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A relaxation result for state constrained inclusions in infinite dimension
Local exact controllability to positive trajectory for parabolic system of chemotaxis
1. | Key Laboratory of System and Control, Academy of Mathematics and Systems Science, Academia Sinica, Beijing 100190, China |
2. | Department of Mathematics, Wuhan University of Technology, Wuhan 430070, China |
References:
[1] |
W. Arendt, Semigroups and evolution equations: functional calculus, regularity and kernel estimates, in Handbook of Differential Equations: Evolutionary Equations, Elsevier, 1 (2004), 1-85. |
[2] |
V. Barbu, Controllability of parabolic and Navier-Stokes equations, Sci. Math. Japon., 56 (2002), 143-211. |
[3] |
V. Barbu, Analysis and Control of Nonlinear Infinite-Dimensional Systems, Academic Press, Boston, 1993. |
[4] |
P. Biler, Local and global solvability of some parabolic systems modelling chemotaxis, Adv. Math. Sci. Appl., 8 (1998), 715-743. |
[5] |
J. -M. Coron, Control and Nonlinearity, AMS, Providence, RI, 2007. |
[6] |
E. Fernández-Cara and E. Zuazua, Null and approximate controllability for weakly blowing up semilinear heat equations, Annales de l'Institut Henri Poincare(C) Non Linear Analysis, 17 (2000), 583-616.
doi: 10.1016/S0294-1449(00)00117-7. |
[7] |
A. Fursikov and O. Yu Imanuvilov, Controllability of Evolution Equations, Lecture Notes Series 34, Seoul National University, Seoul, 1996. |
[8] |
H. Gajewski and K. Zacharias, Global behavior of a reaction-diffusion system modelling chemotaxis, Math. Nathr., 195 (1998), 77-114.
doi: 10.1002/mana.19981950106. |
[9] |
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, Berlin, 1981. |
[10] |
M. A. Herrero and J. J. L. Velázquez, A blow-up mechanism for a chemotaxis model, Ann. Scuola Norm. Sup. Pisa CI. Sci., 24 (1997), 633-683. |
[11] |
T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.
doi: 10.1007/s00285-008-0201-3. |
[12] |
D. Horstmann, From 1970 until present: the Keller-Segel model in chemotaxis and its consequences I, Jahresber. Deutsch. Math.-Verein., 105 (2003), 103-165. |
[13] |
D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations, 215 (2005), 52-107.
doi: 10.1016/j.jde.2004.10.022. |
[14] |
O. Yu. Imanuvilov and M. Yamamoto, Carleman inequalities for parabolic equations in Sobolev spaces of negative order and exact controllability for semilinear parabolic equations, Publ. Res. Inst. Math. Sci., 39 (2003), 227-274.
doi: 10.2977/prims/1145476103. |
[15] |
E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415.
doi: 10.1016/0022-5193(70)90092-5. |
[16] |
O. A. Ladyzhenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-linear Equations of Parabolic Type, AMS, Providence, RI, 1968. |
[17] |
D. Lamberton, Equations d'évolution liné aires associées à les semi-groupes de contractions dans les espaces $L^p$, J. Funct. Anal., 72 (1987), 252-262.
doi: 10.1016/0022-1236(87)90088-7. |
[18] |
G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996.
doi: 10.1142/3302. |
[19] |
M. Ma, C. Ou and Z. A. Wang, Stationary solutions of a volume filling chemotaxis model with logistic growth and their stability, SIAM J. Appl. Math., 72 (2012), 740-766.
doi: 10.1137/110843964. |
[20] |
K. Osaki and A. Yagi, Finite dimensional attractors for one-dimensional Keller-Segel equations, Funkcial. Ekva., 44 (2001), 441-469. |
[21] |
F. Rothe, Global Solutions of Reaction-Diffusion Systems, LNM 1072, Springer-Verlag, 1984. |
[22] |
S.-U. Ryu and A. Yagi, Optimal control of Keller-Segel equations, J. Math. Anal. Appl., 256 (2001), 45-66.
doi: 10.1006/jmaa.2000.7254. |
[23] |
A. Yagi, Norm behavior of solutions to a parabolic system of chemotaxis, Math. Japon., 45 (1997), 241-265. |
[24] |
G. Wang and C. Zhang, Observability estimate from measurable sets in time for some evolution equations,, , ().
|
[25] |
G. Wang and L. Zhang, Exact local controllability of a one-control reaction-diffusion system, J. Optim. Theory Appl., 131 (2006), 453-467.
doi: 10.1007/s10957-006-9161-1. |
[26] |
Z.-A. Wang, Mathematics of traveling waves in chemotaxis, Discrete Contin. Dyn. Syst-Series B., 18 (2013), 601-641.
doi: 10.3934/dcdsb.2013.18.601. |
show all references
References:
[1] |
W. Arendt, Semigroups and evolution equations: functional calculus, regularity and kernel estimates, in Handbook of Differential Equations: Evolutionary Equations, Elsevier, 1 (2004), 1-85. |
[2] |
V. Barbu, Controllability of parabolic and Navier-Stokes equations, Sci. Math. Japon., 56 (2002), 143-211. |
[3] |
V. Barbu, Analysis and Control of Nonlinear Infinite-Dimensional Systems, Academic Press, Boston, 1993. |
[4] |
P. Biler, Local and global solvability of some parabolic systems modelling chemotaxis, Adv. Math. Sci. Appl., 8 (1998), 715-743. |
[5] |
J. -M. Coron, Control and Nonlinearity, AMS, Providence, RI, 2007. |
[6] |
E. Fernández-Cara and E. Zuazua, Null and approximate controllability for weakly blowing up semilinear heat equations, Annales de l'Institut Henri Poincare(C) Non Linear Analysis, 17 (2000), 583-616.
doi: 10.1016/S0294-1449(00)00117-7. |
[7] |
A. Fursikov and O. Yu Imanuvilov, Controllability of Evolution Equations, Lecture Notes Series 34, Seoul National University, Seoul, 1996. |
[8] |
H. Gajewski and K. Zacharias, Global behavior of a reaction-diffusion system modelling chemotaxis, Math. Nathr., 195 (1998), 77-114.
doi: 10.1002/mana.19981950106. |
[9] |
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, Berlin, 1981. |
[10] |
M. A. Herrero and J. J. L. Velázquez, A blow-up mechanism for a chemotaxis model, Ann. Scuola Norm. Sup. Pisa CI. Sci., 24 (1997), 633-683. |
[11] |
T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.
doi: 10.1007/s00285-008-0201-3. |
[12] |
D. Horstmann, From 1970 until present: the Keller-Segel model in chemotaxis and its consequences I, Jahresber. Deutsch. Math.-Verein., 105 (2003), 103-165. |
[13] |
D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations, 215 (2005), 52-107.
doi: 10.1016/j.jde.2004.10.022. |
[14] |
O. Yu. Imanuvilov and M. Yamamoto, Carleman inequalities for parabolic equations in Sobolev spaces of negative order and exact controllability for semilinear parabolic equations, Publ. Res. Inst. Math. Sci., 39 (2003), 227-274.
doi: 10.2977/prims/1145476103. |
[15] |
E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415.
doi: 10.1016/0022-5193(70)90092-5. |
[16] |
O. A. Ladyzhenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-linear Equations of Parabolic Type, AMS, Providence, RI, 1968. |
[17] |
D. Lamberton, Equations d'évolution liné aires associées à les semi-groupes de contractions dans les espaces $L^p$, J. Funct. Anal., 72 (1987), 252-262.
doi: 10.1016/0022-1236(87)90088-7. |
[18] |
G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996.
doi: 10.1142/3302. |
[19] |
M. Ma, C. Ou and Z. A. Wang, Stationary solutions of a volume filling chemotaxis model with logistic growth and their stability, SIAM J. Appl. Math., 72 (2012), 740-766.
doi: 10.1137/110843964. |
[20] |
K. Osaki and A. Yagi, Finite dimensional attractors for one-dimensional Keller-Segel equations, Funkcial. Ekva., 44 (2001), 441-469. |
[21] |
F. Rothe, Global Solutions of Reaction-Diffusion Systems, LNM 1072, Springer-Verlag, 1984. |
[22] |
S.-U. Ryu and A. Yagi, Optimal control of Keller-Segel equations, J. Math. Anal. Appl., 256 (2001), 45-66.
doi: 10.1006/jmaa.2000.7254. |
[23] |
A. Yagi, Norm behavior of solutions to a parabolic system of chemotaxis, Math. Japon., 45 (1997), 241-265. |
[24] |
G. Wang and C. Zhang, Observability estimate from measurable sets in time for some evolution equations,, , ().
|
[25] |
G. Wang and L. Zhang, Exact local controllability of a one-control reaction-diffusion system, J. Optim. Theory Appl., 131 (2006), 453-467.
doi: 10.1007/s10957-006-9161-1. |
[26] |
Z.-A. Wang, Mathematics of traveling waves in chemotaxis, Discrete Contin. Dyn. Syst-Series B., 18 (2013), 601-641.
doi: 10.3934/dcdsb.2013.18.601. |
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