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Well-posedness and stabilization of an Euler-Bernoulli equation with a localized nonlinear dissipation involving the $p$-Laplacian
Traveling curved fronts in monotone bistable systems
1. | School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000 |
References:
[1] |
E. O. Alcahrani, F. A. Davidson and N. Dodds, Travelling waves in near-degenerate bistable competition models, Math. Model. Nat. Phenom., 5 (2010), 13-35.
doi: 10.1051/mmnp/20105502. |
[2] |
A. Bonnet and F. Hamel, Existence of nonplanar solutions of a simple model of premixed Bunsen flames, SIAM J. Math. Anal., 31 (1999), 80-118.
doi: 10.1137/S0036141097316391. |
[3] |
P. K. Brazhnik and J. J. Tyson, On traveling wave solutions of Fisher's equation in two spatial dimensions, SIAM J. Appl. Math., 60 (2000), 371-391.
doi: 10.1137/S0036139997325497. |
[4] |
G. Chapuisat, Existence and nonexistence of curved front solution of a biological equation, J. Differential Equations, 236 (2007), 237-279.
doi: 10.1016/j.jde.2007.01.021. |
[5] |
X. Chen, J.-S. Guo, F. Hamel, H. Ninomiya and J.-M. Roquejoffre, Traveling waves with paraboloid like interfaces for balanced bistable dynamics, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 369-393. |
[6] |
C. Conley and R. Gardner, An application of the generalized Morse index to travelling wave solutions of a competitive reaction-diffusion model, Indiana Univ. Math. J., 33 (1984), 319-343.
doi: 10.1512/iumj.1984.33.33018. |
[7] |
M. Feinberg and D. Terman, Traveling composition waves on isothermal catalyst surfaces, Arch. Rational Mech. Anal., 116 (1991), 35-69.
doi: 10.1007/BF00375602. |
[8] |
P. C. Fife, "Dynamics of Internal Layers and Diffusive Interfaces," CBMS-NSF Regional Conference Series in Applied Mathematics, 53, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1988. |
[9] |
P. C. Fife and J. B. McLeod, The approach of solutions of nonlinear diffusion equations to travelling wave solutions, Arch. Rational Mech. Anal., 65 (1977), 355-361. |
[10] |
P. C. Fife and J. B. McLeod, A phase plane discussion of convergence to travelling fronts for nonlinear diffusions,, Arch. Rational Mech. Anal., 75 (): 281.
|
[11] |
A. Friedman, "Partial Differential Equations Of Parabolic Type," Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. |
[12] |
R. A. Gardner, Existence and stability of travelling wave solutions of competition models: A degree theoretical approach, J. Differential Equations, 44 (1982), 343-364.
doi: 10.1016/0022-0396(82)90001-8. |
[13] |
S. A. Gourley and S. Ruan, Convergence and travelling fronts in functional differential equations with nonlocal terms: A competition model, SIAM J. Math. Anal., 35 (2003), 806-822.
doi: 10.1137/S003614100139991. |
[14] |
C. Gui, Symmetry of travelling wave solutions to the Allen-Cahn equation in $\mathbbR^2$,, preprint, ().
|
[15] |
F. Hamel, R. Monneau and J.-M. Roquejoffre, Stability of travelling waves in a model for conical flames in two space dimensions, Ann. Sci. École Norm. Sup. (4), 37 (2004), 469-506. |
[16] |
F. Hamel, R. Monneau and J.-M. Roquejoffre, Existence and qualitative properties of multidimensional conical bistable fronts, Discrete Contin. Dyn. Syst., 13 (2005), 1069-1096.
doi: 10.3934/dcds.2005.13.1069. |
[17] |
F. Hamel, R. Monneau and J.-M. Roquejoffre, Asymptotic properties and classification of bistable fronts with Lipschitz level sets, Discrete Contin. Dyn. Syst., 14 (2006), 75-92. |
[18] |
F. Hamel and N. Nadirashvili, Travelling fronts and entire solutions of the Fisher-KPP equation in $\mathbbR^N$, Arch. Rational Mech. Anal., 157 (2001), 91-163.
doi: 10.1007/PL00004238. |
[19] |
F. Hamel and J.-M. Roquejoffre, Heteroclinic connections for multidimensional bistable reaction-diffusion equations, Disc. Cont. Dyn. Systems Ser. S, 4 (2011), 101-123.
doi: 10.3934/dcdss.2011.4.101. |
[20] |
M. Haragus and A. Scheel, A bifurcation approach to non-planar traveling waves in reaction-diffusion systems, GAMM-Mitt., 30 (2007), 75-95.
doi: 10.1002/gamm.200790012. |
[21] |
M. Haragus and A. Scheel, Almost planar waves in anisotropic media, Comm. Partial Differential Equations, 31 (2006), 791-815. |
[22] |
M. Haragus and A. Scheel, Corner defects in almost planar interface propagation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23 (2006), 283-329. |
[23] |
R. Huang, Stability of travelling fronts of the Fisher-KPP equation in $\mathbbR^N$, Nonlinear Diff. Eq. Appl., 15 (2008), 599-622. |
[24] |
T. Kapitula, Multidimensional stability of planar travelling waves, Trans. Amer. Math. Soc., 349 (1997), 257-269.
doi: 10.1090/S0002-9947-97-01668-1. |
[25] |
Y. Kan-on, Parameter dependence of propagation speed of travelling waves for competition-diffusion equations, SIAM J. Math. Anal., 26 (1995), 340-363.
doi: 10.1137/S0036141093244556. |
[26] |
Y. Kan-on and Q. Fang, Stability of monotone travelling waves for competition-diffusion equations, Japan J. Indust. Appl. Math., 13 (1996), 343-349.
doi: 10.1007/BF03167252. |
[27] |
C. D. Levermore and J. X. Xin, Multidimensional stability of traveling waves in a bistable reaction-diffusion equation. II, Comm. Partial Differential Equations, 17 (1992), 1901-1924. |
[28] |
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. |
[29] |
G. Lin and W.-T. Li, Bistable wavefronts in a diffusive and competitive Lotka-Volterra type system with nonlocal delays, J. Differential Equations, 244 (2008), 487-513.
doi: 10.1016/j.jde.2007.10.019. |
[30] |
R. H. Martin, Jr. and H. L. Smith, Abstract functional-differential equations and reaction-diffusion systems, Trans. Amer. Math. Soc., 321 (1990), 1-44.
doi: 10.2307/2001590. |
[31] |
K. Mischaikow and V. Hutson, Travelling waves for mutualist species, SIAM J. Math. Anal., 24 (1993), 987-1008.
doi: 10.1137/0524059. |
[32] |
Y. Morita and H. Ninomiya, Monostable-type traveling waves of bistable reaction-diffusion equations in the multi-dimensional space, Bull. Inst. Math. Acad. Sinica (N.S.), 3 (2008), 567-584. |
[33] |
Y. Morita and K. Tachibana, An entire solution to the Lotka-Volterra competition-diffusion equations, SIAM J. Math. Anal., 40 (2009), 2217-2240.
doi: 10.1137/080723715. |
[34] |
J. D. Murray, "Mathematical Biology,'' Biomathematics, 19, Springer-Verlag, Berlin, 1989. |
[35] |
H. Ninomiya and M. Taniguchi, Existence and global stability of traveling curved fronts in the Allen-Cahn equations, J. Differential Equations, 213 (2005), 204-233.
doi: 10.1016/j.jde.2004.06.011. |
[36] |
H. Ninomiya and M. Taniguchi, Global stability of traveling curved fronts in the Allen-Cahn equations, Discrete Contnu. Dyn. Syst., 15 (2006), 829-832. |
[37] |
T. Ogiwara and H. Matano, Monotonicity and convergence results in order-preserving systems in the presence of symmetry, Discrete Contnu. Dyn. Syst., 5 (1999), 1-34. |
[38] |
M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations,'' Prentice-Hall, Inc., Englewood Cliffs, N.J., 1967. |
[39] |
J.-M. Roquejoffre and V. Roussier-Michon, Nontrivial large-time behaviour in bistable reaction-diffusion equations, Ann. Mat. Pura Appl. (4), 188 (2009), 207-233. |
[40] |
D. H. Sattinger, Monotone methods in nonlinear elliptic and parabolic boundary value problems,, Indiana University Math. J., 21 (): 979.
doi: 10.1512/iumj.1972.21.21079. |
[41] |
M. Taniguchi, Traveling fronts of pyramidal shapes in the Allen-Cahn equations, SIAM J. Math. Anal., 39 (2007), 319-344.
doi: 10.1137/060661788. |
[42] |
M. Taniguchi, The uniqueness and asymptotic stability of pyramidal traveling fronts in the Allen-Cahn equations, J. Differential Euqations, 246 (2009), 2103-2130. |
[43] |
J.-C. Tsai, Global exponential stability of traveling waves in monotone bistable systems, Discrete Contnu. Dyn. Syst., 21 (2008), 601-623.
doi: 10.3934/dcds.2008.21.601. |
[44] |
A. I. Volpert, V. A. Volpert and V. A. Volpert, "Traveling Wave Solutions of Parabolic Systems,'' Translations of Mathematical Monographs, 140, Amer. Math. Soc., Providence, RI, 1994. |
[45] |
M. Wang and G. Lv, Entire solutions of a diffusive and competitive Lotka-Volterra type system with nonlocal delays, Nonlinearity, 23 (2010), 1609-1630.
doi: 10.1088/0951-7715/23/7/005. |
[46] |
Z.-C. Wang and J. Wu, Periodic traveling curved fronts in reaction-diffusion equation with bistable time-periodic nonlinearity, J. Differential Equations, 250 (2011), 3196-3229.
doi: 10.1016/j.jde.2011.01.017. |
[47] |
J. X. Xin, Multidimensional stability of traveling waves in a bistable reaction-diffusion equation. I, Comm. Partial Differential Equations, 17 (1992), 1889-1899. |
show all references
References:
[1] |
E. O. Alcahrani, F. A. Davidson and N. Dodds, Travelling waves in near-degenerate bistable competition models, Math. Model. Nat. Phenom., 5 (2010), 13-35.
doi: 10.1051/mmnp/20105502. |
[2] |
A. Bonnet and F. Hamel, Existence of nonplanar solutions of a simple model of premixed Bunsen flames, SIAM J. Math. Anal., 31 (1999), 80-118.
doi: 10.1137/S0036141097316391. |
[3] |
P. K. Brazhnik and J. J. Tyson, On traveling wave solutions of Fisher's equation in two spatial dimensions, SIAM J. Appl. Math., 60 (2000), 371-391.
doi: 10.1137/S0036139997325497. |
[4] |
G. Chapuisat, Existence and nonexistence of curved front solution of a biological equation, J. Differential Equations, 236 (2007), 237-279.
doi: 10.1016/j.jde.2007.01.021. |
[5] |
X. Chen, J.-S. Guo, F. Hamel, H. Ninomiya and J.-M. Roquejoffre, Traveling waves with paraboloid like interfaces for balanced bistable dynamics, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 369-393. |
[6] |
C. Conley and R. Gardner, An application of the generalized Morse index to travelling wave solutions of a competitive reaction-diffusion model, Indiana Univ. Math. J., 33 (1984), 319-343.
doi: 10.1512/iumj.1984.33.33018. |
[7] |
M. Feinberg and D. Terman, Traveling composition waves on isothermal catalyst surfaces, Arch. Rational Mech. Anal., 116 (1991), 35-69.
doi: 10.1007/BF00375602. |
[8] |
P. C. Fife, "Dynamics of Internal Layers and Diffusive Interfaces," CBMS-NSF Regional Conference Series in Applied Mathematics, 53, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1988. |
[9] |
P. C. Fife and J. B. McLeod, The approach of solutions of nonlinear diffusion equations to travelling wave solutions, Arch. Rational Mech. Anal., 65 (1977), 355-361. |
[10] |
P. C. Fife and J. B. McLeod, A phase plane discussion of convergence to travelling fronts for nonlinear diffusions,, Arch. Rational Mech. Anal., 75 (): 281.
|
[11] |
A. Friedman, "Partial Differential Equations Of Parabolic Type," Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. |
[12] |
R. A. Gardner, Existence and stability of travelling wave solutions of competition models: A degree theoretical approach, J. Differential Equations, 44 (1982), 343-364.
doi: 10.1016/0022-0396(82)90001-8. |
[13] |
S. A. Gourley and S. Ruan, Convergence and travelling fronts in functional differential equations with nonlocal terms: A competition model, SIAM J. Math. Anal., 35 (2003), 806-822.
doi: 10.1137/S003614100139991. |
[14] |
C. Gui, Symmetry of travelling wave solutions to the Allen-Cahn equation in $\mathbbR^2$,, preprint, ().
|
[15] |
F. Hamel, R. Monneau and J.-M. Roquejoffre, Stability of travelling waves in a model for conical flames in two space dimensions, Ann. Sci. École Norm. Sup. (4), 37 (2004), 469-506. |
[16] |
F. Hamel, R. Monneau and J.-M. Roquejoffre, Existence and qualitative properties of multidimensional conical bistable fronts, Discrete Contin. Dyn. Syst., 13 (2005), 1069-1096.
doi: 10.3934/dcds.2005.13.1069. |
[17] |
F. Hamel, R. Monneau and J.-M. Roquejoffre, Asymptotic properties and classification of bistable fronts with Lipschitz level sets, Discrete Contin. Dyn. Syst., 14 (2006), 75-92. |
[18] |
F. Hamel and N. Nadirashvili, Travelling fronts and entire solutions of the Fisher-KPP equation in $\mathbbR^N$, Arch. Rational Mech. Anal., 157 (2001), 91-163.
doi: 10.1007/PL00004238. |
[19] |
F. Hamel and J.-M. Roquejoffre, Heteroclinic connections for multidimensional bistable reaction-diffusion equations, Disc. Cont. Dyn. Systems Ser. S, 4 (2011), 101-123.
doi: 10.3934/dcdss.2011.4.101. |
[20] |
M. Haragus and A. Scheel, A bifurcation approach to non-planar traveling waves in reaction-diffusion systems, GAMM-Mitt., 30 (2007), 75-95.
doi: 10.1002/gamm.200790012. |
[21] |
M. Haragus and A. Scheel, Almost planar waves in anisotropic media, Comm. Partial Differential Equations, 31 (2006), 791-815. |
[22] |
M. Haragus and A. Scheel, Corner defects in almost planar interface propagation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23 (2006), 283-329. |
[23] |
R. Huang, Stability of travelling fronts of the Fisher-KPP equation in $\mathbbR^N$, Nonlinear Diff. Eq. Appl., 15 (2008), 599-622. |
[24] |
T. Kapitula, Multidimensional stability of planar travelling waves, Trans. Amer. Math. Soc., 349 (1997), 257-269.
doi: 10.1090/S0002-9947-97-01668-1. |
[25] |
Y. Kan-on, Parameter dependence of propagation speed of travelling waves for competition-diffusion equations, SIAM J. Math. Anal., 26 (1995), 340-363.
doi: 10.1137/S0036141093244556. |
[26] |
Y. Kan-on and Q. Fang, Stability of monotone travelling waves for competition-diffusion equations, Japan J. Indust. Appl. Math., 13 (1996), 343-349.
doi: 10.1007/BF03167252. |
[27] |
C. D. Levermore and J. X. Xin, Multidimensional stability of traveling waves in a bistable reaction-diffusion equation. II, Comm. Partial Differential Equations, 17 (1992), 1901-1924. |
[28] |
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. |
[29] |
G. Lin and W.-T. Li, Bistable wavefronts in a diffusive and competitive Lotka-Volterra type system with nonlocal delays, J. Differential Equations, 244 (2008), 487-513.
doi: 10.1016/j.jde.2007.10.019. |
[30] |
R. H. Martin, Jr. and H. L. Smith, Abstract functional-differential equations and reaction-diffusion systems, Trans. Amer. Math. Soc., 321 (1990), 1-44.
doi: 10.2307/2001590. |
[31] |
K. Mischaikow and V. Hutson, Travelling waves for mutualist species, SIAM J. Math. Anal., 24 (1993), 987-1008.
doi: 10.1137/0524059. |
[32] |
Y. Morita and H. Ninomiya, Monostable-type traveling waves of bistable reaction-diffusion equations in the multi-dimensional space, Bull. Inst. Math. Acad. Sinica (N.S.), 3 (2008), 567-584. |
[33] |
Y. Morita and K. Tachibana, An entire solution to the Lotka-Volterra competition-diffusion equations, SIAM J. Math. Anal., 40 (2009), 2217-2240.
doi: 10.1137/080723715. |
[34] |
J. D. Murray, "Mathematical Biology,'' Biomathematics, 19, Springer-Verlag, Berlin, 1989. |
[35] |
H. Ninomiya and M. Taniguchi, Existence and global stability of traveling curved fronts in the Allen-Cahn equations, J. Differential Equations, 213 (2005), 204-233.
doi: 10.1016/j.jde.2004.06.011. |
[36] |
H. Ninomiya and M. Taniguchi, Global stability of traveling curved fronts in the Allen-Cahn equations, Discrete Contnu. Dyn. Syst., 15 (2006), 829-832. |
[37] |
T. Ogiwara and H. Matano, Monotonicity and convergence results in order-preserving systems in the presence of symmetry, Discrete Contnu. Dyn. Syst., 5 (1999), 1-34. |
[38] |
M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations,'' Prentice-Hall, Inc., Englewood Cliffs, N.J., 1967. |
[39] |
J.-M. Roquejoffre and V. Roussier-Michon, Nontrivial large-time behaviour in bistable reaction-diffusion equations, Ann. Mat. Pura Appl. (4), 188 (2009), 207-233. |
[40] |
D. H. Sattinger, Monotone methods in nonlinear elliptic and parabolic boundary value problems,, Indiana University Math. J., 21 (): 979.
doi: 10.1512/iumj.1972.21.21079. |
[41] |
M. Taniguchi, Traveling fronts of pyramidal shapes in the Allen-Cahn equations, SIAM J. Math. Anal., 39 (2007), 319-344.
doi: 10.1137/060661788. |
[42] |
M. Taniguchi, The uniqueness and asymptotic stability of pyramidal traveling fronts in the Allen-Cahn equations, J. Differential Euqations, 246 (2009), 2103-2130. |
[43] |
J.-C. Tsai, Global exponential stability of traveling waves in monotone bistable systems, Discrete Contnu. Dyn. Syst., 21 (2008), 601-623.
doi: 10.3934/dcds.2008.21.601. |
[44] |
A. I. Volpert, V. A. Volpert and V. A. Volpert, "Traveling Wave Solutions of Parabolic Systems,'' Translations of Mathematical Monographs, 140, Amer. Math. Soc., Providence, RI, 1994. |
[45] |
M. Wang and G. Lv, Entire solutions of a diffusive and competitive Lotka-Volterra type system with nonlocal delays, Nonlinearity, 23 (2010), 1609-1630.
doi: 10.1088/0951-7715/23/7/005. |
[46] |
Z.-C. Wang and J. Wu, Periodic traveling curved fronts in reaction-diffusion equation with bistable time-periodic nonlinearity, J. Differential Equations, 250 (2011), 3196-3229.
doi: 10.1016/j.jde.2011.01.017. |
[47] |
J. X. Xin, Multidimensional stability of traveling waves in a bistable reaction-diffusion equation. I, Comm. Partial Differential Equations, 17 (1992), 1889-1899. |
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