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A generalization of Douady's formula
Spreading speeds and traveling waves of a parabolic-elliptic chemotaxis system with logistic source on $\mathbb{R}^N$
Department of Mathematics and Statistics, Auburn University, Auburn University, AL 36849, USA |
$\label{IntroEq0-2}\begin{cases}u_{t}=Δ{u}-χ\nabla·(u\nabla{v})+u(1-u),{x}∈\mathbb{R}^N,\\{0}=Δ{v}-v+u,{x}∈\mathbb{R}^N,\end{cases}$ |
$\mathop {\lim }\limits_{t \to \infty } \mathop {\sup }\limits_{|x| \le ct} [|u(x,t;{u_0}) - 1| + |v(x,t;{u_0}) - 1|] = 0\quad \forall {\mkern 1mu} {\mkern 1mu} 0 < c < c_ - ^*(\chi )$ |
$\mathop {\lim }\limits_{t \to \infty } \mathop {\sup }\limits_{|x| \le ct} [u(x,t;{u_0}) + v(x,t;{u_0})] = 0\quad \forall {\mkern 1mu} {\mkern 1mu} c > c_ + ^*(\chi ).$ |
$\mathop {\lim }\limits_{\chi \to 0} {c^*}(\chi ) = \mathop {\lim }\limits_{\chi \to 0} c_ + ^*(\chi ) = \mathop {\lim }\limits_{\chi \to 0} c_ - ^*(\chi ) = 2.$ |
References:
[1] |
S. Ai, W. Huang and Z.-A. Wang,
Reaction, diffusion and chemotaxis in wave propagation, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 1-21.
doi: 10.3934/dcdsb.2015.20.1. |
[2] |
S. Ai and Z.-A. Wang,
Traveling bands for the Keller-Segel model with population growth, Math. Biosci. Eng., 12 (2015), 717-737.
doi: 10.3934/mbe.2015.12.717. |
[3] |
N. Bellomo, A. Bellouquid, Y. Tao and M. Winkler,
Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Math.Models Methods Appl.Sci., 25 (2015), 1663-1763.
doi: 10.1142/S021820251550044X. |
[4] |
H. Berestycki, F. Hamel and G. Nadin,
Asymptotic spreading in heterogeneous diffusive excita media, Journal of Functional Analysis, 255 (2008), 2146-2189.
doi: 10.1016/j.jfa.2008.06.030. |
[5] |
H. Berestycki, F. Hamel and N. Nadirashvili,
The speed of propagation for KPP type problems, Ⅰ -Periodic framework, J. Eur. Math. Soc., 7 (2005), 172-213.
doi: 10.4171/JEMS/26. |
[6] |
H. Berestycki, F. Hamel and N. Nadirashvili,
The speed of propagation for KPP type problems, Ⅱ -General domains, J. Amer. Math. Soc., 23 (2010), 1-34.
doi: 10.1090/S0894-0347-09-00633-X. |
[7] |
H. Berestycki and G. Nadin, Asymptotic spreading for general heterogeneous Fisher-KPP type, preprint. |
[8] |
M. Bramson, Convergence of solutions of the Kolmogorov equation to traveling waves, Mem. Amer. Math. Soc., 44 (1983), iv+190 pp.
doi: 10.1090/memo/0285. |
[9] |
J. I. Diaz and T. Nagai,
Symmetrization in a parabolic-elliptic system related to chemotaxis, Advances in Mathematical Sciences and Applications, 5 (1995), 659-680.
|
[10] |
J. I. Diaz, T. Nagai and J.-M. Rakotoson,
Symmetrization techniques on unbounded domains: Application to a chemotaxis system on $\mathbb{R}^{N}$, J. Differential Equations, 145 (1998), 156-183.
doi: 10.1006/jdeq.1997.3389. |
[11] |
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. |
[12] |
M. Freidlin,
On wave front propagation in periodic media, In: Stochastic analysis and applications, ed. M. Pinsky, Advances in probablity and related topics, 7 (1984), 147-166.
|
[13] |
M. Freidlin and J. Gärtner,
On the propagation of concentration waves in periodic and ramdom media, Soviet Math. Dokl., 20 (1979), 1282-1286.
|
[14] |
A. Friedman,
Partial Differential Equation of Parabolic Type, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1964. |
[15] |
M. Funaki, M. Mimura and T. Tsujikawa,
Travelling front solutions arising in the chemotaxis-growth model, Interfaces Free Bound., 8 (2006), 223-245.
doi: 10.4171/IFB/141. |
[16] |
E. Galakhov, O. Salieva and J. I. Tello,
On a parabolic-elliptic system with chemotaxis and logistic type growth, J. Differential Equations, 261 (2016), 4631-4647.
doi: 10.1016/j.jde.2016.07.008. |
[17] |
D. Henry,
Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag Berlin Heidelberg New York, 1981. |
[18] |
D. Horstmann and A. Stevens,
A constructive approach to traveling waves in chemotaxis, J. Nonlin. Sci., 14 (2004), 1-25.
doi: 10.1007/s00332-003-0548-y. |
[19] |
K. Kanga,
Angela Steven Blowup and global solutions in a chemotaxis-growth system, Nonlinear Analysis, 135 (2016), 57-72.
doi: 10.1016/j.na.2016.01.017. |
[20] |
E. F. Keller and L. A. Segel,
Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415.
doi: 10.1016/0022-5193(70)90092-5. |
[21] |
E. F. Keller and L. A. Segel,
A Model for chemotaxis, J. Theoret. Biol., 30 (1971), 225-234.
doi: 10.1016/0022-5193(71)90050-6. |
[22] |
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.
|
[23] |
J. Lankeit,
Eventual smoothness and asymptotics in a three-dimensional chemotaxis system with logistic source, J.Differential Eq., 258 (2015), 1158-1191.
doi: 10.1016/j.jde.2014.10.016. |
[24] |
J. Li, T. Li and Z.-A. Wang,
Stability of traveling waves of the Keller-Segel system with logarithmic sensitivity, Math. Models Methods Appl. Sci., 24 (2014), 2819-2849.
doi: 10.1142/S0218202514500389. |
[25] |
X. Liang and X.-Q. Zhao,
Asymptotic speeds of spread and traveling waves for monotone semiflows with applications, Comm. Pure Appl. Math., 60 (2007), 1-40.
doi: 10.1002/cpa.20154. |
[26] |
X. Liang and X.-Q. Zhao,
Spreading speeds and traveling waves for abstract monostable evolution systems, Journal of Functional Analysis, 259 (2010), 857-903.
doi: 10.1016/j.jfa.2010.04.018. |
[27] |
B. P. Marchant, J. Norbury and J. A. Sherratt,
Travelling wave solutions to a haptotaxis-dominated model of malignant invasion, Nonlinearity, 14 (2001), 1653-1671.
doi: 10.1088/0951-7715/14/6/313. |
[28] |
G. Nadin,
Traveling fronts in space-time periodic media, J. Math. Pures Anal., 92 (2009), 232-262.
doi: 10.1016/j.matpur.2009.04.002. |
[29] |
G. Nadin, B. Perthame and L. Ryzhik,
Traveling waves for the Keller-Segel system with Fisher birth terms, Interfaces Free Bound, 10 (2008), 517-538.
doi: 10.4171/IFB/200. |
[30] |
T. Nagai, T. Senba and K. Yoshida,
Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcialaj Ekvacioj, 40 (1997), 411-433.
|
[31] |
J. Nolen, M. Rudd and J. Xin,
Existence of KPP fronts in spatially-temporally periodic adevction and variational principle for propagation speeds, Dynamics of PDE, 2 (2005), 1-24.
doi: 10.4310/DPDE.2005.v2.n1.a1. |
[32] |
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 and Continuous Dynamical Systems, 13 (2005), 1217-1234.
doi: 10.3934/dcds.2005.13.1217. |
[33] |
R. B. Salako and W. Shen,
Global existence and asymptotic behavior of classical solutions to a parabolic-elliptic chemotaxis system with logistic source on $\mathbb{R}^{N}$, J. Differential Equations, 262 (2017), 5635-5690.
doi: 10.1016/j.jde.2017.02.011. |
[34] |
D. H. Sattinger,
On the stability of waves of nonlinear parabolic systems, Advances in Math., 22 (1976), 312-355.
doi: 10.1016/0001-8708(76)90098-0. |
[35] |
W. Shen,
Variational principle for spatial spreading speeds and generalized propgating speeds in time almost and space periodic KPP models, Trans. Amer. Math. Soc., 362 (2010), 5125-5168.
doi: 10.1090/S0002-9947-10-04950-0. |
[36] |
W. Shen,
Existence of generalized traveling waves in time recurrent and space periodic monostable equations, J. Appl. Anal. Comput., 1 (2011), 69-93.
|
[37] |
Y. Sugiyama,
Global existence in sub-critical cases and finite time blow up in super critical cases to degenerate Keller-Segel systems, Differential Integral Equations, 19 (2006), 841-876.
|
[38] |
Y. Sugiyama and H. Kunii,
Global Existence and decay properties for a degenerate keller-Segel model with a power factor in drift term, J. Differential Equations, 227 (2006), 333-364.
doi: 10.1016/j.jde.2006.03.003. |
[39] |
J. I. Tello and M. Winkler,
A Chemotaxis System with Logistic Source, Communications in Partial Differential Equations, 32 (2007), 849-877.
doi: 10.1080/03605300701319003. |
[40] |
K. Uchiyama,
The behavior of solutions of some nonlinear diffusion equations for large time, J. Math. Kyoto Univ., 18 (1978), 453-508.
doi: 10.1215/kjm/1250522506. |
[41] |
L. Wang, C. Mu and P. Zheng,
On a quasilinear parabolic-elliptic chemotaxis system with logistic source, J. Differential Equations, 256 (2014), 1847-1872.
doi: 10.1016/j.jde.2013.12.007. |
[42] |
Z.-A. Wang,
Mathematics of traveling waves in chemotaxis-review paper, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 601-641.
doi: 10.3934/dcdsb.2013.18.601. |
[43] |
H. F. Weinberger,
Long-time behavior of a class of biology models, SIAM J. Math. Anal., 13 (1982), 353-396.
doi: 10.1137/0513028. |
[44] |
H. F. Weinberger,
On spreading speeds and traveling waves for growth and migration models in a periodic habitat, J. Math. Biol., 45 (2002), 511-548.
doi: 10.1007/s00285-002-0169-3. |
[45] |
M. Winkler,
Chemotaxis with logistic source: Very weak global solutions and their boundedness properties, J. Math. Anal. Appl., 348 (2008), 708-729.
doi: 10.1016/j.jmaa.2008.07.071. |
[46] |
M. Winkler,
Aggregation vs.global diffusive behavior in the higher-dimensional Keller-Segel model, Journal of Differential Equations, 248 (2010), 2889-2905.
doi: 10.1016/j.jde.2010.02.008. |
[47] |
M. Winkler,
Blow-up in a higher-dimensional chemotaxis system despite logistic growth restriction, Journal of Mathematical Analysis and Applications, 384 (2011), 261-272.
doi: 10.1016/j.jmaa.2011.05.057. |
[48] |
M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl., 100 (2013), 748-767, arXiv: 1112.4156v1.
doi: 10.1016/j.matpur.2013.01.020. |
[49] |
M. Winkler,
How far can chemotactic cross-diffusion enforce exceeding carrying capacities, J. Nonlinear Sci., 24 (2014), 809-855.
doi: 10.1007/s00332-014-9205-x. |
[50] |
T. Yokota and N. Yoshino,
Existence of solutions to chemotaxis dynamics with logistic source, Discrete Contin. Dyn. Syst, (2015), 1125-1133.
doi: 10.3934/proc.2015.1125. |
[51] |
P. Zheng, C. Mu, X. Hu and Y. Tian,
Boundedness of solutions in a chemotaxis system with nonlinear sensitivity and logistic source, J. Math. Anal. Appl., 424 (2015), 509-522.
doi: 10.1016/j.jmaa.2014.11.031. |
[52] |
A. Zlatoš,
Transition fronts in inhomogeneous Fisher-KPP reaction-diffusion equations, J. Math. Pures Appl.(9), 98 (2012), 89-102.
doi: 10.1016/j.matpur.2011.11.007. |
show all references
References:
[1] |
S. Ai, W. Huang and Z.-A. Wang,
Reaction, diffusion and chemotaxis in wave propagation, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 1-21.
doi: 10.3934/dcdsb.2015.20.1. |
[2] |
S. Ai and Z.-A. Wang,
Traveling bands for the Keller-Segel model with population growth, Math. Biosci. Eng., 12 (2015), 717-737.
doi: 10.3934/mbe.2015.12.717. |
[3] |
N. Bellomo, A. Bellouquid, Y. Tao and M. Winkler,
Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Math.Models Methods Appl.Sci., 25 (2015), 1663-1763.
doi: 10.1142/S021820251550044X. |
[4] |
H. Berestycki, F. Hamel and G. Nadin,
Asymptotic spreading in heterogeneous diffusive excita media, Journal of Functional Analysis, 255 (2008), 2146-2189.
doi: 10.1016/j.jfa.2008.06.030. |
[5] |
H. Berestycki, F. Hamel and N. Nadirashvili,
The speed of propagation for KPP type problems, Ⅰ -Periodic framework, J. Eur. Math. Soc., 7 (2005), 172-213.
doi: 10.4171/JEMS/26. |
[6] |
H. Berestycki, F. Hamel and N. Nadirashvili,
The speed of propagation for KPP type problems, Ⅱ -General domains, J. Amer. Math. Soc., 23 (2010), 1-34.
doi: 10.1090/S0894-0347-09-00633-X. |
[7] |
H. Berestycki and G. Nadin, Asymptotic spreading for general heterogeneous Fisher-KPP type, preprint. |
[8] |
M. Bramson, Convergence of solutions of the Kolmogorov equation to traveling waves, Mem. Amer. Math. Soc., 44 (1983), iv+190 pp.
doi: 10.1090/memo/0285. |
[9] |
J. I. Diaz and T. Nagai,
Symmetrization in a parabolic-elliptic system related to chemotaxis, Advances in Mathematical Sciences and Applications, 5 (1995), 659-680.
|
[10] |
J. I. Diaz, T. Nagai and J.-M. Rakotoson,
Symmetrization techniques on unbounded domains: Application to a chemotaxis system on $\mathbb{R}^{N}$, J. Differential Equations, 145 (1998), 156-183.
doi: 10.1006/jdeq.1997.3389. |
[11] |
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. |
[12] |
M. Freidlin,
On wave front propagation in periodic media, In: Stochastic analysis and applications, ed. M. Pinsky, Advances in probablity and related topics, 7 (1984), 147-166.
|
[13] |
M. Freidlin and J. Gärtner,
On the propagation of concentration waves in periodic and ramdom media, Soviet Math. Dokl., 20 (1979), 1282-1286.
|
[14] |
A. Friedman,
Partial Differential Equation of Parabolic Type, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1964. |
[15] |
M. Funaki, M. Mimura and T. Tsujikawa,
Travelling front solutions arising in the chemotaxis-growth model, Interfaces Free Bound., 8 (2006), 223-245.
doi: 10.4171/IFB/141. |
[16] |
E. Galakhov, O. Salieva and J. I. Tello,
On a parabolic-elliptic system with chemotaxis and logistic type growth, J. Differential Equations, 261 (2016), 4631-4647.
doi: 10.1016/j.jde.2016.07.008. |
[17] |
D. Henry,
Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag Berlin Heidelberg New York, 1981. |
[18] |
D. Horstmann and A. Stevens,
A constructive approach to traveling waves in chemotaxis, J. Nonlin. Sci., 14 (2004), 1-25.
doi: 10.1007/s00332-003-0548-y. |
[19] |
K. Kanga,
Angela Steven Blowup and global solutions in a chemotaxis-growth system, Nonlinear Analysis, 135 (2016), 57-72.
doi: 10.1016/j.na.2016.01.017. |
[20] |
E. F. Keller and L. A. Segel,
Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415.
doi: 10.1016/0022-5193(70)90092-5. |
[21] |
E. F. Keller and L. A. Segel,
A Model for chemotaxis, J. Theoret. Biol., 30 (1971), 225-234.
doi: 10.1016/0022-5193(71)90050-6. |
[22] |
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.
|
[23] |
J. Lankeit,
Eventual smoothness and asymptotics in a three-dimensional chemotaxis system with logistic source, J.Differential Eq., 258 (2015), 1158-1191.
doi: 10.1016/j.jde.2014.10.016. |
[24] |
J. Li, T. Li and Z.-A. Wang,
Stability of traveling waves of the Keller-Segel system with logarithmic sensitivity, Math. Models Methods Appl. Sci., 24 (2014), 2819-2849.
doi: 10.1142/S0218202514500389. |
[25] |
X. Liang and X.-Q. Zhao,
Asymptotic speeds of spread and traveling waves for monotone semiflows with applications, Comm. Pure Appl. Math., 60 (2007), 1-40.
doi: 10.1002/cpa.20154. |
[26] |
X. Liang and X.-Q. Zhao,
Spreading speeds and traveling waves for abstract monostable evolution systems, Journal of Functional Analysis, 259 (2010), 857-903.
doi: 10.1016/j.jfa.2010.04.018. |
[27] |
B. P. Marchant, J. Norbury and J. A. Sherratt,
Travelling wave solutions to a haptotaxis-dominated model of malignant invasion, Nonlinearity, 14 (2001), 1653-1671.
doi: 10.1088/0951-7715/14/6/313. |
[28] |
G. Nadin,
Traveling fronts in space-time periodic media, J. Math. Pures Anal., 92 (2009), 232-262.
doi: 10.1016/j.matpur.2009.04.002. |
[29] |
G. Nadin, B. Perthame and L. Ryzhik,
Traveling waves for the Keller-Segel system with Fisher birth terms, Interfaces Free Bound, 10 (2008), 517-538.
doi: 10.4171/IFB/200. |
[30] |
T. Nagai, T. Senba and K. Yoshida,
Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcialaj Ekvacioj, 40 (1997), 411-433.
|
[31] |
J. Nolen, M. Rudd and J. Xin,
Existence of KPP fronts in spatially-temporally periodic adevction and variational principle for propagation speeds, Dynamics of PDE, 2 (2005), 1-24.
doi: 10.4310/DPDE.2005.v2.n1.a1. |
[32] |
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 and Continuous Dynamical Systems, 13 (2005), 1217-1234.
doi: 10.3934/dcds.2005.13.1217. |
[33] |
R. B. Salako and W. Shen,
Global existence and asymptotic behavior of classical solutions to a parabolic-elliptic chemotaxis system with logistic source on $\mathbb{R}^{N}$, J. Differential Equations, 262 (2017), 5635-5690.
doi: 10.1016/j.jde.2017.02.011. |
[34] |
D. H. Sattinger,
On the stability of waves of nonlinear parabolic systems, Advances in Math., 22 (1976), 312-355.
doi: 10.1016/0001-8708(76)90098-0. |
[35] |
W. Shen,
Variational principle for spatial spreading speeds and generalized propgating speeds in time almost and space periodic KPP models, Trans. Amer. Math. Soc., 362 (2010), 5125-5168.
doi: 10.1090/S0002-9947-10-04950-0. |
[36] |
W. Shen,
Existence of generalized traveling waves in time recurrent and space periodic monostable equations, J. Appl. Anal. Comput., 1 (2011), 69-93.
|
[37] |
Y. Sugiyama,
Global existence in sub-critical cases and finite time blow up in super critical cases to degenerate Keller-Segel systems, Differential Integral Equations, 19 (2006), 841-876.
|
[38] |
Y. Sugiyama and H. Kunii,
Global Existence and decay properties for a degenerate keller-Segel model with a power factor in drift term, J. Differential Equations, 227 (2006), 333-364.
doi: 10.1016/j.jde.2006.03.003. |
[39] |
J. I. Tello and M. Winkler,
A Chemotaxis System with Logistic Source, Communications in Partial Differential Equations, 32 (2007), 849-877.
doi: 10.1080/03605300701319003. |
[40] |
K. Uchiyama,
The behavior of solutions of some nonlinear diffusion equations for large time, J. Math. Kyoto Univ., 18 (1978), 453-508.
doi: 10.1215/kjm/1250522506. |
[41] |
L. Wang, C. Mu and P. Zheng,
On a quasilinear parabolic-elliptic chemotaxis system with logistic source, J. Differential Equations, 256 (2014), 1847-1872.
doi: 10.1016/j.jde.2013.12.007. |
[42] |
Z.-A. Wang,
Mathematics of traveling waves in chemotaxis-review paper, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 601-641.
doi: 10.3934/dcdsb.2013.18.601. |
[43] |
H. F. Weinberger,
Long-time behavior of a class of biology models, SIAM J. Math. Anal., 13 (1982), 353-396.
doi: 10.1137/0513028. |
[44] |
H. F. Weinberger,
On spreading speeds and traveling waves for growth and migration models in a periodic habitat, J. Math. Biol., 45 (2002), 511-548.
doi: 10.1007/s00285-002-0169-3. |
[45] |
M. Winkler,
Chemotaxis with logistic source: Very weak global solutions and their boundedness properties, J. Math. Anal. Appl., 348 (2008), 708-729.
doi: 10.1016/j.jmaa.2008.07.071. |
[46] |
M. Winkler,
Aggregation vs.global diffusive behavior in the higher-dimensional Keller-Segel model, Journal of Differential Equations, 248 (2010), 2889-2905.
doi: 10.1016/j.jde.2010.02.008. |
[47] |
M. Winkler,
Blow-up in a higher-dimensional chemotaxis system despite logistic growth restriction, Journal of Mathematical Analysis and Applications, 384 (2011), 261-272.
doi: 10.1016/j.jmaa.2011.05.057. |
[48] |
M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl., 100 (2013), 748-767, arXiv: 1112.4156v1.
doi: 10.1016/j.matpur.2013.01.020. |
[49] |
M. Winkler,
How far can chemotactic cross-diffusion enforce exceeding carrying capacities, J. Nonlinear Sci., 24 (2014), 809-855.
doi: 10.1007/s00332-014-9205-x. |
[50] |
T. Yokota and N. Yoshino,
Existence of solutions to chemotaxis dynamics with logistic source, Discrete Contin. Dyn. Syst, (2015), 1125-1133.
doi: 10.3934/proc.2015.1125. |
[51] |
P. Zheng, C. Mu, X. Hu and Y. Tian,
Boundedness of solutions in a chemotaxis system with nonlinear sensitivity and logistic source, J. Math. Anal. Appl., 424 (2015), 509-522.
doi: 10.1016/j.jmaa.2014.11.031. |
[52] |
A. Zlatoš,
Transition fronts in inhomogeneous Fisher-KPP reaction-diffusion equations, J. Math. Pures Appl.(9), 98 (2012), 89-102.
doi: 10.1016/j.matpur.2011.11.007. |
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