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On the optimal control for a semilinear equation with cost depending on the free boundary
Spreading speed revisited: Analysis of a free boundary model
1. | School of Science and Technology, University of New England, Armidale, NSW 2351, Australia, Australia, Australia |
References:
[1] |
D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation,, in, 446 (1975), 5.
|
[2] |
D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusions arising in population genetics,, Adv. Math., 30 (1978), 33.
doi: 10.1016/0001-8708(78)90130-5. |
[3] |
H. Berestycki, F. Hamel and H. Matano, Bistable traveling waves around an obstacle,, Comm. Pure Appl. Math., 62 (2009), 729.
doi: 10.1002/cpa.20275. |
[4] |
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. |
[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), 173.
doi: 10.4171/JEMS/26. |
[6] |
X. F. Chen and A. Friedman, A free boundary problem arising in a model of wound healing,, SIAM J. Math. Anal., 32 (2000), 778.
doi: 10.1137/S0036141099351693. |
[7] |
Y. Du, "Order Structure and Topological Methods in Nonlinear Partial Differential Equations,", 1, 1 (2006).
doi: 10.1142/9789812774446. |
[8] |
Y. Du and Z. M. Guo, Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary, II,, J. Diff. Eqns., 250 (2011), 4336.
doi: 10.1016/j.jde.2011.02.011. |
[9] |
Y. Du and Z. M. Guo, The Stefan problem for the Fisher-KPP equation,, J. Diff. Eqns., 253 (2012), 996.
doi: 10.1016/j.jde.2012.04.014. |
[10] |
Y. Du, Z. M. Guo and R. Peng, A diffusive logistic model with a free boundary in time-periodic environment,, preprint, (2011). |
[11] |
Y. Du and Z. G. Lin, Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary,, SIAM J. Math. Anal., 42 (2010), 1305.
doi: 10.1137/090771089. |
[12] |
Y. Du and B. Lou, Spreading and vanishing in nonlinear diffusion problems with free boundaries,, preprint, (2011). |
[13] |
Y. Du and H. Matano, Convergence and sharp thresholds for propagation in nonlinear diffusion problems,, J. European Math. Soc., 12 (2010), 279.
doi: 10.4171/JEMS/198. |
[14] |
X. Fauvergue, J-C. Malausa, L. Giuge and F. Courchamp, Invading parasitoids suffer no Allee effect: A manipulative field experiment,, Ecology, 88 (2008), 2392. |
[15] |
I. Filin, R. D. Holt and M. Barfield, The relation of density regulation to habitat specialization, evolution of a speciesrange, and the dynamics of biological invasions,, Am. Nat., 172 (2008), 233. |
[16] |
R. A. Fisher, The wave of advance of advantageous genes,, Ann. Eugenics, 7 (1937), 335. |
[17] |
K. P. Hadeler and F. Rothe, Travelling fronts in nonlinear diffusion equations,, J. Math. Biol., 2 (1975), 251.
|
[18] |
D. Hilhorst, M. Iida, M. Mimura and H. Ninomiya, A competition-diffusion system approximation to the classical two-phase Stefan problem,, Japan J. Indust. Appl. Math., 18 (2001), 161.
doi: 10.1007/BF03168569. |
[19] |
A. N. Kolmogorov, I. G. Petrovsky and N. S. Piskunov, Ètude de l'équation de la diffusion avec croissance de la quantitéde matière et son application à un problème biologique,, Bull. Univ. Moscou Sér. Internat. A1 (1937), A1 (1937), 1. |
[20] |
A. M. Kramer, B. Dennis, A. M. Liebhold and J. M. Drake, The evidence for Allee effects,, Popul. Ecol., 51 (2009), 341. |
[21] |
M. A. Lewis and P. Kareiva, Allee dynamics and the spreading of invasive organisms,, Theor. Population Bio., 43 (1993), 141. |
[22] |
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.
doi: 10.1002/cpa.20154. |
[23] |
Z. G. Lin, A free boundary problem for a predator-prey model,, Nonlinearity, 20 (2007), 1883.
doi: 10.1088/0951-7715/20/8/004. |
[24] |
J. L. Lockwood, M. F. Hoopes and M. P. Marchetti, "Invasion Ecology,", Blackwell Publishing, (2007). |
[25] |
M. Mimura, Y. Yamada and S. Yotsutani, A free boundary problem in ecology,, Japan J. Appl. Math., 2 (1985), 151.
doi: 10.1007/BF03167042. |
[26] |
R. Peng and X. Q. Zhao, The diffusive logistic model with a free boundary and seasonal succession,, Discrete Cont. Dyn. Syst. A., (). |
[27] |
L. I. Rubinstein, "The Stefan Problem,", Amer. Math. Soc., (1971). |
[28] |
N. Shigesada and K. Kawasaki, "Biological Invasions: Theory and Practice,", Oxford Series in Ecology and Evolution, (1997). |
[29] |
J. G. Skellam, Random dispersal in theoretical populations,, Biometrika, 38 (1951), 196.
|
[30] |
H. F. Weinberger, On spreading speeds and traveling waves for growth and migration models in a periodic habitat,, J. Math. Biol., 45 (2002), 511.
doi: 10.1007/s00285-002-0169-3. |
[31] |
H. F. Weinberger, M. A. Lewis and B. Li, Anomalous spreading speeds of cooperative recursion systems,, J. Math. Biol., 55 (2007), 207.
doi: 10.1007/s00285-007-0078-6. |
[32] |
J. X. Xin, Front propagation in heterogeneous media,, SIAM Rev., 42 (2000), 161.
doi: 10.1137/S0036144599364296. |
show all references
References:
[1] |
D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation,, in, 446 (1975), 5.
|
[2] |
D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusions arising in population genetics,, Adv. Math., 30 (1978), 33.
doi: 10.1016/0001-8708(78)90130-5. |
[3] |
H. Berestycki, F. Hamel and H. Matano, Bistable traveling waves around an obstacle,, Comm. Pure Appl. Math., 62 (2009), 729.
doi: 10.1002/cpa.20275. |
[4] |
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. |
[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), 173.
doi: 10.4171/JEMS/26. |
[6] |
X. F. Chen and A. Friedman, A free boundary problem arising in a model of wound healing,, SIAM J. Math. Anal., 32 (2000), 778.
doi: 10.1137/S0036141099351693. |
[7] |
Y. Du, "Order Structure and Topological Methods in Nonlinear Partial Differential Equations,", 1, 1 (2006).
doi: 10.1142/9789812774446. |
[8] |
Y. Du and Z. M. Guo, Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary, II,, J. Diff. Eqns., 250 (2011), 4336.
doi: 10.1016/j.jde.2011.02.011. |
[9] |
Y. Du and Z. M. Guo, The Stefan problem for the Fisher-KPP equation,, J. Diff. Eqns., 253 (2012), 996.
doi: 10.1016/j.jde.2012.04.014. |
[10] |
Y. Du, Z. M. Guo and R. Peng, A diffusive logistic model with a free boundary in time-periodic environment,, preprint, (2011). |
[11] |
Y. Du and Z. G. Lin, Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary,, SIAM J. Math. Anal., 42 (2010), 1305.
doi: 10.1137/090771089. |
[12] |
Y. Du and B. Lou, Spreading and vanishing in nonlinear diffusion problems with free boundaries,, preprint, (2011). |
[13] |
Y. Du and H. Matano, Convergence and sharp thresholds for propagation in nonlinear diffusion problems,, J. European Math. Soc., 12 (2010), 279.
doi: 10.4171/JEMS/198. |
[14] |
X. Fauvergue, J-C. Malausa, L. Giuge and F. Courchamp, Invading parasitoids suffer no Allee effect: A manipulative field experiment,, Ecology, 88 (2008), 2392. |
[15] |
I. Filin, R. D. Holt and M. Barfield, The relation of density regulation to habitat specialization, evolution of a speciesrange, and the dynamics of biological invasions,, Am. Nat., 172 (2008), 233. |
[16] |
R. A. Fisher, The wave of advance of advantageous genes,, Ann. Eugenics, 7 (1937), 335. |
[17] |
K. P. Hadeler and F. Rothe, Travelling fronts in nonlinear diffusion equations,, J. Math. Biol., 2 (1975), 251.
|
[18] |
D. Hilhorst, M. Iida, M. Mimura and H. Ninomiya, A competition-diffusion system approximation to the classical two-phase Stefan problem,, Japan J. Indust. Appl. Math., 18 (2001), 161.
doi: 10.1007/BF03168569. |
[19] |
A. N. Kolmogorov, I. G. Petrovsky and N. S. Piskunov, Ètude de l'équation de la diffusion avec croissance de la quantitéde matière et son application à un problème biologique,, Bull. Univ. Moscou Sér. Internat. A1 (1937), A1 (1937), 1. |
[20] |
A. M. Kramer, B. Dennis, A. M. Liebhold and J. M. Drake, The evidence for Allee effects,, Popul. Ecol., 51 (2009), 341. |
[21] |
M. A. Lewis and P. Kareiva, Allee dynamics and the spreading of invasive organisms,, Theor. Population Bio., 43 (1993), 141. |
[22] |
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.
doi: 10.1002/cpa.20154. |
[23] |
Z. G. Lin, A free boundary problem for a predator-prey model,, Nonlinearity, 20 (2007), 1883.
doi: 10.1088/0951-7715/20/8/004. |
[24] |
J. L. Lockwood, M. F. Hoopes and M. P. Marchetti, "Invasion Ecology,", Blackwell Publishing, (2007). |
[25] |
M. Mimura, Y. Yamada and S. Yotsutani, A free boundary problem in ecology,, Japan J. Appl. Math., 2 (1985), 151.
doi: 10.1007/BF03167042. |
[26] |
R. Peng and X. Q. Zhao, The diffusive logistic model with a free boundary and seasonal succession,, Discrete Cont. Dyn. Syst. A., (). |
[27] |
L. I. Rubinstein, "The Stefan Problem,", Amer. Math. Soc., (1971). |
[28] |
N. Shigesada and K. Kawasaki, "Biological Invasions: Theory and Practice,", Oxford Series in Ecology and Evolution, (1997). |
[29] |
J. G. Skellam, Random dispersal in theoretical populations,, Biometrika, 38 (1951), 196.
|
[30] |
H. F. Weinberger, On spreading speeds and traveling waves for growth and migration models in a periodic habitat,, J. Math. Biol., 45 (2002), 511.
doi: 10.1007/s00285-002-0169-3. |
[31] |
H. F. Weinberger, M. A. Lewis and B. Li, Anomalous spreading speeds of cooperative recursion systems,, J. Math. Biol., 55 (2007), 207.
doi: 10.1007/s00285-007-0078-6. |
[32] |
J. X. Xin, Front propagation in heterogeneous media,, SIAM Rev., 42 (2000), 161.
doi: 10.1137/S0036144599364296. |
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