-N. |
[32] |
T. Tao, B. Zhu and A. Zlatoš, Transition fronts for inhomogeneous monostable reaction-diffusion equations via linearization at zero, Nonlinearity, 27 (2014), 2409-2416.
doi: 10.1088/0951-7715/27/9/2409. |
[33] |
A. C. Thompson, On certain contraction mappings in a partially ordered vector space, in Proceedings of the American Mathematical Society, 14, 1963, 438-443. |
[34] |
H. Weinberger, On spreading speed and travelling waves for growth and migration models in a periodic habitat, J. Math. Biol., 45 (2002), 511-548.
doi: 10.1007/s00285-002-0169-3. |
[35] |
A. Zlatoš, Transition fronts in inhomogeneous Fisher-KPP reaction-diffusion equations, J. Math. Pures Appl., 98 (2012), 89-102.
doi: 10.1016/j.matpur.2011.11.007. |
[36] |
X.-Q. Zhao, Global attractivity in monotone and subhomogeneous almost periodic systems, J. Differential Equations, 187 (2003), 494-509.
doi: 10.1016/S0022-0396(02)00054-2. |
show all references
References:
[1] |
H. Berestycki and F. Hamel, Generalized transition waves and their properties, Comm. Pure Appl. Math., 65 (2012), 592-648.
doi: 10.1002/cpa.21389. |
[2] |
H. Berestycki, F. Hamel and G. Nadin, Asymptotic spreading in heterogeneous diffusive excitable media, J. Funct. Anal., 255 (2008), 2146-2189.
doi: 10.1016/j.jfa.2008.06.030. |
[3] |
H. Berestycki and G. Nadin, Spreading speeds for one-dimensional monostable reaction-diffusion equations, J. Math. Phys., 53 (2012), 115619, 23pp.
doi: 10.1063/1.4764932. |
[4] |
Y. Du and Z. Guo, Spreading-vanishing dichotomy in a diffusive logistic model with a free boundary, II, J. Differential Equations, 250 (2011), 4336-4366.
doi: 10.1016/j.jde.2011.02.011. |
[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), 2089-2142.
doi: 10.1016/j.jfa.2013.07.016. |
[6] |
Y. Du and Z. Lin, Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary, SIAM J. Math. Anal., 42 (2010), 377-405.
doi: 10.1137/090771089. |
[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), 279-305.
doi: 10.1016/j.anihpc.2013.11.004. |
[8] |
Y. Du and B. Lou, Spreading and vanishing in nonlinear diffusion problems with free boundaries, J. Eur. Math. Soc., 17 (2015), 2673-2724.
doi: 10.4171/JEMS/568. |
[9] |
A. M. Fink, Almost Periodic Differential Equations, Lectures Notes in Mathematics, Springer-Verlag, Berlin-New York, 1974. |
[10] |
R. Fisher, The wave of advance of advantageous genes, Ann. of Eugenics, 7 (1937), 355-369.
doi: 10.1111/j.1469-1809.1937.tb02153.x. |
[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), 790-821.
doi: 10.1137/080723259. |
[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), 1-26. |
[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), 181-215.
doi: 10.1007/s10884-014-9351-8. |
[14] |
X. Liang and X.-Q. Zhao, Spreading speeds and traveling waves for abstract monostable evolution systems, J. Funct. Anal., 259 (2010), 857-903.
doi: 10.1016/j.jfa.2010.04.018. |
[15] |
J. L. Lockwood, M. F. Hoppes and M. P. Marchetti, Invasion Ecology, Blackwell Publishing, 2007. |
[16] |
J. Mierczynski and W. Shen, Exponential separation and principal Lyapunov exponent/spectrum for random/nonautonomous parabolic equations, J. Differential Equations, 191 (2003), 175-205.
doi: 10.1016/S0022-0396(03)00016-0. |
[17] |
J. Mierczynski and W. Shen, Lyapunov exponents and asymptotic dynamics in random kolmogorov models, J. Evolution Equations, 4 (2004), 371-390.
doi: 10.1007/s00028-004-0160-0. |
[18] |
J. Mierczynski and W. Shen, Spectral Theory for Random and Nonautonomous Parabolic Equations and Applications, Chapman & Hall/CRC Monogr. Surv. Pure Appl. Math., Chapman & Hall/CRC, Boca Raton, FL, 2008.
doi: 10.1201/9781584888963. |
[19] |
G. Nadin, Traveling fronts in space-time periodic media, J. Math. Pures Appl., 92 (2009), 232-262.
doi: 10.1016/j.matpur.2009.04.002. |
[20] |
G. Nadin and L. Rossi, Propagation phenomena for time heterogeneous KPP reaction-diffusion equations, J. Math. Pures Appl., 98 (2012), 633-653.
doi: 10.1016/j.matpur.2012.05.005. |
[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), 217-246.
doi: 10.1007/s00205-011-0449-4. |
[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), 1217-1234.
doi: 10.3934/dcds.2005.13.1217. |
[23] |
J. Nolen and J. Xin, A variational principle based study of KPP minimal front speeds in random shears, Nonlinearity, 18 (2005), 1655-1675.
doi: 10.1088/0951-7715/18/4/013. |
[24] |
A. Pazy, Semigroups of Linear Operators and Application to Partial Differential Equations, Springer-Verlag, New York, 1983.
doi: 10.1007/978-1-4612-5561-1. |
[25] |
R. Peng and X.-Q. Zhao, The diffusive logistic model with a free boundary and seasonal succession, Discrete and Continuous Dynamical Systems, Ser. A, 33 (2013), 2007-2031.
doi: 10.3934/dcds.2013.33.2007. |
[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), 5125-5168.
doi: 10.1090/S0002-9947-10-04950-0. |
[27] |
W. Shen, Existence, uniqueness, and stability of generalized traveling waves in time dependent monostable equations, J. Dynam. Differential Equations, 23 (2011), 1-44.
doi: 10.1007/s10884-010-9200-3. |
[28] |
W. Shen, Existence of generalized traveling waves in time recurrent and space periodic monostable equations, J. Appl. Anal. Comput., 1 (2011), 69-93. |
[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. |
[30] |
N. Shigesada and K. Kawasaki, Biological Invasions: Theory and Practice, Oxford Series in Ecology and Evolution, Oxford, Oxford University Press, 1997. |
[31] |
P. Takáč, Convergence in the part metric for discrete dynamical systems in ordered topological cones, Nonlinear Anal., 26 (1996), 1753-1777.
doi: 10.1016/0362-546X(95)00015-N. |
[32] |
T. Tao, B. Zhu and A. Zlatoš, Transition fronts for inhomogeneous monostable reaction-diffusion equations via linearization at zero, Nonlinearity, 27 (2014), 2409-2416.
doi: 10.1088/0951-7715/27/9/2409. |
[33] |
A. C. Thompson, On certain contraction mappings in a partially ordered vector space, in Proceedings of the American Mathematical Society, 14, 1963, 438-443. |
[34] |
H. Weinberger, On spreading speed and travelling waves for growth and migration models in a periodic habitat, J. Math. Biol., 45 (2002), 511-548.
doi: 10.1007/s00285-002-0169-3. |
[35] |
A. Zlatoš, Transition fronts in inhomogeneous Fisher-KPP reaction-diffusion equations, J. Math. Pures Appl., 98 (2012), 89-102.
doi: 10.1016/j.matpur.2011.11.007. |
[36] |
X.-Q. Zhao, Global attractivity in monotone and subhomogeneous almost periodic systems, J. Differential Equations, 187 (2003), 494-509.
doi: 10.1016/S0022-0396(02)00054-2. |
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