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Multiple positive solutions for Kirchhoff type problems with singularity
Long-time dynamics of the parabolic $p$-Laplacian equation
1. | Department of Mathematics, Faculty of Science, Hacettepe University, Beytepe 06800, Ankara, Turkey, Turkey |
References:
[1] |
R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics," Applied Mathematical Sciences, 68, Springer-Verlag, New York, 1988. |
[2] |
A. V. Babin and M. I. Vishik, Attractors of differential evolution equations in unbounded domain, Proc. Roy. Soc. Edinburg, 116A (1990), 221-243.
doi: 10.1017/S0308210500031498. |
[3] |
A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations," Studies in Mathematics and its Applications, 25, North-Holland Publishing Co., Amsterdam, 1992. |
[4] |
E. Feireisl, Ph. Laurencot, F. Simondon and H. Toure, Compact attractors for reaction diffusion equations in $R^n$, C. R. Acad. Sci. Paris Ser. I, 319 (1994), 147-151. |
[5] |
B. Wang, Attractors for reaction diffusion equations in unbounded domains, Physica D, 128 (1999), 41-52. |
[6] |
M. Efendiev and S. Zelik, The attractor for a nonlinear reaction-diffusion system in an unbounded domain, Comm. Pure Appl. Math., 54 (2001), 625-688.
doi: 10.1002/cpa.1011. |
[7] |
J. M. Arrieta, J. W. Cholewa, T. Dlotko and A. Rodriguez-Bernal, Asymptotic behavior and attractors for reaction diffusion equations in unbounded domains, Nonlinear Analysis: Theory, Methods & Applications, 56 (2004), 515 - 554.
doi: 10.1016/j.na.2003.09.023. |
[8] |
A. N. Carvalho, J. W. Cholewa and T. Dlotko, Global attractors for problems with monotone operators, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat., 2 (1999), 693-706. |
[9] |
A. N. Carvalho and C. B. Gentile, Asymptotic behavior of non-linear parabolic equations with monotone principial part, J. Math. Anal. Appl., 280 (2003), 252-272.
doi: 10.1016/S0022-247X(03)00037-4. |
[10] |
M. Nakao and N. Aris, On global attractor for nonlinear parabolic equation of $m$-Laplacian type, J. Math. Anal. Appl., 331 (2007), 793-809.
doi: 10.1016/j.jmaa.2006.08.044. |
[11] |
M. Yang, C. Sun and C. Zhong, Global attractors for $p$-Laplacian equation, J. Math. Anal. Appl., 337 (2007), 1130-1142.
doi: 10.1016/j.jmaa.2006.04.085. |
[12] |
M. Nakao and C. Chen, On global attractor for a nonlinear parabolic equation of $m$-Laplacian type in $R^n$, Funkcialaj Ekvacioj, 50 (2007), 449-468.
doi: 10.1619/fesi.50.449. |
[13] |
C. Chen, L. Shi and H. Wang, Existence of a global attractors in $L^p$ for $m$-Laplacian parabolic equation in $R^n$, Boundary Value Problems, 2009 (2009), 1-17.
doi: 10.1155/2009/563767. |
[14] |
A. Kh. Khanmamedov, Existence of a global attractor for the parabolic equation with nonlinear Laplacian principal part in an unbounded domain, J. Math. Anal. Appl., 316 (2006), 601-615.
doi: 10.1016/j.jmaa.2005.05.003. |
[15] |
A. Kh. Khanmamedov, Global attractors for one dimensional $p$-Laplacian equation, Nonlinear Analysis: Theory, Methods & Applications, 71 (2009), 155-171.
doi: 10.1016/j.na.2008.10.037. |
[16] |
M. Yang, C. Sun and C. Zhong, Existence of a global attractor for a $p$-Laplacian equation in $R^n$, Nonlinear Analysis: Theory, Methods and Applications, 66 (2007), 1-13.
doi: 10.1016/j.na.2005.11.004. |
[17] |
C. T. Anh and T. D. Ke, Long time behavior for quasilinear parabolic equations involving weighted p-Laplacian operators, Nonlinear Analysis: Theory, Methods & Applications, 71 (2009), 4415-4422.
doi: 10.1016/j.na.2009.02.125. |
[18] |
C. T. Anh and T. D. Ke, On quasilinear parabolic equations involving weighted p-Laplacian operators, Nonlinear Differential Equations and Applications, 17 (2010), 195-212.
doi: 10.1007/s00030-009-0048-3. |
[19] |
A. Kh. Khanmamedov, Global attractors for 2-D wave equations with displacement-dependent damping, Math. Methods Appl. Sci., 33 (2010), 177-187.
doi: 10.1002/mma.1161. |
[20] |
R. E. Showalter, "Monotone Operators in Banach Space and Nonlinear Partial Differential Equations," Mathematical Surveys Monographs, 49, American Mathematical Society, 1997. |
[21] |
J. Simon, Compact sets in the space $L_p(0, T;B)$, Annali Mat. Pura Appl., 146 (1987), 65-96.
doi: 10.1007/BF01762360. |
[22] |
M. A. Krasnoselskii and Y. B. Rutickii, "Convex Functions and Orlicz Spaces," P. Noordhoff Ltd., Groningen, 1961. |
[23] |
J.-L. Lions and E. Magenes, "Non-homogeneous Boundary Value Problems and Applications," 1, Springer-Verlag, New York-Heidelberg, 1972. |
[24] |
O. A. Ladyzhenskaya, On the determination of minimal global attractors for the Navier-Stokes equations and other partial differential equations, Uspekhi Mat. Nauk, 42 (1987), 25- 60; Russian Math. Surveys, 42 (1987), 27-73 (English Transl.).
doi: 10.1070/RM1987v042n06ABEH001503. |
show all references
References:
[1] |
R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics," Applied Mathematical Sciences, 68, Springer-Verlag, New York, 1988. |
[2] |
A. V. Babin and M. I. Vishik, Attractors of differential evolution equations in unbounded domain, Proc. Roy. Soc. Edinburg, 116A (1990), 221-243.
doi: 10.1017/S0308210500031498. |
[3] |
A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations," Studies in Mathematics and its Applications, 25, North-Holland Publishing Co., Amsterdam, 1992. |
[4] |
E. Feireisl, Ph. Laurencot, F. Simondon and H. Toure, Compact attractors for reaction diffusion equations in $R^n$, C. R. Acad. Sci. Paris Ser. I, 319 (1994), 147-151. |
[5] |
B. Wang, Attractors for reaction diffusion equations in unbounded domains, Physica D, 128 (1999), 41-52. |
[6] |
M. Efendiev and S. Zelik, The attractor for a nonlinear reaction-diffusion system in an unbounded domain, Comm. Pure Appl. Math., 54 (2001), 625-688.
doi: 10.1002/cpa.1011. |
[7] |
J. M. Arrieta, J. W. Cholewa, T. Dlotko and A. Rodriguez-Bernal, Asymptotic behavior and attractors for reaction diffusion equations in unbounded domains, Nonlinear Analysis: Theory, Methods & Applications, 56 (2004), 515 - 554.
doi: 10.1016/j.na.2003.09.023. |
[8] |
A. N. Carvalho, J. W. Cholewa and T. Dlotko, Global attractors for problems with monotone operators, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat., 2 (1999), 693-706. |
[9] |
A. N. Carvalho and C. B. Gentile, Asymptotic behavior of non-linear parabolic equations with monotone principial part, J. Math. Anal. Appl., 280 (2003), 252-272.
doi: 10.1016/S0022-247X(03)00037-4. |
[10] |
M. Nakao and N. Aris, On global attractor for nonlinear parabolic equation of $m$-Laplacian type, J. Math. Anal. Appl., 331 (2007), 793-809.
doi: 10.1016/j.jmaa.2006.08.044. |
[11] |
M. Yang, C. Sun and C. Zhong, Global attractors for $p$-Laplacian equation, J. Math. Anal. Appl., 337 (2007), 1130-1142.
doi: 10.1016/j.jmaa.2006.04.085. |
[12] |
M. Nakao and C. Chen, On global attractor for a nonlinear parabolic equation of $m$-Laplacian type in $R^n$, Funkcialaj Ekvacioj, 50 (2007), 449-468.
doi: 10.1619/fesi.50.449. |
[13] |
C. Chen, L. Shi and H. Wang, Existence of a global attractors in $L^p$ for $m$-Laplacian parabolic equation in $R^n$, Boundary Value Problems, 2009 (2009), 1-17.
doi: 10.1155/2009/563767. |
[14] |
A. Kh. Khanmamedov, Existence of a global attractor for the parabolic equation with nonlinear Laplacian principal part in an unbounded domain, J. Math. Anal. Appl., 316 (2006), 601-615.
doi: 10.1016/j.jmaa.2005.05.003. |
[15] |
A. Kh. Khanmamedov, Global attractors for one dimensional $p$-Laplacian equation, Nonlinear Analysis: Theory, Methods & Applications, 71 (2009), 155-171.
doi: 10.1016/j.na.2008.10.037. |
[16] |
M. Yang, C. Sun and C. Zhong, Existence of a global attractor for a $p$-Laplacian equation in $R^n$, Nonlinear Analysis: Theory, Methods and Applications, 66 (2007), 1-13.
doi: 10.1016/j.na.2005.11.004. |
[17] |
C. T. Anh and T. D. Ke, Long time behavior for quasilinear parabolic equations involving weighted p-Laplacian operators, Nonlinear Analysis: Theory, Methods & Applications, 71 (2009), 4415-4422.
doi: 10.1016/j.na.2009.02.125. |
[18] |
C. T. Anh and T. D. Ke, On quasilinear parabolic equations involving weighted p-Laplacian operators, Nonlinear Differential Equations and Applications, 17 (2010), 195-212.
doi: 10.1007/s00030-009-0048-3. |
[19] |
A. Kh. Khanmamedov, Global attractors for 2-D wave equations with displacement-dependent damping, Math. Methods Appl. Sci., 33 (2010), 177-187.
doi: 10.1002/mma.1161. |
[20] |
R. E. Showalter, "Monotone Operators in Banach Space and Nonlinear Partial Differential Equations," Mathematical Surveys Monographs, 49, American Mathematical Society, 1997. |
[21] |
J. Simon, Compact sets in the space $L_p(0, T;B)$, Annali Mat. Pura Appl., 146 (1987), 65-96.
doi: 10.1007/BF01762360. |
[22] |
M. A. Krasnoselskii and Y. B. Rutickii, "Convex Functions and Orlicz Spaces," P. Noordhoff Ltd., Groningen, 1961. |
[23] |
J.-L. Lions and E. Magenes, "Non-homogeneous Boundary Value Problems and Applications," 1, Springer-Verlag, New York-Heidelberg, 1972. |
[24] |
O. A. Ladyzhenskaya, On the determination of minimal global attractors for the Navier-Stokes equations and other partial differential equations, Uspekhi Mat. Nauk, 42 (1987), 25- 60; Russian Math. Surveys, 42 (1987), 27-73 (English Transl.).
doi: 10.1070/RM1987v042n06ABEH001503. |
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