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Inertial manifolds for the 3D Cahn-Hilliard equations with periodic boundary conditions
A nonlocal diffusion population model with age structure and Dirichlet boundary condition
1. | School of Mathematics and Information Sciences, Guangzhou University, Guangzhou, 510006, China |
2. | Department of Mathematics, Michigan State University, East Lansing, MI 48824, United States |
References:
[1] |
H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach space, SIAM Review, 18 (1976), 620-709. |
[2] |
N. F. Britton, Spatial structures and periodic travelling waves in an integro-differential reaction-diffusion population model, SIAM J. Appl. Math., 50 (1990), 1663-1688.
doi: 10.1137/0150099. |
[3] |
L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, Vol. 19, American Mathematical Society, Providence, RI, 1998. |
[4] |
S. A. Gourley and J. Wu, Delayed non-local diffusive systems in biological invasion and disease spread, in Nonlinear Dynamics and Evolution Equations (H. Brunner, X.-Q. Zhao and X. Zou eds.), Fields Inst. Commun., 48 (2006), 137-200. |
[5] |
Z. M. Guo, Z. C. Yang and X. Zou, Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition: a non-monotone case, Commun. Pure Appl. Anal., 11 (2012), 1825-1838.
doi: 10.3934/cpaa.2012.11.1825. |
[6] |
J. K. Hale, Asymptotic Behavior of Dissipative Systems, Math. Surveys and Monographs 25, Amer. Math. Soc., Providence, RI, 1988. |
[7] |
P. Li and S. T. Yau, On the Schrödinger equation and the eigenvalue problem, Commun. Math. Phys., 88 (1983), 309-318. |
[8] |
D. Liang, J. W. -H. So, F. Zhang and X. Zou, Population dynamic models with nonlocal delay on bounded fields and their numerical computations, Diff. Eqns. Dynam. Syst., 11 (2003), 117-139. |
[9] |
M. C. Mackey and L. Glass, Oscillation and chaos in physiological control systems, Science, 197 (1977), 287-289. |
[10] |
J. A. J. Metz and O. Diekmann, The Dynamics of Physiologically Structured Populations, (J. A. J. Metz and O. Diekmann eds.), Springer-Verlag, New York, 1986.
doi: 10.1007/978-3-662-13159-6. |
[11] |
M. H. Protter and H. F. Weinberger, Maximum Principle in Differential Equations, Springer-Verlag, New York, 1984.
doi: 10.1007/978-1-4612-5282-5. |
[12] |
D. H. Sattinger, Monotone methods in nonlinear elliptic and parabolic boundary value problems, Indiana Univ. Math. J., 21 (1972), 979-1000. |
[13] |
H. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences, Texts in Applied Mathematics 57, Springer, New York, 2011.
doi: 10.1007/978-1-4419-7646-8. |
[14] |
H. Smith and H. Thieme, Strongly order preserving semi-flows generated by functional differential equations, J. Diff. Eqns., 93 (1991), 332-363.
doi: 10.1016/0022-0396(91)90016-3. |
[15] |
J. W. -H. So, J. Wu and Y. Yang, Numerical steady state and hopf bifurcation analysis on the diffusive Nicholson's blowflies equation, Appl. Math. Comput., 111 (2000), 33-51.
doi: 10.1016/S0096-3003(99)00047-8. |
[16] |
J. W. -H. So, J. Wu and X. Zou, A reaction diffusion model for a single species with age structure-I. Traveling wave fronts on unbounded domains, Proc. Royal Soc. London. A, 457 (2001), 1841-1853.
doi: 10.1098/rspa.2001.0789. |
[17] |
H. R. Thieme and X.-Q. Zhao, A non-local delayed and diffusive predator-prey model, Nonlinear Anal. RWA., 2 (2001), 145-160.
doi: 10.1016/S0362-546X(00)00112-7. |
[18] |
H. R. Thieme and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction-diffusion models, J. Diff. Eqns., 195 (2003), 430-470.
doi: 10.1016/S0022-0396(03)00175-X. |
[19] |
J. Wu, Theory and Applications of Partial Functional Differential Equations, Appl. Math. Sci. 119, Springer-Verlag, New York, 1996.
doi: 10.1007/978-1-4612-4050-1. |
[20] |
D. Xu and X.-Q. Zhao, A nonlocal reaction-diffusion population model with stage structure, Canad. Appl. Math. Quart., 11 (2003), 303-319. |
[21] |
S. T. Yau and R. Schoen, Lectures on Differential Geometry, Higher Education Press, Beijing, 2004. |
[22] |
T. Yi and X. Zou, Global dynamics of a delay differential equation with spatial non-locality in an unbounded domain, J. Diff. Eqns., 251 (2011), 2598-2611.
doi: 10.1016/j.jde.2011.04.027. |
[23] |
T. Yi and X. Zou, On Dirichlet problem for a class of delayed reaction-diffusion equations with spatial non-locality, J. Dyn. Diff. Equat., 25 (2013), 959-979.
doi: 10.1007/s10884-013-9324-3. |
[24] |
X.-Q. Zhao, Global attractivity in a class of nonmonotone reaction diffusion equations with time delay, Canad. Appl. Math. Quart., 17 (2009), 271-281. |
show all references
References:
[1] |
H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach space, SIAM Review, 18 (1976), 620-709. |
[2] |
N. F. Britton, Spatial structures and periodic travelling waves in an integro-differential reaction-diffusion population model, SIAM J. Appl. Math., 50 (1990), 1663-1688.
doi: 10.1137/0150099. |
[3] |
L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, Vol. 19, American Mathematical Society, Providence, RI, 1998. |
[4] |
S. A. Gourley and J. Wu, Delayed non-local diffusive systems in biological invasion and disease spread, in Nonlinear Dynamics and Evolution Equations (H. Brunner, X.-Q. Zhao and X. Zou eds.), Fields Inst. Commun., 48 (2006), 137-200. |
[5] |
Z. M. Guo, Z. C. Yang and X. Zou, Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition: a non-monotone case, Commun. Pure Appl. Anal., 11 (2012), 1825-1838.
doi: 10.3934/cpaa.2012.11.1825. |
[6] |
J. K. Hale, Asymptotic Behavior of Dissipative Systems, Math. Surveys and Monographs 25, Amer. Math. Soc., Providence, RI, 1988. |
[7] |
P. Li and S. T. Yau, On the Schrödinger equation and the eigenvalue problem, Commun. Math. Phys., 88 (1983), 309-318. |
[8] |
D. Liang, J. W. -H. So, F. Zhang and X. Zou, Population dynamic models with nonlocal delay on bounded fields and their numerical computations, Diff. Eqns. Dynam. Syst., 11 (2003), 117-139. |
[9] |
M. C. Mackey and L. Glass, Oscillation and chaos in physiological control systems, Science, 197 (1977), 287-289. |
[10] |
J. A. J. Metz and O. Diekmann, The Dynamics of Physiologically Structured Populations, (J. A. J. Metz and O. Diekmann eds.), Springer-Verlag, New York, 1986.
doi: 10.1007/978-3-662-13159-6. |
[11] |
M. H. Protter and H. F. Weinberger, Maximum Principle in Differential Equations, Springer-Verlag, New York, 1984.
doi: 10.1007/978-1-4612-5282-5. |
[12] |
D. H. Sattinger, Monotone methods in nonlinear elliptic and parabolic boundary value problems, Indiana Univ. Math. J., 21 (1972), 979-1000. |
[13] |
H. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences, Texts in Applied Mathematics 57, Springer, New York, 2011.
doi: 10.1007/978-1-4419-7646-8. |
[14] |
H. Smith and H. Thieme, Strongly order preserving semi-flows generated by functional differential equations, J. Diff. Eqns., 93 (1991), 332-363.
doi: 10.1016/0022-0396(91)90016-3. |
[15] |
J. W. -H. So, J. Wu and Y. Yang, Numerical steady state and hopf bifurcation analysis on the diffusive Nicholson's blowflies equation, Appl. Math. Comput., 111 (2000), 33-51.
doi: 10.1016/S0096-3003(99)00047-8. |
[16] |
J. W. -H. So, J. Wu and X. Zou, A reaction diffusion model for a single species with age structure-I. Traveling wave fronts on unbounded domains, Proc. Royal Soc. London. A, 457 (2001), 1841-1853.
doi: 10.1098/rspa.2001.0789. |
[17] |
H. R. Thieme and X.-Q. Zhao, A non-local delayed and diffusive predator-prey model, Nonlinear Anal. RWA., 2 (2001), 145-160.
doi: 10.1016/S0362-546X(00)00112-7. |
[18] |
H. R. Thieme and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction-diffusion models, J. Diff. Eqns., 195 (2003), 430-470.
doi: 10.1016/S0022-0396(03)00175-X. |
[19] |
J. Wu, Theory and Applications of Partial Functional Differential Equations, Appl. Math. Sci. 119, Springer-Verlag, New York, 1996.
doi: 10.1007/978-1-4612-4050-1. |
[20] |
D. Xu and X.-Q. Zhao, A nonlocal reaction-diffusion population model with stage structure, Canad. Appl. Math. Quart., 11 (2003), 303-319. |
[21] |
S. T. Yau and R. Schoen, Lectures on Differential Geometry, Higher Education Press, Beijing, 2004. |
[22] |
T. Yi and X. Zou, Global dynamics of a delay differential equation with spatial non-locality in an unbounded domain, J. Diff. Eqns., 251 (2011), 2598-2611.
doi: 10.1016/j.jde.2011.04.027. |
[23] |
T. Yi and X. Zou, On Dirichlet problem for a class of delayed reaction-diffusion equations with spatial non-locality, J. Dyn. Diff. Equat., 25 (2013), 959-979.
doi: 10.1007/s10884-013-9324-3. |
[24] |
X.-Q. Zhao, Global attractivity in a class of nonmonotone reaction diffusion equations with time delay, Canad. Appl. Math. Quart., 17 (2009), 271-281. |
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