• Previous Article
    Oscillations induced by quiescent adult female in a model of wild aedes aegypti mosquitoes
  • DCDS-S Home
  • This Issue
  • Next Article
    A fractional-order delay differential model with optimal control for cancer treatment based on synergy between anti-angiogenic and immune cell therapies
September  2020, 13(9): 2425-2442. doi: 10.3934/dcdss.2020193

Dynamics of solutions of a reaction-diffusion equation with delayed inhibition

1. 

Laboratoire d'Analyse Non linéaire et Mathématiques Appliquées, Département de Mathématiques, Université Aboubekr Belkaïd Tlemcen, 13000 Tlemcen, Algeria

2. 

Institute of Problems of Mechanical Engineering, Russian Academy of Sciences, 199178 Saint Petersburg, Russia

3. 

Peoples' Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya St, Moscow, 117198, Russian Federation

4. 

Institut Camille Jordan, UMR 5208 CNRS, University Lyon 1, 69622 Villeurbanne, France

5. 

INRIA, Université de Lyon, Université Lyon 1, Institut Camille Jordan, 43 Bd. du 11 Novembre 1918, 69200 Villeurbanne Cedex, France

6. 

Marchuk Institute of Numerical Mathematics of the RAS, ul. Gubkina 8, 119333 Moscow, Russian Federation

* Corresponding author: Vitaly Volpert

Received  October 2018 Revised  June 2019 Published  September 2020 Early access  December 2019

Reaction-diffusion equation with a logistic production term and a delayed inhibition term is studied. Global stability of the homogeneous in space equilibrium is proved under some conditions on the delay term. In the case where these conditions are not satisfied, this solution can become unstable resulting in the emergence of spatiotemporal pattern formation studied in numerical simulations.

Citation: Tarik Mohammed Touaoula, Mohammed Nor Frioui, Nikolay Bessonov, Vitaly Volpert. Dynamics of solutions of a reaction-diffusion equation with delayed inhibition. Discrete and Continuous Dynamical Systems - S, 2020, 13 (9) : 2425-2442. doi: 10.3934/dcdss.2020193
References:
[1]

L. BerezanskyE. Braverman and L. Idels, Mackey-Glass model of hematopoiesis with non-monotone feedback: Stability, oscillation and control, Appl. Math. Comput., 219 (2013), 6268-6283.  doi: 10.1016/j.amc.2012.12.043.

[2]

N. BessonovG. BocharovT. M. TouaoulaS. Trofimchuk and V. Volpert, Delay reaction-diffusion equation for infection dynamics, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 2073-2091.  doi: 10.3934/dcdsb.2019085.

[3]

N. BessonovN. Reinberg and V. Volpert, Mathematics of Darwin's diagram, Math. Model. Nat. Phenom., 9 (2014), 5-25.  doi: 10.1051/mmnp/20149302.

[4]

N. BessonovN. ReinbergM. Banerjee and V. Volpert, The origin of species by means of mathematical modelling, Acta Bioteoretica, 66 (2018), 333-344.  doi: 10.1007/s10441-018-9328-9.

[5]

G. Bocharov, A. Meyerhans, N. Bessonov, S. Trofimchuk and V. Volpert, Spatiotemporal dynamics of virus infection spreading in tissues, PLoS ONE, (2016). doi: 10.1371/journal.pone.0168576.

[6]

G. Bocharov, A. Meyerhans, N. Bessonov, S. Trofimchuk and V. Volpert, Modelling the dynamics of virus infection and immune response in space and time, Internat. J. Parallel Emergent Distributed Syst., (2017), 341–355. doi: 10.1080/17445760.2017.1363203.

[7]

E. Braverman and S. Zhukovskiy, Absolute and delay-dependent stability of equations with a distributed delay, Discrete Contin. Dyn. Syst., 32 (2012), 2041-2061.  doi: 10.3934/dcds.2012.32.2041.

[8]

K. Deng and Y. Wu, On the diffusive Nicholson's blowflies equation with distributed delay, Appl. Math. Lett., 50 (2015), 126-132.  doi: 10.1016/j.aml.2015.06.013.

[9]

W. E. Fitzgibbon, Semilinear functional differential equations in Banach space, J. Differential Equations, 29 (1978), 1-14.  doi: 10.1016/0022-0396(78)90037-2.

[10]

Z. LingL. Zhang and Z. Lin, Turing pattern formation in a predator-prey system with cross diffusion, Appl. Math. Model., 38 (2014), 5022-5032.  doi: 10.1016/j.apm.2014.04.015.

[11]

E. LizM. PintoV. Tkachenko and S. Tromichuk, A global stability criterion for a family of delayed population models, Quart. Appl. Math., 63 (2005), 56-70.  doi: 10.1090/S0033-569X-05-00951-3.

[12]

E. Liz and G. Rost, On the global attractor of delay differential equations with unimodal feedback, Discrete Contin. Dyn. Syst., 24 (2009), 1215-1224.  doi: 10.3934/dcds.2009.24.1215.

[13]

E. LizV. Tkachenko and S. Tromichuk, A global stability criterion for scalar functional differential equations, SIAM. J. Math. Anal., 35 (2003), 596-622.  doi: 10.1137/S0036141001399222.

[14]

J. Mallet-Paret and R. Nussbaum, A differential-delay equation arising in optics and physiology, SIAM. J. Math. Anal., 20 (1989), 249-292.  doi: 10.1137/0520019.

[15]

J. Mallet-Paret and G. R. Sell., The Poincaré-Bendixson theorem for monotone cyclic feedback systems with delay, J. Differential Equations, 125 (1996), 441-489.  doi: 10.1006/jdeq.1996.0037.

[16]

R. Martin and H. L. Smith, Abstract functional differential equations and reaction-diffusion systems, Trans. Amer. Math. Soc., 321 (1990), 1-44.  doi: 10.2307/2001590.

[17]

R. Martin and H. L. Smith, Reaction-diffusion systems with time delay: Monotonicity, invariance, comparison and convergence, J. Reine Angew. Math., 413 (1991), 1-35.  doi: 10.1515/crll.1991.413.1.

[18] C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992.  doi: 10.1007/978-1-4615-3034-3.
[19]

G. Rost and J. Wu, Domain-decomposition method for the global dynamics of delay differential equations with unimodal feedback, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 463 (2007), 2655-2669.  doi: 10.1098/rspa.2007.1890.

[20]

H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs, 41, American Mathematical Society, Providence, RI, 1995.

[21]

G. Q. SunC. H. WangL. L. ChangY. P. Wu and L. L. Zhen Jin, Effects of feedback regulation on vegetation patterns in semi-arid environments, Appl. Math. Model., 61 (2018), 200-215.  doi: 10.1016/j.apm.2018.04.010.

[22]

H. R. Thieme, Mathematics in Population Biology, Princeton Series in Theoretical and Computational Biology, Princeton University Press, Princeton, NJ, 2003. doi: 10.2307/j.ctv301f9v.

[23]

H. R. Thieme and X.-Q. Zhao, A non-local delayed and diffusive predator-prey model, Nonlinear Anal. Real World Appl., 2 (2001), 145-160.  doi: 10.1016/S0362-546X(00)00112-7.

[24]

T. M. Touaoula, Global stability for a class of functional differential equations (Application to Nicholson's blowflies and Mackey-Glass models), Discrete Contin. Dyn. Syst., 38 (2018), 4391-4419.  doi: 10.3934/dcds.2018191.

[25]

S. Trofimchuk and V. Volpert, Traveling waves for a bistable reaction-diffusion equation with delay, SIAM J. Math. Anal., 50 (2018), 1175-1190.  doi: 10.1137/17M1115587.

[26]

A. I. Volpert, V. A. Volpert and V. A. Volpert, Traveling Wave Solutions of Parabolic Systems: Translation of Mathematical Monographs, Translations of Mathematical Monographs, 140, American Mathematical Society, Providence, RI, 1994.

[27]

J. Wu, Theory and applications of partial functional differential equations, Applied Mathematical Sciences, 119, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-4050-1.

[28]

D. Xu and X.-Q. Zhao, A nonlocal reaction-diffusion population model with stage structure, Can. Appl. Math. Q., 11 (2003), 303-319. 

[29]

T. YiY. Chen and J. Wu, Global dynamics of delayed reaction-diffusion equations in unbounded domains, Z. Angew. Math. Phys., 63 (2012), 793-812.  doi: 10.1007/s00033-012-0224-x.

[30]

T. Yi and X. Zou, Global attractivity of the diffusive Nicholson blowflies equation with Neumann boundary condition: A non-monotone case, J. Differential Equations, 245 (2008), 3376-3388.  doi: 10.1016/j.jde.2008.03.007.

[31]

T. Yi and X. Zou, Map dynamics versus dynamics of associated delay reaction-diffusion equations with a Neumann condition, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 466 (2010), 2955-2973.  doi: 10.1098/rspa.2009.0650.

[32]

T. Yi and X. Zou, Global dynamics of a delay differential equation with spatial non-locality in an unbounded domain, J. Differential Equations, 251 (2011), 2598-2611.  doi: 10.1016/j.jde.2011.04.027.

[33]

T. Yi and X. Zou, On Dirichlet problem for a class of delayed reaction-diffusion equations with spatial non-locality, J. Dynam. Differential Equations, 25 (2013), 959-979.  doi: 10.1007/s10884-013-9324-3.

[34]

T. Yi and X. Zou, Dirichlet problem for a delayed reaction-diffusion equation on a semi-infinite interval, J. Dynam. Differential Equations, 28 (2016), 1007-1030.  doi: 10.1007/s10884-015-9457-7.

[35]

Y. Yuan and X. Q. Zhao, Global stability for non-monotone delay equations (with application to a model of blood cell production), J. Differential Equations, 252 (2012), 2189-2209.  doi: 10.1016/j.jde.2011.08.026.

[36]

X. Q. Zhao, Global attractivity in a class of nonmonotone reaction-diffusion equations with time delay, Can. Appl. Math. Q., 17 (2009), 271-281. 

show all references

References:
[1]

L. BerezanskyE. Braverman and L. Idels, Mackey-Glass model of hematopoiesis with non-monotone feedback: Stability, oscillation and control, Appl. Math. Comput., 219 (2013), 6268-6283.  doi: 10.1016/j.amc.2012.12.043.

[2]

N. BessonovG. BocharovT. M. TouaoulaS. Trofimchuk and V. Volpert, Delay reaction-diffusion equation for infection dynamics, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 2073-2091.  doi: 10.3934/dcdsb.2019085.

[3]

N. BessonovN. Reinberg and V. Volpert, Mathematics of Darwin's diagram, Math. Model. Nat. Phenom., 9 (2014), 5-25.  doi: 10.1051/mmnp/20149302.

[4]

N. BessonovN. ReinbergM. Banerjee and V. Volpert, The origin of species by means of mathematical modelling, Acta Bioteoretica, 66 (2018), 333-344.  doi: 10.1007/s10441-018-9328-9.

[5]

G. Bocharov, A. Meyerhans, N. Bessonov, S. Trofimchuk and V. Volpert, Spatiotemporal dynamics of virus infection spreading in tissues, PLoS ONE, (2016). doi: 10.1371/journal.pone.0168576.

[6]

G. Bocharov, A. Meyerhans, N. Bessonov, S. Trofimchuk and V. Volpert, Modelling the dynamics of virus infection and immune response in space and time, Internat. J. Parallel Emergent Distributed Syst., (2017), 341–355. doi: 10.1080/17445760.2017.1363203.

[7]

E. Braverman and S. Zhukovskiy, Absolute and delay-dependent stability of equations with a distributed delay, Discrete Contin. Dyn. Syst., 32 (2012), 2041-2061.  doi: 10.3934/dcds.2012.32.2041.

[8]

K. Deng and Y. Wu, On the diffusive Nicholson's blowflies equation with distributed delay, Appl. Math. Lett., 50 (2015), 126-132.  doi: 10.1016/j.aml.2015.06.013.

[9]

W. E. Fitzgibbon, Semilinear functional differential equations in Banach space, J. Differential Equations, 29 (1978), 1-14.  doi: 10.1016/0022-0396(78)90037-2.

[10]

Z. LingL. Zhang and Z. Lin, Turing pattern formation in a predator-prey system with cross diffusion, Appl. Math. Model., 38 (2014), 5022-5032.  doi: 10.1016/j.apm.2014.04.015.

[11]

E. LizM. PintoV. Tkachenko and S. Tromichuk, A global stability criterion for a family of delayed population models, Quart. Appl. Math., 63 (2005), 56-70.  doi: 10.1090/S0033-569X-05-00951-3.

[12]

E. Liz and G. Rost, On the global attractor of delay differential equations with unimodal feedback, Discrete Contin. Dyn. Syst., 24 (2009), 1215-1224.  doi: 10.3934/dcds.2009.24.1215.

[13]

E. LizV. Tkachenko and S. Tromichuk, A global stability criterion for scalar functional differential equations, SIAM. J. Math. Anal., 35 (2003), 596-622.  doi: 10.1137/S0036141001399222.

[14]

J. Mallet-Paret and R. Nussbaum, A differential-delay equation arising in optics and physiology, SIAM. J. Math. Anal., 20 (1989), 249-292.  doi: 10.1137/0520019.

[15]

J. Mallet-Paret and G. R. Sell., The Poincaré-Bendixson theorem for monotone cyclic feedback systems with delay, J. Differential Equations, 125 (1996), 441-489.  doi: 10.1006/jdeq.1996.0037.

[16]

R. Martin and H. L. Smith, Abstract functional differential equations and reaction-diffusion systems, Trans. Amer. Math. Soc., 321 (1990), 1-44.  doi: 10.2307/2001590.

[17]

R. Martin and H. L. Smith, Reaction-diffusion systems with time delay: Monotonicity, invariance, comparison and convergence, J. Reine Angew. Math., 413 (1991), 1-35.  doi: 10.1515/crll.1991.413.1.

[18] C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992.  doi: 10.1007/978-1-4615-3034-3.
[19]

G. Rost and J. Wu, Domain-decomposition method for the global dynamics of delay differential equations with unimodal feedback, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 463 (2007), 2655-2669.  doi: 10.1098/rspa.2007.1890.

[20]

H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs, 41, American Mathematical Society, Providence, RI, 1995.

[21]

G. Q. SunC. H. WangL. L. ChangY. P. Wu and L. L. Zhen Jin, Effects of feedback regulation on vegetation patterns in semi-arid environments, Appl. Math. Model., 61 (2018), 200-215.  doi: 10.1016/j.apm.2018.04.010.

[22]

H. R. Thieme, Mathematics in Population Biology, Princeton Series in Theoretical and Computational Biology, Princeton University Press, Princeton, NJ, 2003. doi: 10.2307/j.ctv301f9v.

[23]

H. R. Thieme and X.-Q. Zhao, A non-local delayed and diffusive predator-prey model, Nonlinear Anal. Real World Appl., 2 (2001), 145-160.  doi: 10.1016/S0362-546X(00)00112-7.

[24]

T. M. Touaoula, Global stability for a class of functional differential equations (Application to Nicholson's blowflies and Mackey-Glass models), Discrete Contin. Dyn. Syst., 38 (2018), 4391-4419.  doi: 10.3934/dcds.2018191.

[25]

S. Trofimchuk and V. Volpert, Traveling waves for a bistable reaction-diffusion equation with delay, SIAM J. Math. Anal., 50 (2018), 1175-1190.  doi: 10.1137/17M1115587.

[26]

A. I. Volpert, V. A. Volpert and V. A. Volpert, Traveling Wave Solutions of Parabolic Systems: Translation of Mathematical Monographs, Translations of Mathematical Monographs, 140, American Mathematical Society, Providence, RI, 1994.

[27]

J. Wu, Theory and applications of partial functional differential equations, Applied Mathematical Sciences, 119, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-4050-1.

[28]

D. Xu and X.-Q. Zhao, A nonlocal reaction-diffusion population model with stage structure, Can. Appl. Math. Q., 11 (2003), 303-319. 

[29]

T. YiY. Chen and J. Wu, Global dynamics of delayed reaction-diffusion equations in unbounded domains, Z. Angew. Math. Phys., 63 (2012), 793-812.  doi: 10.1007/s00033-012-0224-x.

[30]

T. Yi and X. Zou, Global attractivity of the diffusive Nicholson blowflies equation with Neumann boundary condition: A non-monotone case, J. Differential Equations, 245 (2008), 3376-3388.  doi: 10.1016/j.jde.2008.03.007.

[31]

T. Yi and X. Zou, Map dynamics versus dynamics of associated delay reaction-diffusion equations with a Neumann condition, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 466 (2010), 2955-2973.  doi: 10.1098/rspa.2009.0650.

[32]

T. Yi and X. Zou, Global dynamics of a delay differential equation with spatial non-locality in an unbounded domain, J. Differential Equations, 251 (2011), 2598-2611.  doi: 10.1016/j.jde.2011.04.027.

[33]

T. Yi and X. Zou, On Dirichlet problem for a class of delayed reaction-diffusion equations with spatial non-locality, J. Dynam. Differential Equations, 25 (2013), 959-979.  doi: 10.1007/s10884-013-9324-3.

[34]

T. Yi and X. Zou, Dirichlet problem for a delayed reaction-diffusion equation on a semi-infinite interval, J. Dynam. Differential Equations, 28 (2016), 1007-1030.  doi: 10.1007/s10884-015-9457-7.

[35]

Y. Yuan and X. Q. Zhao, Global stability for non-monotone delay equations (with application to a model of blood cell production), J. Differential Equations, 252 (2012), 2189-2209.  doi: 10.1016/j.jde.2011.08.026.

[36]

X. Q. Zhao, Global attractivity in a class of nonmonotone reaction-diffusion equations with time delay, Can. Appl. Math. Q., 17 (2009), 271-281. 

Figure 1.  Example of numerical simulations of equation (22) with the maximum map of solution (left) and solution $ u(x,t) $ plotted as a function of two variables 3D (right)
Figure 2.  Maximum maps for one-maximum (1M) symmetric mode, $ \tau = 1.1 $, D = 0.001 (left), D = 0.00058 (middle) and D = 0.00045 (right)
Figure 8.  Maximum maps for the functions $ v(x,t) = \cos (t) \cdot \cos (\alpha x - \beta ) $ (left), $ \cos(\alpha x-ct) $ (middle), $ \cos(\alpha |x-x_0|-ct) $ (right)
Figure 3.  Maximum maps for one-maximum (1M) symmetric mode, $ \tau = 1.1 $, D = 0.0004 (left) and D = 0.00038 (right)
Figure 4.  Maximum maps for one-maximum (1M) spinning mode, $ \tau = 1.1 $, D = 0.00031 (left), D = 0.00026 (middle) and D = 0.000125 (right)
Figure 5.  Maximum maps for 2M symmetric mode, $ \tau = 1.1 $, D = 0.000195 (left) and D = 0.000175 (right)
Figure 6.  Maximum maps for 1M symmetric mode, $ \tau = 1.1, D = 0.0005 $, $ \delta = 10^{-6} $ (left) and $ \delta = 10^{-5} $ (right)
Figure 7.  Maximum maps for 1M spinning mode, $ \tau = 1.1, D = 0.0001 $, $ \delta = 10^{-5} $ (left) and $ \delta = 10^{-4} $ (right)
Figure 9.  Maximum maps for the functions $ v(x,t) = \cos (\alpha x - \beta - \epsilon (t) ) $ (left), $ \cos(\alpha |x-x_0| + \epsilon (t) (ct-\beta)) $ (middle), $ \cos (t) \cdot \cos (\alpha |x-x_0|-ct-\epsilon (t)) $ (right)
Figure 10.  Maximum maps for the functions $ v(x,t) = \cos (t) \cdot \cos (\alpha |x-x_0|-cos(ct)+\beta) $ (left), $ \cos(t) \cdot \cos(\alpha x-c_1t) + \cos(\alpha x+c_2 t) $ (middle), $ \cos(t) \cdot \cos(\alpha_1 x+c_1t) + \cos(\alpha x-c_2 t) $ (right)
[1]

Yansu Ji, Jianwei Shen, Xiaochen Mao. Pattern formation of Brusselator in the reaction-diffusion system. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022103

[2]

Oleksiy V. Kapustyan, Pavlo O. Kasyanov, José Valero. Structure and regularity of the global attractor of a reaction-diffusion equation with non-smooth nonlinear term. Discrete and Continuous Dynamical Systems, 2014, 34 (10) : 4155-4182. doi: 10.3934/dcds.2014.34.4155

[3]

Joseph G. Yan, Dong-Ming Hwang. Pattern formation in reaction-diffusion systems with $D_2$-symmetric kinetics. Discrete and Continuous Dynamical Systems, 1996, 2 (2) : 255-270. doi: 10.3934/dcds.1996.2.255

[4]

Nick Bessonov, Gennady Bocharov, Tarik Mohammed Touaoula, Sergei Trofimchuk, Vitaly Volpert. Delay reaction-diffusion equation for infection dynamics. Discrete and Continuous Dynamical Systems - B, 2019, 24 (5) : 2073-2091. doi: 10.3934/dcdsb.2019085

[5]

Jia-Cheng Zhao, Zhong-Xin Ma. Global attractor for a partly dissipative reaction-diffusion system with discontinuous nonlinearity. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022103

[6]

Elena Trofimchuk, Sergei Trofimchuk. Admissible wavefront speeds for a single species reaction-diffusion equation with delay. Discrete and Continuous Dynamical Systems, 2008, 20 (2) : 407-423. doi: 10.3934/dcds.2008.20.407

[7]

Alexandra Köthe, Anna Marciniak-Czochra, Izumi Takagi. Hysteresis-driven pattern formation in reaction-diffusion-ODE systems. Discrete and Continuous Dynamical Systems, 2020, 40 (6) : 3595-3627. doi: 10.3934/dcds.2020170

[8]

Yejuan Wang, Peter E. Kloeden. The uniform attractor of a multi-valued process generated by reaction-diffusion delay equations on an unbounded domain. Discrete and Continuous Dynamical Systems, 2014, 34 (10) : 4343-4370. doi: 10.3934/dcds.2014.34.4343

[9]

Fengqi Yi, Eamonn A. Gaffney, Sungrim Seirin-Lee. The bifurcation analysis of turing pattern formation induced by delay and diffusion in the Schnakenberg system. Discrete and Continuous Dynamical Systems - B, 2017, 22 (2) : 647-668. doi: 10.3934/dcdsb.2017031

[10]

B. Ambrosio, M. A. Aziz-Alaoui, V. L. E. Phan. Global attractor of complex networks of reaction-diffusion systems of Fitzhugh-Nagumo type. Discrete and Continuous Dynamical Systems - B, 2018, 23 (9) : 3787-3797. doi: 10.3934/dcdsb.2018077

[11]

Hongyan Zhang, Siyu Liu, Yue Zhang. Dynamics and spatiotemporal pattern formations of a homogeneous reaction-diffusion Thomas model. Discrete and Continuous Dynamical Systems - S, 2017, 10 (5) : 1149-1164. doi: 10.3934/dcdss.2017062

[12]

M. Grasselli, V. Pata. A reaction-diffusion equation with memory. Discrete and Continuous Dynamical Systems, 2006, 15 (4) : 1079-1088. doi: 10.3934/dcds.2006.15.1079

[13]

Vladimir V. Chepyzhov, Mark I. Vishik. Trajectory attractor for reaction-diffusion system with diffusion coefficient vanishing in time. Discrete and Continuous Dynamical Systems, 2010, 27 (4) : 1493-1509. doi: 10.3934/dcds.2010.27.1493

[14]

Tarik Mohammed Touaoula. Global dynamics for a class of reaction-diffusion equations with distributed delay and neumann condition. Communications on Pure and Applied Analysis, 2020, 19 (5) : 2473-2490. doi: 10.3934/cpaa.2020108

[15]

Boris Andreianov, Halima Labani. Preconditioning operators and $L^\infty$ attractor for a class of reaction-diffusion systems. Communications on Pure and Applied Analysis, 2012, 11 (6) : 2179-2199. doi: 10.3934/cpaa.2012.11.2179

[16]

Guo Lin, Haiyan Wang. Traveling wave solutions of a reaction-diffusion equation with state-dependent delay. Communications on Pure and Applied Analysis, 2016, 15 (2) : 319-334. doi: 10.3934/cpaa.2016.15.319

[17]

Maria do Carmo Pacheco de Toledo, Sergio Muniz Oliva. A discretization scheme for an one-dimensional reaction-diffusion equation with delay and its dynamics. Discrete and Continuous Dynamical Systems, 2009, 23 (3) : 1041-1060. doi: 10.3934/dcds.2009.23.1041

[18]

Angelo Favini, Atsushi Yagi. Global existence for Laplace reaction-diffusion equations. Discrete and Continuous Dynamical Systems - S, 2020, 13 (5) : 1473-1493. doi: 10.3934/dcdss.2020083

[19]

Zhaosheng Feng. Traveling waves to a reaction-diffusion equation. Conference Publications, 2007, 2007 (Special) : 382-390. doi: 10.3934/proc.2007.2007.382

[20]

Yuan Lou, Wei-Ming Ni, Shoji Yotsutani. Pattern formation in a cross-diffusion system. Discrete and Continuous Dynamical Systems, 2015, 35 (4) : 1589-1607. doi: 10.3934/dcds.2015.35.1589

2021 Impact Factor: 1.865

Metrics

  • PDF downloads (531)
  • HTML views (277)
  • Cited by (3)

[Back to Top]