
-
Previous Article
Flocking of non-identical Cucker-Smale models on general coupling network
- DCDS-B Home
- This Issue
-
Next Article
Boundary dynamics of the replicator equations for neutral models of cyclic dominance
Numerical study of vanishing and spreading dynamics of chemotaxis systems with logistic source and a free boundary
1. | School of Mathematics, Jilin University, Changchun, Jilin 130012, China |
2. | School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, P. R. China |
3. | Department of Mathematics and Statistics, Auburn University, AL 36849, USA |
The current paper is to investigate the numerical approximation of logistic type chemotaxis models in one space dimension with a free boundary. Such a model with a free boundary describes the spreading of a new or invasive species subject to the influence of some chemical substances in an environment with a free boundary representing the spreading front (see Bao and Shen [
References:
[1] |
L. Bao and W. Shen,
Logistic type attraction-repulsion chemotaxis systems with a free boundary or unbounded boundary. I. Asymptotic dynamics in fixed unbounded domain, Discrete Contin. Dyn. Syst. Ser. A, 40 (2020), 1107-1130.
doi: 10.3934/dcds.2020072. |
[2] |
L. Bao and W. Shen, Logistic type attraction-repulsion chemotaxis systems with a free boundary or unbounded boundary. II, Spreading-vanishing dichotomy in a domain with a free boundary, preprint. Google Scholar |
[3] |
N. Bellomo, A. Bellouquid, Y. Tao and M. Winkler,
Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci., 25 (2015), 1663-1763.
doi: 10.1142/S021820251550044X. |
[4] |
C. C. Chiu and J. L. Yu,
An optimal adaptive time-stepping scheme for solving reaction-diffusion-chemotaxis systems, Math. Biosci. Eng., 4 (2007), 187-203.
doi: 10.3934/mbe.2007.4.187. |
[5] |
J. I. Diaz and T. Nagai,
Symmetrization in a parabolic-elliptic system related to chemotaxis, Advances in Mathematical Science and Applications, 5 (1995), 659-680.
|
[6] |
J. I. Diaz, T. Nagai and J.-M. Rakotoson,
Symmetrization techniques on unbounded domains: Application to a chemotaxis system on $\mathbb{R}^N$, J. Differential Equations, 145 (1998), 156-183.
doi: 10.1006/jdeq.1997.3389. |
[7] |
Y.-H. Du and Z.-G. Lin,
Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary, SIAM J. Math. Anal., 42 (2010), 377-405.
doi: 10.1137/090771089. |
[8] |
Y.-H. Du and X. Liang, Pulsating semi-waves in periodic media and spreading speed determined by a free boundary model, Ann. Inst. H. Poincar$\acute{e}$ Anal. Non Lin$\acute{e}$aire, 32 (2015), 279–305.
doi: 10.1016/j.anihpc.2013.11.004. |
[9] |
E. Galakhov, O. Salieva and J. I. Tello,
On a parabolic-elliptic system with chemotaxis and logistic type growth, J. Differential Equations, 261 (2016), 4631-4647.
doi: 10.1016/j.jde.2016.07.008. |
[10] |
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. |
[11] |
T. B. Issa and W. Shen,
Dynamics in chemotaxis models of parabolic-elliptic type on bounded domain with time and space dependent logistic sources, SIAM J. Appl. Dyn. Syst., 16 (2017), 926-973.
doi: 10.1137/16M1092428. |
[12] |
H. Jin and Z. A. Wang,
Boundedness, blowup and critical mass phenomenon in competing chemotaxis, J. Differential Equations, 260 (2016), 162-196.
doi: 10.1016/j.jde.2015.08.040. |
[13] |
K. Kanga and A. Steven,
Blowup and global solutions in a chemotaxis-growth system, Nonlinear Analysis, 135 (2016), 57-72.
doi: 10.1016/j.na.2016.01.017. |
[14] |
E. F. Keller and L. A. Segel,
Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415.
doi: 10.1016/0022-5193(70)90092-5. |
[15] |
E. F. Keller and L. A. Segel,
A Model for chemotaxis, J. Theoret. Biol., 30 (1971), 225-234.
doi: 10.1016/0022-5193(71)90050-6. |
[16] |
H. G. Landau,
Heat conduction in a melting solid, Quaterly of Applied Mathematics, 8 (1950), 81-94.
doi: 10.1090/qam/33441. |
[17] |
F. Li, X. Liang and W. Shen,
Diffusive KPP equations with free boundaries in time almost periodic environments: I. Spreading and vanishing dichotomy, Discrete Contin. Dyn. Syst, 36 (2016), 3317-3338.
doi: 10.3934/dcds.2016.36.3317. |
[18] |
F. Li, X. Liang and W. Shen,
Diffusive KPP equations with free boundaries in time almost periodic environments: II. Spreading speeds and semi-wave solutions, J. Differential Equations, 261 (2016), 2403-2445.
doi: 10.1016/j.jde.2016.04.035. |
[19] |
R. H. Li, Z. Y. Chen and W. Wu, Generalized Difference Methods for Differential Equations- Numerical Analysis of Finite Volume Methods, Marcel Dekker, Inc, 2000. |
[20] |
X. J. Li, C. W. Shu and Y. Yang,
Local discontinuous Galerkin method for the Keller-Segel chemotaxis model, J. Sci. Comput., 73 (2017), 943-967.
doi: 10.1007/s10915-016-0354-y. |
[21] |
J. G. Liu, L. Wang and Z. N. Zhou,
Positivity-preserving and asymptotic preserving method for 2D Keller-Segal equations, Math. Comp., 87 (2018), 1165-1189.
doi: 10.1090/mcom/3250. |
[22] |
S. Liu and X. F. Liu, Numerical methods for a wwo-species competition-diffusion model with free boundaries, Mathematics, 6 (2018), 72-96. Google Scholar |
[23] |
S. Liu, Y. H. Du and X. F. Liu,
Numerical studies of a class of reaction-diffusion equations with stefan conditions, International Journal of Computer Mathematics, 97 (2020), 959-979.
doi: 10.1080/00207160.2019.1599868. |
[24] |
J. L. Lockwood, M. F. Hoopes and M. P. Marchetti, Invasion Ecology, Blackwell Publishing, 2007. Google Scholar |
[25] |
M. Luca, A. Chavez-Ross, L. Edelstein-Keshet and A. Mogilner,
Chemotactic signaling, microglia, and Alzheimer's disease senile plaques: Is there a connection?, Bulletin of Mathematical Biology, 65 (2003), 693-730.
doi: 10.1016/S0092-8240(03)00030-2. |
[26] |
M.-A. Piqueras, R. Company and L. L$\acute{o}$dar,
A front-fixing numerical method for a free boundary nonlinear diffusion logistic population model, J. Comput. Appl. Math., 309 (2017), 473-481.
doi: 10.1016/j.cam.2016.02.029. |
[27] |
T. Nagai, T. Senba and K. Yoshida,
Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcialaj Ekvacioj, 40 (1997), 411-433.
|
[28] |
N. Saito and T. Suzuki,
Notes on finite difference schemes to a parabolic-elliptic system modelling chemotaxis, Appl. Math. Comput., 171 (2005), 72-90.
doi: 10.1016/j.amc.2005.01.037. |
[29] |
R. B. Salako and W. Shen,
Spreading Speeds and Traveling waves of a parabolic-elliptic chemotaxis system with logistic source on $\mathbb{R}^N$, Discrete Contin. Dyn. Syst., 37 (2017), 6189-6225.
doi: 10.3934/dcds.2017268. |
[30] |
R. B. Salako, W. Shen and S. W. Xue,
Can chemotaxis speed up or slow down the spatial spreading in parabolic-elliptic chemotaxis systems with logistic source?, J. Math. Biol., 79 (2019), 1455-1490.
doi: 10.1007/s00285-019-01400-0. |
[31] |
N. Shigesada and K. Kawasaki, Biological Invasions: Theory and Practice, Oxford Series in Ecology and Evolution, Oxford Univ. Press., Oxford, 1997. Google Scholar |
[32] |
Y. Sugiyama,
Global existence in sub-critical cases and finite time blow up in super critical cases to degenerate Keller-Segel systems, Differential Integral Equations, 19 (2006), 841-876.
|
[33] |
Y. Sugiyama and H. Kunii,
Global Existence and decay properties for a degenerate keller-Segel model with a power factor in drift term, J. Differential Equations, 227 (2006), 333-364.
doi: 10.1016/j.jde.2006.03.003. |
[34] |
Y.-S. Tao and Z. A. Wang,
Competing effects of attraction vs. repulsion in chemotaxis, Math. Models Methods Appl. Sci., 23 (2013), 1-36.
doi: 10.1142/S0218202512500443. |
[35] |
J. I. Tello and M. Winkler,
A chemotaxis system with logistic source, Communications in Partial Differential Equations, 32 (2007), 849-877.
doi: 10.1080/03605300701319003. |
[36] |
L. Wang, C. Mu and P. Zheng,
On a quasilinear parabolic-elliptic chemotaxis system with logistic source, J. Differential Equations, 256 (2014), 1847-1872.
doi: 10.1016/j.jde.2013.12.007. |
[37] |
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. |
[38] |
M. Winkler,
Blow-up in a higher-dimensional chemotaxis system despite logistic growth restriction, Journal of Mathematical Analysis and Applications, 384 (2011), 261-272.
doi: 10.1016/j.jmaa.2011.05.057. |
[39] |
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. |
[40] |
M. Winkler,
Global asymptotic stability of constant equilibria in a fully parabolic chemotaxis system with strong logistic dampening, J. Differential Equations, 257 (2014), 1056-1077.
doi: 10.1016/j.jde.2014.04.023. |
[41] |
M. Winkler,
How far can chemotactic cross-diffusion enforce exceeding carrying capacities?, J. Nonlinear Sci., 24 (2014), 809-855.
doi: 10.1007/s00332-014-9205-x. |
[42] |
T. Yokota and N. Yoshino,
Existence of solutions to chemotaxis dynamics with logistic source, Discrete Contin. Dyn. Syst. Dynamical systems, differential equations and applications. 10th AIMS Conference. Suppl., 2015 (2015), 1125-1133.
doi: 10.3934/proc.2015.1125. |
[43] |
P. Zheng, C. Mu, X. Hu and Y. Tian,
Boundedness of solutions in a chemotaxis system with nonlinear sensitivity and logistic source, Math. Anal. Appl., 424 (2015), 509-522.
doi: 10.1016/j.jmaa.2014.11.031. |
show all references
References:
[1] |
L. Bao and W. Shen,
Logistic type attraction-repulsion chemotaxis systems with a free boundary or unbounded boundary. I. Asymptotic dynamics in fixed unbounded domain, Discrete Contin. Dyn. Syst. Ser. A, 40 (2020), 1107-1130.
doi: 10.3934/dcds.2020072. |
[2] |
L. Bao and W. Shen, Logistic type attraction-repulsion chemotaxis systems with a free boundary or unbounded boundary. II, Spreading-vanishing dichotomy in a domain with a free boundary, preprint. Google Scholar |
[3] |
N. Bellomo, A. Bellouquid, Y. Tao and M. Winkler,
Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci., 25 (2015), 1663-1763.
doi: 10.1142/S021820251550044X. |
[4] |
C. C. Chiu and J. L. Yu,
An optimal adaptive time-stepping scheme for solving reaction-diffusion-chemotaxis systems, Math. Biosci. Eng., 4 (2007), 187-203.
doi: 10.3934/mbe.2007.4.187. |
[5] |
J. I. Diaz and T. Nagai,
Symmetrization in a parabolic-elliptic system related to chemotaxis, Advances in Mathematical Science and Applications, 5 (1995), 659-680.
|
[6] |
J. I. Diaz, T. Nagai and J.-M. Rakotoson,
Symmetrization techniques on unbounded domains: Application to a chemotaxis system on $\mathbb{R}^N$, J. Differential Equations, 145 (1998), 156-183.
doi: 10.1006/jdeq.1997.3389. |
[7] |
Y.-H. Du and Z.-G. Lin,
Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary, SIAM J. Math. Anal., 42 (2010), 377-405.
doi: 10.1137/090771089. |
[8] |
Y.-H. Du and X. Liang, Pulsating semi-waves in periodic media and spreading speed determined by a free boundary model, Ann. Inst. H. Poincar$\acute{e}$ Anal. Non Lin$\acute{e}$aire, 32 (2015), 279–305.
doi: 10.1016/j.anihpc.2013.11.004. |
[9] |
E. Galakhov, O. Salieva and J. I. Tello,
On a parabolic-elliptic system with chemotaxis and logistic type growth, J. Differential Equations, 261 (2016), 4631-4647.
doi: 10.1016/j.jde.2016.07.008. |
[10] |
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. |
[11] |
T. B. Issa and W. Shen,
Dynamics in chemotaxis models of parabolic-elliptic type on bounded domain with time and space dependent logistic sources, SIAM J. Appl. Dyn. Syst., 16 (2017), 926-973.
doi: 10.1137/16M1092428. |
[12] |
H. Jin and Z. A. Wang,
Boundedness, blowup and critical mass phenomenon in competing chemotaxis, J. Differential Equations, 260 (2016), 162-196.
doi: 10.1016/j.jde.2015.08.040. |
[13] |
K. Kanga and A. Steven,
Blowup and global solutions in a chemotaxis-growth system, Nonlinear Analysis, 135 (2016), 57-72.
doi: 10.1016/j.na.2016.01.017. |
[14] |
E. F. Keller and L. A. Segel,
Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415.
doi: 10.1016/0022-5193(70)90092-5. |
[15] |
E. F. Keller and L. A. Segel,
A Model for chemotaxis, J. Theoret. Biol., 30 (1971), 225-234.
doi: 10.1016/0022-5193(71)90050-6. |
[16] |
H. G. Landau,
Heat conduction in a melting solid, Quaterly of Applied Mathematics, 8 (1950), 81-94.
doi: 10.1090/qam/33441. |
[17] |
F. Li, X. Liang and W. Shen,
Diffusive KPP equations with free boundaries in time almost periodic environments: I. Spreading and vanishing dichotomy, Discrete Contin. Dyn. Syst, 36 (2016), 3317-3338.
doi: 10.3934/dcds.2016.36.3317. |
[18] |
F. Li, X. Liang and W. Shen,
Diffusive KPP equations with free boundaries in time almost periodic environments: II. Spreading speeds and semi-wave solutions, J. Differential Equations, 261 (2016), 2403-2445.
doi: 10.1016/j.jde.2016.04.035. |
[19] |
R. H. Li, Z. Y. Chen and W. Wu, Generalized Difference Methods for Differential Equations- Numerical Analysis of Finite Volume Methods, Marcel Dekker, Inc, 2000. |
[20] |
X. J. Li, C. W. Shu and Y. Yang,
Local discontinuous Galerkin method for the Keller-Segel chemotaxis model, J. Sci. Comput., 73 (2017), 943-967.
doi: 10.1007/s10915-016-0354-y. |
[21] |
J. G. Liu, L. Wang and Z. N. Zhou,
Positivity-preserving and asymptotic preserving method for 2D Keller-Segal equations, Math. Comp., 87 (2018), 1165-1189.
doi: 10.1090/mcom/3250. |
[22] |
S. Liu and X. F. Liu, Numerical methods for a wwo-species competition-diffusion model with free boundaries, Mathematics, 6 (2018), 72-96. Google Scholar |
[23] |
S. Liu, Y. H. Du and X. F. Liu,
Numerical studies of a class of reaction-diffusion equations with stefan conditions, International Journal of Computer Mathematics, 97 (2020), 959-979.
doi: 10.1080/00207160.2019.1599868. |
[24] |
J. L. Lockwood, M. F. Hoopes and M. P. Marchetti, Invasion Ecology, Blackwell Publishing, 2007. Google Scholar |
[25] |
M. Luca, A. Chavez-Ross, L. Edelstein-Keshet and A. Mogilner,
Chemotactic signaling, microglia, and Alzheimer's disease senile plaques: Is there a connection?, Bulletin of Mathematical Biology, 65 (2003), 693-730.
doi: 10.1016/S0092-8240(03)00030-2. |
[26] |
M.-A. Piqueras, R. Company and L. L$\acute{o}$dar,
A front-fixing numerical method for a free boundary nonlinear diffusion logistic population model, J. Comput. Appl. Math., 309 (2017), 473-481.
doi: 10.1016/j.cam.2016.02.029. |
[27] |
T. Nagai, T. Senba and K. Yoshida,
Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcialaj Ekvacioj, 40 (1997), 411-433.
|
[28] |
N. Saito and T. Suzuki,
Notes on finite difference schemes to a parabolic-elliptic system modelling chemotaxis, Appl. Math. Comput., 171 (2005), 72-90.
doi: 10.1016/j.amc.2005.01.037. |
[29] |
R. B. Salako and W. Shen,
Spreading Speeds and Traveling waves of a parabolic-elliptic chemotaxis system with logistic source on $\mathbb{R}^N$, Discrete Contin. Dyn. Syst., 37 (2017), 6189-6225.
doi: 10.3934/dcds.2017268. |
[30] |
R. B. Salako, W. Shen and S. W. Xue,
Can chemotaxis speed up or slow down the spatial spreading in parabolic-elliptic chemotaxis systems with logistic source?, J. Math. Biol., 79 (2019), 1455-1490.
doi: 10.1007/s00285-019-01400-0. |
[31] |
N. Shigesada and K. Kawasaki, Biological Invasions: Theory and Practice, Oxford Series in Ecology and Evolution, Oxford Univ. Press., Oxford, 1997. Google Scholar |
[32] |
Y. Sugiyama,
Global existence in sub-critical cases and finite time blow up in super critical cases to degenerate Keller-Segel systems, Differential Integral Equations, 19 (2006), 841-876.
|
[33] |
Y. Sugiyama and H. Kunii,
Global Existence and decay properties for a degenerate keller-Segel model with a power factor in drift term, J. Differential Equations, 227 (2006), 333-364.
doi: 10.1016/j.jde.2006.03.003. |
[34] |
Y.-S. Tao and Z. A. Wang,
Competing effects of attraction vs. repulsion in chemotaxis, Math. Models Methods Appl. Sci., 23 (2013), 1-36.
doi: 10.1142/S0218202512500443. |
[35] |
J. I. Tello and M. Winkler,
A chemotaxis system with logistic source, Communications in Partial Differential Equations, 32 (2007), 849-877.
doi: 10.1080/03605300701319003. |
[36] |
L. Wang, C. Mu and P. Zheng,
On a quasilinear parabolic-elliptic chemotaxis system with logistic source, J. Differential Equations, 256 (2014), 1847-1872.
doi: 10.1016/j.jde.2013.12.007. |
[37] |
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. |
[38] |
M. Winkler,
Blow-up in a higher-dimensional chemotaxis system despite logistic growth restriction, Journal of Mathematical Analysis and Applications, 384 (2011), 261-272.
doi: 10.1016/j.jmaa.2011.05.057. |
[39] |
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. |
[40] |
M. Winkler,
Global asymptotic stability of constant equilibria in a fully parabolic chemotaxis system with strong logistic dampening, J. Differential Equations, 257 (2014), 1056-1077.
doi: 10.1016/j.jde.2014.04.023. |
[41] |
M. Winkler,
How far can chemotactic cross-diffusion enforce exceeding carrying capacities?, J. Nonlinear Sci., 24 (2014), 809-855.
doi: 10.1007/s00332-014-9205-x. |
[42] |
T. Yokota and N. Yoshino,
Existence of solutions to chemotaxis dynamics with logistic source, Discrete Contin. Dyn. Syst. Dynamical systems, differential equations and applications. 10th AIMS Conference. Suppl., 2015 (2015), 1125-1133.
doi: 10.3934/proc.2015.1125. |
[43] |
P. Zheng, C. Mu, X. Hu and Y. Tian,
Boundedness of solutions in a chemotaxis system with nonlinear sensitivity and logistic source, Math. Anal. Appl., 424 (2015), 509-522.
doi: 10.1016/j.jmaa.2014.11.031. |
























[1] |
Bo Duan, Zhengce Zhang. A reaction-diffusion-advection two-species competition system with a free boundary in heterogeneous environment. Discrete & Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021067 |
[2] |
Marita Holtmannspötter, Arnd Rösch, Boris Vexler. A priori error estimates for the space-time finite element discretization of an optimal control problem governed by a coupled linear PDE-ODE system. Mathematical Control & Related Fields, 2021 doi: 10.3934/mcrf.2021014 |
[3] |
Kuan-Hsiang Wang. An eigenvalue problem for nonlinear Schrödinger-Poisson system with steep potential well. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021030 |
[4] |
Carlos Fresneda-Portillo, Sergey E. Mikhailov. Analysis of Boundary-Domain Integral Equations to the mixed BVP for a compressible stokes system with variable viscosity. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3059-3088. doi: 10.3934/cpaa.2019137 |
[5] |
Nikolaos Roidos. Expanding solutions of quasilinear parabolic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021026 |
[6] |
Zhouxin Li, Yimin Zhang. Ground states for a class of quasilinear Schrödinger equations with vanishing potentials. Communications on Pure & Applied Analysis, 2021, 20 (2) : 933-954. doi: 10.3934/cpaa.2020298 |
[7] |
M. Grasselli, V. Pata. Asymptotic behavior of a parabolic-hyperbolic system. Communications on Pure & Applied Analysis, 2004, 3 (4) : 849-881. doi: 10.3934/cpaa.2004.3.849 |
[8] |
Yizhuo Wang, Shangjiang Guo. A SIS reaction-diffusion model with a free boundary condition and nonhomogeneous coefficients. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1627-1652. doi: 10.3934/dcdsb.2018223 |
[9] |
Qiang Guo, Dong Liang. An adaptive wavelet method and its analysis for parabolic equations. Numerical Algebra, Control & Optimization, 2013, 3 (2) : 327-345. doi: 10.3934/naco.2013.3.327 |
[10] |
Bernold Fiedler, Carlos Rocha, Matthias Wolfrum. Sturm global attractors for $S^1$-equivariant parabolic equations. Networks & Heterogeneous Media, 2012, 7 (4) : 617-659. doi: 10.3934/nhm.2012.7.617 |
[11] |
Marco Ghimenti, Anna Maria Micheletti. Compactness results for linearly perturbed Yamabe problem on manifolds with boundary. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1757-1778. doi: 10.3934/dcdss.2020453 |
[12] |
Yanqin Fang, Jihui Zhang. Multiplicity of solutions for the nonlinear Schrödinger-Maxwell system. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1267-1279. doi: 10.3934/cpaa.2011.10.1267 |
[13] |
Manoel J. Dos Santos, Baowei Feng, Dilberto S. Almeida Júnior, Mauro L. Santos. Global and exponential attractors for a nonlinear porous elastic system with delay term. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2805-2828. doi: 10.3934/dcdsb.2020206 |
[14] |
Madalina Petcu, Roger Temam. The one dimensional shallow water equations with Dirichlet boundary conditions on the velocity. Discrete & Continuous Dynamical Systems - S, 2011, 4 (1) : 209-222. doi: 10.3934/dcdss.2011.4.209 |
[15] |
Nizami A. Gasilov. Solving a system of linear differential equations with interval coefficients. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2739-2747. doi: 10.3934/dcdsb.2020203 |
[16] |
Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1693-1716. doi: 10.3934/dcdss.2020450 |
[17] |
Jiangxing Wang. Convergence analysis of an accurate and efficient method for nonlinear Maxwell's equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2429-2440. doi: 10.3934/dcdsb.2020185 |
[18] |
Hong Yi, Chunlai Mu, Guangyu Xu, Pan Dai. A blow-up result for the chemotaxis system with nonlinear signal production and logistic source. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2537-2559. doi: 10.3934/dcdsb.2020194 |
[19] |
Xiaoming Wang. Quasi-periodic solutions for a class of second order differential equations with a nonlinear damping term. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 543-556. doi: 10.3934/dcdss.2017027 |
[20] |
Marion Darbas, Jérémy Heleine, Stephanie Lohrengel. Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations. Inverse Problems & Imaging, 2020, 14 (6) : 1107-1133. doi: 10.3934/ipi.2020056 |
2019 Impact Factor: 1.27
Tools
Metrics
Other articles
by authors
[Back to Top]