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Estimates on the number of limit cycles of a generalized Abel equation
Non-trivial non-negative periodic solutions of a system of doubly degenerate parabolic equations with nonlocal terms
1. | Dipartimento di Matematica, Università di Bari, Via E. Orabona 4, 70125 Bari, Italy |
2. | Dipartimento di Ingegneria dell' Informazione, Università di Siena, Via Roma 56, 53100, Siena |
3. | Dipartimento di Ingegneria dell’Informazione, Università di Siena, Via Roma 56, 53100 Siena, Italy |
References:
[1] |
G. A. Afrouzi and S. H. Rasouli, Population models involving the $p$-Laplacian with indefinite weight and constant yield harvesting, Chaos Solitons Fractals, 31 (2007), 404-408.
doi: 10.1016/j.chaos.2005.09.067. |
[2] |
W. Allegretto and P. Nistri, Existence and optimal control for periodic parabolic equations with nonlocal terms, IMA J. Math. Control Inform., 16 (1999), 43-58.
doi: 10.1093/imamci/16.1.43. |
[3] |
P. Benilan and P. Wittbold, On mild and weak solutions of elliptic-parabolic problems, Adv. Differential Equations, 1 (1996), 1053-1073. |
[4] |
E. DiBenedetto, "Degenerate Parabolic Equations,'' Universitext, Springer-Verlag, New York, 1993. |
[5] |
G. Fragnelli, P. Nistri and D. Papini, Positive periodic solutions and optimal control for a distributed biological model of two interacting species, Nonlinear Anal. Real World Appl., 12 (2011), 1410-1428.
doi: 10.1016/j.nonrwa.2010.10.002. |
[6] |
G. Fragnelli, Positive periodic solutions for a system of anisotropic parabolic equations, J. Math. Anal. Appl., 367 (2010), 204-228.
doi: 10.1016/j.jmaa.2009.12.039. |
[7] |
E. Gurtin and R. C. MacCamy, On the diffusion of biological populations, Math. Biosci., 33 (1977), 35-49.
doi: 10.1016/0025-5564(77)90062-1. |
[8] |
E. Gurtin and R. C. MacCamy, Diffusion models for age-structured populations, Math. Biosci., 54 (1981), 49-59.
doi: 10.1016/0025-5564(81)90075-4. |
[9] |
R. Huang, Y. Wang and Y. Ke, Existence of non-trivial nonnegative periodic solutions for a class of degenerate parabolic equations with nonlocal terms, Discrete Contin. Dyn. Syst. Ser. B, 5 (2005), 1005-1014.
doi: 10.3934/dcdsb.2005.5.1005. |
[10] |
A. V. Ivanov, Hölder estimates for equations of fast diffusion type, St. Petersburg Math. J., 6 (1995), 791-825. |
[11] |
A. V. Ivanov, Hölder estimates for equations of slow and normal diffusion type, J. Math. Sci. (New York), 85 (1997), 1640-1644.
doi: 10.1007/BF02355324. |
[12] |
B. Kawohl and P. Lindqvist, Positive eigenfunctions for the $p$-Laplace operator revisited, Analysis (Munich), 26 (2006), 545-550. |
[13] |
O. Ladyženskaja, V. Solonnikov and N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type,'' Translations of Mathematical Monographs, 23, American Mathematical Society, Providence, RI, 1967. |
[14] |
D. Li and X. Zhang, On a nonlocal aggregation model with nonlinear diffusion, Discrete Contin. Dyn. Syst., 27 (2010), 301-323.
doi: 10.3934/dcds.2010.27.301. |
[15] |
N. Mizoguchi, Periodic solutions for degenerate diffusion equations, Indiana Univ. Math. J., 44 (1995), 413-432.
doi: 10.1512/iumj.1995.44.1994. |
[16] |
M. Nakao, Periodic solutions of some nonlinear degenerate parabolic equations, J. Math. Anal. Appl., 104 (1984), 554-567.
doi: 10.1016/0022-247X(84)90020-9. |
[17] |
A. Okubo, "Diffusion and Ecological Problems: Mathematical Models,'' Ecology and Diffusion, Biomathematics, 10, Springer-Verlag, Berlin-New York, 1980. |
[18] |
S. Oruganti, J. Shi and R. Shivaji, Logistic equation with the $p$-Laplacian and constant yield harvesting,, Abstr. Appl. Anal., 2004 (): 723.
doi: 10.1155/S1085337504311097. |
[19] |
M. M. Porzio and V. Vespri, Hölder estimates for local solution of some doubly nonlinear degenerate parabolic equation, J. Differential Equations, 103 (1993), 146-178.
doi: 10.1006/jdeq.1993.1045. |
[20] |
R. E. Showalter, "Monotone Operators in Banach Space and Nonlinear Partial Differential Equations,'' Mathematical Surveys and Monographs, 49, American Mathematical Society, Providence, RI, 1997. |
[21] |
J. L. Vázquez, "The Porous Medium Equation. Mathematical Theory,'' Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2007. |
[22] |
C. Wang, J. Yin and M. Wen, Periodic optimal control for a degenerate nonlinear diffusion equation, Applied Mathematics and Information Science, Comput. Math. Model., 17 (2006), 364-375.
doi: 10.1007/s10598-006-0030-4. |
[23] |
J. Wang and W. Gao, Existence of nontrivial nonnegative periodic solutions for a class of doubly degenerate parabolic equation with nonlocal terms, J. Math. Anal. Appl., 331 (2007), 481-498.
doi: 10.1016/j.jmaa.2006.08.059. |
[24] |
Y. Wang, J. Yin and Z. Wu, Periodic solutions of porous medium equations with weakly nonlinear sources, Northeast. Math. J., 16 (2000), 475-483. |
[25] |
Y. Wang, J. Yin and Z. Wu, Periodic solutions of evolution $p$-Laplacian equations with nonlinear sources, J. Math. Anal. Appl., 219 (1998), 76-96.
doi: 10.1006/jmaa.1997.5783. |
[26] |
Q. Zhou, Y. Ke, Y. Wang and J. Yin, Periodic $p$-Laplacian with nonlocal terms, Nonlinear Anal., 66 (2007), 442-453.
doi: 10.1016/j.na.2005.11.038. |
show all references
References:
[1] |
G. A. Afrouzi and S. H. Rasouli, Population models involving the $p$-Laplacian with indefinite weight and constant yield harvesting, Chaos Solitons Fractals, 31 (2007), 404-408.
doi: 10.1016/j.chaos.2005.09.067. |
[2] |
W. Allegretto and P. Nistri, Existence and optimal control for periodic parabolic equations with nonlocal terms, IMA J. Math. Control Inform., 16 (1999), 43-58.
doi: 10.1093/imamci/16.1.43. |
[3] |
P. Benilan and P. Wittbold, On mild and weak solutions of elliptic-parabolic problems, Adv. Differential Equations, 1 (1996), 1053-1073. |
[4] |
E. DiBenedetto, "Degenerate Parabolic Equations,'' Universitext, Springer-Verlag, New York, 1993. |
[5] |
G. Fragnelli, P. Nistri and D. Papini, Positive periodic solutions and optimal control for a distributed biological model of two interacting species, Nonlinear Anal. Real World Appl., 12 (2011), 1410-1428.
doi: 10.1016/j.nonrwa.2010.10.002. |
[6] |
G. Fragnelli, Positive periodic solutions for a system of anisotropic parabolic equations, J. Math. Anal. Appl., 367 (2010), 204-228.
doi: 10.1016/j.jmaa.2009.12.039. |
[7] |
E. Gurtin and R. C. MacCamy, On the diffusion of biological populations, Math. Biosci., 33 (1977), 35-49.
doi: 10.1016/0025-5564(77)90062-1. |
[8] |
E. Gurtin and R. C. MacCamy, Diffusion models for age-structured populations, Math. Biosci., 54 (1981), 49-59.
doi: 10.1016/0025-5564(81)90075-4. |
[9] |
R. Huang, Y. Wang and Y. Ke, Existence of non-trivial nonnegative periodic solutions for a class of degenerate parabolic equations with nonlocal terms, Discrete Contin. Dyn. Syst. Ser. B, 5 (2005), 1005-1014.
doi: 10.3934/dcdsb.2005.5.1005. |
[10] |
A. V. Ivanov, Hölder estimates for equations of fast diffusion type, St. Petersburg Math. J., 6 (1995), 791-825. |
[11] |
A. V. Ivanov, Hölder estimates for equations of slow and normal diffusion type, J. Math. Sci. (New York), 85 (1997), 1640-1644.
doi: 10.1007/BF02355324. |
[12] |
B. Kawohl and P. Lindqvist, Positive eigenfunctions for the $p$-Laplace operator revisited, Analysis (Munich), 26 (2006), 545-550. |
[13] |
O. Ladyženskaja, V. Solonnikov and N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type,'' Translations of Mathematical Monographs, 23, American Mathematical Society, Providence, RI, 1967. |
[14] |
D. Li and X. Zhang, On a nonlocal aggregation model with nonlinear diffusion, Discrete Contin. Dyn. Syst., 27 (2010), 301-323.
doi: 10.3934/dcds.2010.27.301. |
[15] |
N. Mizoguchi, Periodic solutions for degenerate diffusion equations, Indiana Univ. Math. J., 44 (1995), 413-432.
doi: 10.1512/iumj.1995.44.1994. |
[16] |
M. Nakao, Periodic solutions of some nonlinear degenerate parabolic equations, J. Math. Anal. Appl., 104 (1984), 554-567.
doi: 10.1016/0022-247X(84)90020-9. |
[17] |
A. Okubo, "Diffusion and Ecological Problems: Mathematical Models,'' Ecology and Diffusion, Biomathematics, 10, Springer-Verlag, Berlin-New York, 1980. |
[18] |
S. Oruganti, J. Shi and R. Shivaji, Logistic equation with the $p$-Laplacian and constant yield harvesting,, Abstr. Appl. Anal., 2004 (): 723.
doi: 10.1155/S1085337504311097. |
[19] |
M. M. Porzio and V. Vespri, Hölder estimates for local solution of some doubly nonlinear degenerate parabolic equation, J. Differential Equations, 103 (1993), 146-178.
doi: 10.1006/jdeq.1993.1045. |
[20] |
R. E. Showalter, "Monotone Operators in Banach Space and Nonlinear Partial Differential Equations,'' Mathematical Surveys and Monographs, 49, American Mathematical Society, Providence, RI, 1997. |
[21] |
J. L. Vázquez, "The Porous Medium Equation. Mathematical Theory,'' Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2007. |
[22] |
C. Wang, J. Yin and M. Wen, Periodic optimal control for a degenerate nonlinear diffusion equation, Applied Mathematics and Information Science, Comput. Math. Model., 17 (2006), 364-375.
doi: 10.1007/s10598-006-0030-4. |
[23] |
J. Wang and W. Gao, Existence of nontrivial nonnegative periodic solutions for a class of doubly degenerate parabolic equation with nonlocal terms, J. Math. Anal. Appl., 331 (2007), 481-498.
doi: 10.1016/j.jmaa.2006.08.059. |
[24] |
Y. Wang, J. Yin and Z. Wu, Periodic solutions of porous medium equations with weakly nonlinear sources, Northeast. Math. J., 16 (2000), 475-483. |
[25] |
Y. Wang, J. Yin and Z. Wu, Periodic solutions of evolution $p$-Laplacian equations with nonlinear sources, J. Math. Anal. Appl., 219 (1998), 76-96.
doi: 10.1006/jmaa.1997.5783. |
[26] |
Q. Zhou, Y. Ke, Y. Wang and J. Yin, Periodic $p$-Laplacian with nonlocal terms, Nonlinear Anal., 66 (2007), 442-453.
doi: 10.1016/j.na.2005.11.038. |
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