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On tamed milstein schemes of SDEs driven by Lévy noise
Asymptotic behavior in a chemotaxis-growth system with nonlinear production of signals
1. | College of Information Science & Technology, Dong Hua University, Shanghai 200051, China |
2. | Department of Applied Mathematics, Dong Hua University, Shanghai 200051, China |
$\left\{ {\begin{array}{*{20}{l}} {{u_t} = \Delta u - \chi \nabla \cdot (u\nabla v) + \mu u(1 - u),}&{x \in \Omega ,{\mkern 1mu} t > 0,}\\ {{v_t} = \Delta v - v + h(u),}&{x \in \Omega ,{\mkern 1mu} t > 0,} \end{array}} \right.$ |
$Ω\subset\mathbb{R}^3$ |
$χ>0$ |
$μ>0$ |
$h(s)$ |
$[0, ∞)$ |
$4|{h}'|<\sqrt{2\mu -7{{\chi }^{2}}},$ |
$u_0∈ C^0(\bar{Ω})$ |
$v_0∈ W^{1, ∞}(Ω)$ |
$Ω× (0, ∞)$ |
$χ |h'| < \sqrt{8μ},$ |
$(1, h(1))$ |
References:
[1] |
M. A. J. Chaplain and J. I. Tello,
On the stability of homogeneous steady states of a chemotaxis system with logistic growth term, Appl. Math. Lett., 57 (2016), 1-6.
doi: 10.1016/j.aml.2015.12.001. |
[2] |
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. |
[3] |
S. Ishida, K. Seki and T. Yokota,
Boundedness in quasilinear Keller-Segel systems of parabolic-parabolic type on non-convex bounded domains, J. Differential Equations, 256 (2014), 2993-3010.
doi: 10.1016/j.jde.2014.01.028. |
[4] |
E. F. Keller and L. A. Segel,
Initiation of slime mold aggregation viewed as an instaility, J. Theor. Biol., 26 (1970), 399-415.
doi: 10.1016/0022-5193(70)90092-5. |
[5] |
P. L. Lions,
Résolution de problémes elliptiques quasilinéaires, Arch. Ration. Mech. Anal., 74 (1980), 335-353.
doi: 10.1007/BF00249679. |
[6] |
M. Mimura and T. Tsujikawa,
Aggregating pattern dynamics in a chemotaxis model including growth, Physica A, 230 (1996), 499-543.
|
[7] |
N. Mizoguchi and P. Souplet,
Nondegeneracy of blow-up points for the parabolic Keller-Segel system, Ann. Inst. H. Poincaré, Analyse Non Linéaire, 31 (2014), 851-875.
doi: 10.1016/j.anihpc.2013.07.007. |
[8] |
M. R. Myerscough, P. K. Maini and J. Painter,
Pattern formation in a generalized chemotactic model, Bull. Math. Biol., 60 (1998), 1-26.
|
[9] |
E. Nakaguchi and K. Osaki,
Global solutions and exponential attractors of a parabolic-parabolic system for chemotaxis with subquadratic degradation, Discrete Contin. Dyn. Syst. B, 18 (2013), 2627-2646.
doi: 10.3934/dcdsb.2013.18.2627. |
[10] |
E. Nakaguchi and K. Osaki,
$L_p$-estimates of solutions to n-dimensional parabolic-parabolic system for chemotaxis with subquadratic degradation, Funkcialaj Ekvacioj, 59 (2016), 51-66.
doi: 10.1619/fesi.59.51. |
[11] |
E. Nakaguchi and K. Osaki, Global existence of solutions to n-diemnsional parabolic-parabolic system for chemotaxis with subquadratic degradation, Preprint. |
[12] |
M. E. Orme and M. A. J. Chaplain,
A mathematical model of the first steps of tumour-related angiogenesis: Capillary sprout formation and secondary branching, IMA J. Math. Appl. Med. Biol., 13 (1996), 73-98.
|
[13] |
K. Osaki, T. Tsujikawa, A. Yagi and M. Mimura,
Exponential attractor for a chemotaxis-growth system of equations, Nonlinear Analysis, 51 (2002), 119-144.
doi: 10.1016/S0362-546X(01)00815-X. |
[14] |
M. M. Porzio and V. Vespri,
Holder estimates for local solutions of some doubly nonlinear degenerate parabolic equations, J. Differential Equations, 103 (1993), 146-178.
doi: 10.1006/jdeq.1993.1045. |
[15] |
C. Stinner, C. Surulescu and M. Winkler,
Global weak solutions in a PDE-ODE system modeling multiscale cancer cell invasion, SIAM J. Math. Anal., 46 (2014), 1969-2007.
doi: 10.1137/13094058X. |
[16] |
Y. Tao and M. Winkler,
Large time behavior in a mutidimensional chemotaxis-haptotaxis model with slow signal diffusion, SIAM J. Math. Anal., 47 (2015), 4229-4250.
doi: 10.1137/15M1014115. |
[17] |
Y. Tao and M. Winkler,
Boundedness and decay enforced by quadratic degradation in a three-dimensional chemotaxis-fluid system, Z. Angew. Math. Phys., 66 (2015), 2555-2573.
doi: 10.1007/s00033-015-0541-y. |
[18] |
Y. Tao and M. Winkler, Boundedness and competitive exclusion in a population model with cross-diffusion for one species, Preprint. |
[19] |
M. Winkler,
Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Commun. Partial Differential Equations, 35 (2010), 1516-1537.
doi: 10.1080/03605300903473426. |
[20] |
M. Winkler,
Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differential Equations, 248 (2010), 2889-2905.
doi: 10.1016/j.jde.2010.02.008. |
show all references
References:
[1] |
M. A. J. Chaplain and J. I. Tello,
On the stability of homogeneous steady states of a chemotaxis system with logistic growth term, Appl. Math. Lett., 57 (2016), 1-6.
doi: 10.1016/j.aml.2015.12.001. |
[2] |
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. |
[3] |
S. Ishida, K. Seki and T. Yokota,
Boundedness in quasilinear Keller-Segel systems of parabolic-parabolic type on non-convex bounded domains, J. Differential Equations, 256 (2014), 2993-3010.
doi: 10.1016/j.jde.2014.01.028. |
[4] |
E. F. Keller and L. A. Segel,
Initiation of slime mold aggregation viewed as an instaility, J. Theor. Biol., 26 (1970), 399-415.
doi: 10.1016/0022-5193(70)90092-5. |
[5] |
P. L. Lions,
Résolution de problémes elliptiques quasilinéaires, Arch. Ration. Mech. Anal., 74 (1980), 335-353.
doi: 10.1007/BF00249679. |
[6] |
M. Mimura and T. Tsujikawa,
Aggregating pattern dynamics in a chemotaxis model including growth, Physica A, 230 (1996), 499-543.
|
[7] |
N. Mizoguchi and P. Souplet,
Nondegeneracy of blow-up points for the parabolic Keller-Segel system, Ann. Inst. H. Poincaré, Analyse Non Linéaire, 31 (2014), 851-875.
doi: 10.1016/j.anihpc.2013.07.007. |
[8] |
M. R. Myerscough, P. K. Maini and J. Painter,
Pattern formation in a generalized chemotactic model, Bull. Math. Biol., 60 (1998), 1-26.
|
[9] |
E. Nakaguchi and K. Osaki,
Global solutions and exponential attractors of a parabolic-parabolic system for chemotaxis with subquadratic degradation, Discrete Contin. Dyn. Syst. B, 18 (2013), 2627-2646.
doi: 10.3934/dcdsb.2013.18.2627. |
[10] |
E. Nakaguchi and K. Osaki,
$L_p$-estimates of solutions to n-dimensional parabolic-parabolic system for chemotaxis with subquadratic degradation, Funkcialaj Ekvacioj, 59 (2016), 51-66.
doi: 10.1619/fesi.59.51. |
[11] |
E. Nakaguchi and K. Osaki, Global existence of solutions to n-diemnsional parabolic-parabolic system for chemotaxis with subquadratic degradation, Preprint. |
[12] |
M. E. Orme and M. A. J. Chaplain,
A mathematical model of the first steps of tumour-related angiogenesis: Capillary sprout formation and secondary branching, IMA J. Math. Appl. Med. Biol., 13 (1996), 73-98.
|
[13] |
K. Osaki, T. Tsujikawa, A. Yagi and M. Mimura,
Exponential attractor for a chemotaxis-growth system of equations, Nonlinear Analysis, 51 (2002), 119-144.
doi: 10.1016/S0362-546X(01)00815-X. |
[14] |
M. M. Porzio and V. Vespri,
Holder estimates for local solutions of some doubly nonlinear degenerate parabolic equations, J. Differential Equations, 103 (1993), 146-178.
doi: 10.1006/jdeq.1993.1045. |
[15] |
C. Stinner, C. Surulescu and M. Winkler,
Global weak solutions in a PDE-ODE system modeling multiscale cancer cell invasion, SIAM J. Math. Anal., 46 (2014), 1969-2007.
doi: 10.1137/13094058X. |
[16] |
Y. Tao and M. Winkler,
Large time behavior in a mutidimensional chemotaxis-haptotaxis model with slow signal diffusion, SIAM J. Math. Anal., 47 (2015), 4229-4250.
doi: 10.1137/15M1014115. |
[17] |
Y. Tao and M. Winkler,
Boundedness and decay enforced by quadratic degradation in a three-dimensional chemotaxis-fluid system, Z. Angew. Math. Phys., 66 (2015), 2555-2573.
doi: 10.1007/s00033-015-0541-y. |
[18] |
Y. Tao and M. Winkler, Boundedness and competitive exclusion in a population model with cross-diffusion for one species, Preprint. |
[19] |
M. Winkler,
Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Commun. Partial Differential Equations, 35 (2010), 1516-1537.
doi: 10.1080/03605300903473426. |
[20] |
M. Winkler,
Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differential Equations, 248 (2010), 2889-2905.
doi: 10.1016/j.jde.2010.02.008. |
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