Non-trivial non-negative periodic solutions of a system of doubly degenerate parabolic equations with nonlocal terms

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March 2011, 31(1): 35-64. doi: 10.3934/dcds.2011.31.35

Non-trivial non-negative periodic solutions of a system of doubly degenerate parabolic equations with nonlocal terms

Genni Fragnelli ^1, , Paolo Nistri ^2, and Duccio Papini ^3,

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

Received February 2010 Revised September 2010 Published June 2011

Abstract
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The aim of the paper is to provide conditions ensuring the existence of non-trivial non-negative periodic solutions to a system of doubly degenerate parabolic equations containing delayed nonlocal terms and satisfying Dirichlet boundary conditions. The employed approach is based on the theory of the Leray-Schauder topological degree theory, thus a crucial purpose of the paper is to obtain a priori bounds in a convenient functional space, here $L^2(Q_T)$, on the solutions of certain homotopies. This is achieved under different assumptions on the sign of the kernels of the nonlocal terms. The considered system is a possible model of the interactions between two biological species sharing the same territory where such interactions are modeled by the kernels of the nonlocal terms. To this regard the obtained results can be viewed as coexistence results of the two biological populations under different intra and inter specific interferences on their natural growth rates.

Keywords: topological degree., Doubly degenerate parabolic equations, non-negative periodic solutions.

Mathematics Subject Classification: Primary: 35K65, 35B10; Secondary: 47H1.

Citation: Genni Fragnelli, Paolo Nistri, Duccio Papini. Non-trivial non-negative periodic solutions of a system of doubly degenerate parabolic equations with nonlocal terms. Discrete and Continuous Dynamical Systems, 2011, 31 (1) : 35-64. doi: 10.3934/dcds.2011.31.35

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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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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