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Boundedness and global solvability to a chemotaxis-haptotaxis model with slow and fast diffusion
School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, China |
$\left\{ \begin{array}{l}{u_t} = \Delta {u^m} - \chi \nabla \cdot \left( {u \cdot \nabla v} \right) - \xi \nabla \cdot \left( {u \cdot \nabla w} \right) + \mu u\left( {1 - u - w} \right),{\rm{in}}\;\Omega \times {{\mathbb{R}}^ + },\\{v_t} - \nabla v + v = u,\;{\rm{in}}\;\Omega \times {{\mathbb{R}}^ + },\\{w_t} = - vw,\;\;{\rm{in}}\;\Omega \times {{\mathbb{R}}^ + },\end{array} \right.$ |
$Ω\subset\mathbb R^N$ |
$N≥ 3$ |
$m>0$ |
$\frac{χ}{μ}$ |
$m>1$ |
$m>\frac{2N}{N+2}$ |
$m≤ \frac{2N}{N+2}$ |
References:
[1] |
X. Cao, Boundedness in a three-dimensional chemotaxis-haptotaxis model, Z. Angew. Math. Phys. , 67 (2016), Art. 11, 13 pp. |
[2] |
M. A. J. Chaplain and G. Lolas,
Mathematical modelling of cancer invasion of tissue: Dynamic heterogeneity, Networks and Heterogeneous Media, 1 (2006), 399-439.
doi: 10.3934/nhm.2006.1.399. |
[3] |
T. Cieslak and C. Stinner,
Finite-time blowup and global-in-time unbounded solutions to a parabolic-parabolic quasilinear Keller-Segel system in higher dimensions, J. Differential Equations, 252 (2012), 5832-5851.
doi: 10.1016/j.jde.2012.01.045. |
[4] |
D. Horstmann and G. Wang,
Blow-up in a chemotaxis model without symmetry assumptions, European J. Appl. Math., 12 (2001), 159-177.
|
[5] |
C. Jin,
Boundedness and global solvability to a chemotaxis model with nonlinear diffusion, J. Differential Equations, 263 (2017), 5759-5772.
doi: 10.1016/j.jde.2017.06.034. |
[6] |
E. Keller and A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415. Available from: https://www.researchgate.net/publication/17711401_Initiation_of_Slime_Mold_Aggregation_Viewed_as_an_Instability.
doi: 10.1016/0022-5193(70)90092-5. |
[7] |
Y. Li and J. Lankeit,
Boundedness in a chemotaxis-haptotaxis model with nonlinear diffusion, Nonlinearity, 29 (2016), 1564-1595.
doi: 10.1088/0951-7715/29/5/1564. |
[8] |
L. Nirenberg,
On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa, 13 (1959), 115-162.
|
[9] |
Y. Sugiyama,
Time global existence and asymptotic behavior of solutions to degenerate quasilinear parabolic sys-tems of chemotaxis, Differential Integral Equations, 20 (2007), 133-180.
|
[10] |
Z. Szymanska, C. Morales-Rodrigo, M. Lachowicz and M. Chaplain,
Mathematical modelling of cancer invasion of tissue: The role and effect of nonlocal interactions, Math. Models Methods Appl. Sci., 19 (2009), 257-281.
doi: 10.1142/S0218202509003425. |
[11] |
T. Senba and T. Suzuki,
Parabolic system of chemotaxis: Blowup in a finite and the infinite time, Methods Appl. Anal., 8 (2001), 349-367.
doi: 10.4310/MAA.2001.v8.n2.a9. |
[12] |
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. |
[13] |
Y. Tao, Boundedness in a two-dimensional chemotaxis-haptotaxis system, arXiv: 1407.7382v1. Google Scholar |
[14] |
Y. Tao and M. Winkler,
A chemotaxis-haptotaxis model: The roles of nonlinear diffusion and logistic source, SIAM J. Math. Anal., 43 (2011), 685-704.
doi: 10.1137/100802943. |
[15] |
Y. Tao and M. Winkler,
Global existence and boundedness in a Keller-Segel-Stokes model with arbitrary porous medium diffusion, Discrete Contin. Dyn. Syst., 32 (2012), 1901-1914.
doi: 10.3934/dcds.2012.32.1901. |
[16] |
Y. Tao and M. Winkler,
Large time behavior in a multidimensional chemotaxis-haptotaxis model with slow signal diffusion, SIAM J. Math. Anal., 47 (2015), 4229-4250.
doi: 10.1137/15M1014115. |
[17] |
J. Tello and M. Winkler,
A chemotaxis system with logistic source, Comm. Partial Differential Equations, 32 (2007), 849-877.
doi: 10.1080/03605300701319003. |
[18] |
M. Winkler,
Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl., 100 (2013), 748-767.
doi: 10.1016/j.matpur.2013.01.020. |
[19] |
Y. Wang,
Boundedness in the higher-dimensional chemotaxis-haptotaxis model with nonlinear diffusion, J. Differential Equations, 260 (2016), 1975-1989.
doi: 10.1016/j.jde.2015.09.051. |
[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. |
[21] |
M. Winkler,
Chemotaxis with logistic source: Very weak global solutions and their boundedness properties, J.Math. Anal. Appl., 48 (2008), 708-729.
doi: 10.1016/j.jmaa.2008.07.071. |
[22] |
M. Winkler,
Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Comm. Partial Differential Equations, 35 (2010), 1516-1537.
doi: 10.1080/03605300903473426. |
[23] |
J. Zheng,
Boundedness of solutions to a quasilinear higher-dimensional chemotaxis-haptotaxis model with nonlinear diffusion, Discrete Contin. Dyn. Syst., 37 (2017), 627-643.
|
show all references
References:
[1] |
X. Cao, Boundedness in a three-dimensional chemotaxis-haptotaxis model, Z. Angew. Math. Phys. , 67 (2016), Art. 11, 13 pp. |
[2] |
M. A. J. Chaplain and G. Lolas,
Mathematical modelling of cancer invasion of tissue: Dynamic heterogeneity, Networks and Heterogeneous Media, 1 (2006), 399-439.
doi: 10.3934/nhm.2006.1.399. |
[3] |
T. Cieslak and C. Stinner,
Finite-time blowup and global-in-time unbounded solutions to a parabolic-parabolic quasilinear Keller-Segel system in higher dimensions, J. Differential Equations, 252 (2012), 5832-5851.
doi: 10.1016/j.jde.2012.01.045. |
[4] |
D. Horstmann and G. Wang,
Blow-up in a chemotaxis model without symmetry assumptions, European J. Appl. Math., 12 (2001), 159-177.
|
[5] |
C. Jin,
Boundedness and global solvability to a chemotaxis model with nonlinear diffusion, J. Differential Equations, 263 (2017), 5759-5772.
doi: 10.1016/j.jde.2017.06.034. |
[6] |
E. Keller and A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415. Available from: https://www.researchgate.net/publication/17711401_Initiation_of_Slime_Mold_Aggregation_Viewed_as_an_Instability.
doi: 10.1016/0022-5193(70)90092-5. |
[7] |
Y. Li and J. Lankeit,
Boundedness in a chemotaxis-haptotaxis model with nonlinear diffusion, Nonlinearity, 29 (2016), 1564-1595.
doi: 10.1088/0951-7715/29/5/1564. |
[8] |
L. Nirenberg,
On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa, 13 (1959), 115-162.
|
[9] |
Y. Sugiyama,
Time global existence and asymptotic behavior of solutions to degenerate quasilinear parabolic sys-tems of chemotaxis, Differential Integral Equations, 20 (2007), 133-180.
|
[10] |
Z. Szymanska, C. Morales-Rodrigo, M. Lachowicz and M. Chaplain,
Mathematical modelling of cancer invasion of tissue: The role and effect of nonlocal interactions, Math. Models Methods Appl. Sci., 19 (2009), 257-281.
doi: 10.1142/S0218202509003425. |
[11] |
T. Senba and T. Suzuki,
Parabolic system of chemotaxis: Blowup in a finite and the infinite time, Methods Appl. Anal., 8 (2001), 349-367.
doi: 10.4310/MAA.2001.v8.n2.a9. |
[12] |
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. |
[13] |
Y. Tao, Boundedness in a two-dimensional chemotaxis-haptotaxis system, arXiv: 1407.7382v1. Google Scholar |
[14] |
Y. Tao and M. Winkler,
A chemotaxis-haptotaxis model: The roles of nonlinear diffusion and logistic source, SIAM J. Math. Anal., 43 (2011), 685-704.
doi: 10.1137/100802943. |
[15] |
Y. Tao and M. Winkler,
Global existence and boundedness in a Keller-Segel-Stokes model with arbitrary porous medium diffusion, Discrete Contin. Dyn. Syst., 32 (2012), 1901-1914.
doi: 10.3934/dcds.2012.32.1901. |
[16] |
Y. Tao and M. Winkler,
Large time behavior in a multidimensional chemotaxis-haptotaxis model with slow signal diffusion, SIAM J. Math. Anal., 47 (2015), 4229-4250.
doi: 10.1137/15M1014115. |
[17] |
J. Tello and M. Winkler,
A chemotaxis system with logistic source, Comm. Partial Differential Equations, 32 (2007), 849-877.
doi: 10.1080/03605300701319003. |
[18] |
M. Winkler,
Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl., 100 (2013), 748-767.
doi: 10.1016/j.matpur.2013.01.020. |
[19] |
Y. Wang,
Boundedness in the higher-dimensional chemotaxis-haptotaxis model with nonlinear diffusion, J. Differential Equations, 260 (2016), 1975-1989.
doi: 10.1016/j.jde.2015.09.051. |
[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. |
[21] |
M. Winkler,
Chemotaxis with logistic source: Very weak global solutions and their boundedness properties, J.Math. Anal. Appl., 48 (2008), 708-729.
doi: 10.1016/j.jmaa.2008.07.071. |
[22] |
M. Winkler,
Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Comm. Partial Differential Equations, 35 (2010), 1516-1537.
doi: 10.1080/03605300903473426. |
[23] |
J. Zheng,
Boundedness of solutions to a quasilinear higher-dimensional chemotaxis-haptotaxis model with nonlinear diffusion, Discrete Contin. Dyn. Syst., 37 (2017), 627-643.
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