February  2022, 27(2): 1075-1090. doi: 10.3934/dcdsb.2021081

Approximate controllability of the semilinear reaction-diffusion equation governed by a multiplicative control

M2PA Laboratory, ENS, University of Sidi Mohamed Ben Abdellah, P.O. Box 5206, Bensouda, Fès, Morocco

* Corresponding author: Mohamed Ouzahra

Received  April 2020 Revised  August 2020 Published  February 2022 Early access  March 2021

Fund Project: The author is supported by USMBA grant (2020)

In this paper we are concerned with the approximate controllability of a multidimensional semilinear reaction-diffusion equation governed by a multiplicative control, which is locally distributed in the reaction term. For a given initial state we provide sufficient conditions on the desirable state to be approximately reached within an arbitrarily small time interval. Our approaches are based on linear semigroup theory and some results on uniform approximation with smooth functions.

Citation: Mohamed Ouzahra. Approximate controllability of the semilinear reaction-diffusion equation governed by a multiplicative control. Discrete and Continuous Dynamical Systems - B, 2022, 27 (2) : 1075-1090. doi: 10.3934/dcdsb.2021081
References:
[1]

A. R. A. Anderson and M. A. J. Chaplain, Continuous and discrete mathematical models of tumor-induced angiogenesis, Bulletin of Mathematical Biology, 60 (1998), 857-899.  doi: 10.1006/bulm.1998.0042.

[2]

H. Attouch, G. Buttazzo and G. Michaille, Variational Analysis in Sobolev and BV Spaces Applications to PDES and Optimization, SIAM, Society for Industrial and Applied Mathematics, Philadelphia, PA; Mathematical Optimization Society, Philadelphia, PA, 2014. doi: 10.1137/1.9781611973488.

[3]

J. M. Ball, On the asymptotic behaviour of generalized processes, with applications to nonlinear evolution equations, J. Differential Equations, 27 (1978), 224-265.  doi: 10.1016/0022-0396(78)90032-3.

[4]

J. M. Ball and M. Slemrod, Feedback stabilization of distributed semilinear control systems, Appl. Math. Opt., 5 (1979), 169-179.  doi: 10.1007/BF01442552.

[5]

J. M. Ball, Strongly continuous semigroups, weak solutions, and the variation of constants formula, Proceedings of the American Mathematical Society, 63 (1977) 370–373. doi: 10.2307/2041821.

[6]

J. M. BallJ. E. Marsden and M. Slemrod, Controllability for distributed bilinear systems, SIAM J. Control and Optim., 20 (1982), 575-597.  doi: 10.1137/0320042.

[7]

H. T. BanksP. M. Kareiva and L. Zia, Analyzing field studies of insect dispersal using two-dimensional transport equations, Environmental Entomology, 17 (1988), 815-820.  doi: 10.1093/ee/17.5.815.

[8]

K. Beauchard and J.-M. Coron, Controllability of a quantum particle in a moving potential well, J. Funct. Anal., 232 (2006), 328-389.  doi: 10.1016/j.jfa.2005.03.021.

[9]

K. Beauchard, Local controllability of a 1-dimensional beam equation, SIAM J. Control. Optim., 47 (2008), 1219-1273.  doi: 10.1137/050642034.

[10]

K. Beauchard, Local controllability and non-controllability for a $1D$ wave equation with bilinear control, J. Differential Equations., 250 (2011), 2064-2098.  doi: 10.1016/j.jde.2010.10.008.

[11]

H. Brezis, Analyse Fonctionnelle, Théorie et Applications, Masson, Paris, 1983.

[12]

P. Cannarsa and A. Khapalov, Multiplicative controllability for reaction-diffusion equation with target states admitting finitely many changes of sign, Discrete and Continuous Dynamical Systems Series B, 14 (2010), 1293-1311.  doi: 10.3934/dcdsb.2010.14.1293.

[13]

P. CannarsaG. Floridia and A. Y. Khapalov, Multiplicative controllability for semilinear reaction-diffusion equations with finitely many changes of sign, Journal de Mathématiques Pures et Appliquées, 108 (2017), 425-458.  doi: 10.1016/j.matpur.2017.07.002.

[14]

P. J. Davis, Interpolation and Approximation, Dover Publications, INC. New York, 1975.

[15]

Z. Ditzian, Derivatives of Bernstein polynomials and smoothness, Proceedings of the American Mathematical Society, 93 (1985), 25-31.  doi: 10.1090/S0002-9939-1985-0766520-7.

[16]

S. S. Dragomir, Some Gronwall Type Inequalities and Applications, Nova Science Publishers, 2003.

[17]

A. DoubovaE. Fernández-caraGo nzález-Burgos and E. Zuazua, On the controllability of parabolic systems with a nonlinear term involving the state and the gradient, SIAM. J. Control Optim, 41 (2002), 798-819.  doi: 10.1137/S0363012901386465.

[18]

L. C. Evans, Partial Differential Equation, Graduate Studies in Math, 19. American Mathematical Society, Providence, RI, 1998.

[19]

C. FabreJ.-P. Puel and E. Zuazua, Approximate controllability of the semilinear heat equation, Proc. Roy. Soc. Edinburgh A, 125 (1995), 31-61.  doi: 10.1017/S0308210500030742.

[20]

E. Fern$\acute{a}$ndez-Cara and E. Zuazua, Null and approximate controllability for weakly blowing-up semilinear heat equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 17 (2000), 583-616.  doi: 10.1016/S0294-1449(00)00117-7.

[21]

L. A. Fern$\acute{a}$ndez and E. Zuazua, Approximate controllability for the semilinear heat equation involving gradient terms, J. Optim. Theory Appl., 101 (1999), 307-328.  doi: 10.1023/A:1021737526541.

[22]

L. A. Fern$\acute{a}$ndez and A. Y. Khapalov, Controllability properties for the one-dimensional Heat equation under multiplicative or nonnegative additive controls with local mobile support, ESAIM: Control Optim. Calc. Var., 18 (2012), 1207-1224.  doi: 10.1051/cocv/2012004.

[23]

S. C. Ferreira, M. L. Martins and M. J. Vilela, Reaction-diffusion model for the growth of avascular tumor, Phys. Rev. E, 65 (2002), 021907. doi: 10.1103/PhysRevE.65.021907.

[24]

A. Friedman, PDE problems arising in mathematical biology, Networks and Heterogeneousmedia, 7 (2012), 691-703.  doi: 10.3934/nhm.2012.7.691.

[25]

A. V. Fursikov and O. Yu. Imanuvilov, Controllability of Evolution Equations, Lecture Notes 34, Seoul National University, Korea, 1996.

[26]

R. Glowinski and J.-L. Lions, Exact and approximate controllability for distributed parameter systems, Acta Numerica, 3, (1994) 269–378. doi: 10.1017/S0962492900002452.

[27]

A. Y. Khapalov, Controllability of the semilinear parabolic equation governed by a multiplicative control in the reaction term: A qualitative approach, SIAM J. Control. Optim., 41 (2003), 1886-1900.  doi: 10.1137/S0363012901394607.

[28]

A. Khapalov, Controllability properties of a vibrating string with variable axial load, Discrete Cont. Dyn. Syst., 11 (2004), 311-324.  doi: 10.3934/dcds.2004.11.311.

[29]

A. Y. Khapalov, Reachability of nonnegative equilibrium states for the semilinear vibrating string by varying its axial load and the gain of damping, ESAIM: Control Optim. Calc. Var., 12 (2006), 231-252.  doi: 10.1051/cocv:2006001.

[30]

A. Y. Khapalov, Controllability of partial differential equations governed by multiplicative controls, Lecture Notes in Mathematics, 1995, Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-12413-6.

[31]

K. Kime, Simultaneous control of a rod equation and a simple Schr$\ddot{o}$dinger equation, Systems & Control Letters, 24 (1995), 301-306.  doi: 10.1016/0167-6911(94)00022-N.

[32]

G. Lebeau and L. Robbiano, Contrôle exacte de l'équation de la chaleur, Comm. PDE, 20 (1995), 335-356.  doi: 10.1080/03605309508821097.

[33]

P. LinZ. Zhou and H. Gao, Exact controllability of the parabolic system with bilinear control, Applied Mathematics Letters, 19 (2006), 568-575.  doi: 10.1016/j.aml.2005.05.016.

[34]

G. G. Lorentz, Bernstein Polynomials, Chelsea Publishing Co., New York, 1986.

[35]

V. Nersesyan, Growth of Sobolev norms and controllability of the Schr$\ddot{o}$dinger equation, Comm. Math. Phys., 290 (2009) 371–387. doi: 10.1007/s00220-009-0842-0.

[36]

M. Ouzahra, Controllability of the wave equation with bilinear controls, European Journal of Control, 20 (2014), 57-63.  doi: 10.1016/j.ejcon.2013.10.007.

[37]

M. OuzahraA. Tsouli and A. Boutoulout, Exact controllability of the heat equation with bilinear control, Mathematical Methods in the Applied Sciences, 38 (2015), 5074-5084.  doi: 10.1002/mma.3428.

[38]

M. Ouzahra, Approximate and exact controllability of a reaction-diffusion equation governed by bilinear control, European Journal of Control, 32 (2016), 32-38.  doi: 10.1016/j.ejcon.2016.05.004.

[39]

M. Ouzahra, Controllability of the semilinear wave equation governed by a multiplicative control, Evolution Equations & Control Theory, 8 (2019), 669-686.  doi: 10.3934/eect.2019039.

[40]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[41]

B. Perthame, Some mathematical aspects of tumor growth and therapy, Proceedings of the International Congress of Mathematicians—Seoul, Vol. 1, Kyung Moon Sa, Seoul, 2014,529–545.

[42]

T. Roose, S. J. Chapman and P. K. Maini, Mathematical models of avascular tumor growth, SIAM Rev., 49 (2007) 179–208. doi: 10.1137/S0036144504446291.

[43]

S. Salsa, Partial Differential Equations in Action: From Modelling to Theory, Springer-Verlag Italia, Milan, 2008.

[44]

A. Tosin, Initial/boundary-value problems of tumor growth within a host tissue, J. Math.Biol., 66 (2013) 163–202. doi: 10.1007/s00285-012-0505-1.

[45]

H. Ye, J. Gao and Y. Ding, A generalized Gronwall inequality and its application to a fractional differential equation, J. Math. Anal. Appl., 328 (2007) 1075–1081. doi: 10.1016/j.jmaa.2006.05.061.

[46]

E. Zuazua, Approximate controllability for semilinear heat equations with globally Lipschitz nonlinearities, Control Cybern., 28 (1999), 665-683. 

show all references

References:
[1]

A. R. A. Anderson and M. A. J. Chaplain, Continuous and discrete mathematical models of tumor-induced angiogenesis, Bulletin of Mathematical Biology, 60 (1998), 857-899.  doi: 10.1006/bulm.1998.0042.

[2]

H. Attouch, G. Buttazzo and G. Michaille, Variational Analysis in Sobolev and BV Spaces Applications to PDES and Optimization, SIAM, Society for Industrial and Applied Mathematics, Philadelphia, PA; Mathematical Optimization Society, Philadelphia, PA, 2014. doi: 10.1137/1.9781611973488.

[3]

J. M. Ball, On the asymptotic behaviour of generalized processes, with applications to nonlinear evolution equations, J. Differential Equations, 27 (1978), 224-265.  doi: 10.1016/0022-0396(78)90032-3.

[4]

J. M. Ball and M. Slemrod, Feedback stabilization of distributed semilinear control systems, Appl. Math. Opt., 5 (1979), 169-179.  doi: 10.1007/BF01442552.

[5]

J. M. Ball, Strongly continuous semigroups, weak solutions, and the variation of constants formula, Proceedings of the American Mathematical Society, 63 (1977) 370–373. doi: 10.2307/2041821.

[6]

J. M. BallJ. E. Marsden and M. Slemrod, Controllability for distributed bilinear systems, SIAM J. Control and Optim., 20 (1982), 575-597.  doi: 10.1137/0320042.

[7]

H. T. BanksP. M. Kareiva and L. Zia, Analyzing field studies of insect dispersal using two-dimensional transport equations, Environmental Entomology, 17 (1988), 815-820.  doi: 10.1093/ee/17.5.815.

[8]

K. Beauchard and J.-M. Coron, Controllability of a quantum particle in a moving potential well, J. Funct. Anal., 232 (2006), 328-389.  doi: 10.1016/j.jfa.2005.03.021.

[9]

K. Beauchard, Local controllability of a 1-dimensional beam equation, SIAM J. Control. Optim., 47 (2008), 1219-1273.  doi: 10.1137/050642034.

[10]

K. Beauchard, Local controllability and non-controllability for a $1D$ wave equation with bilinear control, J. Differential Equations., 250 (2011), 2064-2098.  doi: 10.1016/j.jde.2010.10.008.

[11]

H. Brezis, Analyse Fonctionnelle, Théorie et Applications, Masson, Paris, 1983.

[12]

P. Cannarsa and A. Khapalov, Multiplicative controllability for reaction-diffusion equation with target states admitting finitely many changes of sign, Discrete and Continuous Dynamical Systems Series B, 14 (2010), 1293-1311.  doi: 10.3934/dcdsb.2010.14.1293.

[13]

P. CannarsaG. Floridia and A. Y. Khapalov, Multiplicative controllability for semilinear reaction-diffusion equations with finitely many changes of sign, Journal de Mathématiques Pures et Appliquées, 108 (2017), 425-458.  doi: 10.1016/j.matpur.2017.07.002.

[14]

P. J. Davis, Interpolation and Approximation, Dover Publications, INC. New York, 1975.

[15]

Z. Ditzian, Derivatives of Bernstein polynomials and smoothness, Proceedings of the American Mathematical Society, 93 (1985), 25-31.  doi: 10.1090/S0002-9939-1985-0766520-7.

[16]

S. S. Dragomir, Some Gronwall Type Inequalities and Applications, Nova Science Publishers, 2003.

[17]

A. DoubovaE. Fernández-caraGo nzález-Burgos and E. Zuazua, On the controllability of parabolic systems with a nonlinear term involving the state and the gradient, SIAM. J. Control Optim, 41 (2002), 798-819.  doi: 10.1137/S0363012901386465.

[18]

L. C. Evans, Partial Differential Equation, Graduate Studies in Math, 19. American Mathematical Society, Providence, RI, 1998.

[19]

C. FabreJ.-P. Puel and E. Zuazua, Approximate controllability of the semilinear heat equation, Proc. Roy. Soc. Edinburgh A, 125 (1995), 31-61.  doi: 10.1017/S0308210500030742.

[20]

E. Fern$\acute{a}$ndez-Cara and E. Zuazua, Null and approximate controllability for weakly blowing-up semilinear heat equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 17 (2000), 583-616.  doi: 10.1016/S0294-1449(00)00117-7.

[21]

L. A. Fern$\acute{a}$ndez and E. Zuazua, Approximate controllability for the semilinear heat equation involving gradient terms, J. Optim. Theory Appl., 101 (1999), 307-328.  doi: 10.1023/A:1021737526541.

[22]

L. A. Fern$\acute{a}$ndez and A. Y. Khapalov, Controllability properties for the one-dimensional Heat equation under multiplicative or nonnegative additive controls with local mobile support, ESAIM: Control Optim. Calc. Var., 18 (2012), 1207-1224.  doi: 10.1051/cocv/2012004.

[23]

S. C. Ferreira, M. L. Martins and M. J. Vilela, Reaction-diffusion model for the growth of avascular tumor, Phys. Rev. E, 65 (2002), 021907. doi: 10.1103/PhysRevE.65.021907.

[24]

A. Friedman, PDE problems arising in mathematical biology, Networks and Heterogeneousmedia, 7 (2012), 691-703.  doi: 10.3934/nhm.2012.7.691.

[25]

A. V. Fursikov and O. Yu. Imanuvilov, Controllability of Evolution Equations, Lecture Notes 34, Seoul National University, Korea, 1996.

[26]

R. Glowinski and J.-L. Lions, Exact and approximate controllability for distributed parameter systems, Acta Numerica, 3, (1994) 269–378. doi: 10.1017/S0962492900002452.

[27]

A. Y. Khapalov, Controllability of the semilinear parabolic equation governed by a multiplicative control in the reaction term: A qualitative approach, SIAM J. Control. Optim., 41 (2003), 1886-1900.  doi: 10.1137/S0363012901394607.

[28]

A. Khapalov, Controllability properties of a vibrating string with variable axial load, Discrete Cont. Dyn. Syst., 11 (2004), 311-324.  doi: 10.3934/dcds.2004.11.311.

[29]

A. Y. Khapalov, Reachability of nonnegative equilibrium states for the semilinear vibrating string by varying its axial load and the gain of damping, ESAIM: Control Optim. Calc. Var., 12 (2006), 231-252.  doi: 10.1051/cocv:2006001.

[30]

A. Y. Khapalov, Controllability of partial differential equations governed by multiplicative controls, Lecture Notes in Mathematics, 1995, Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-12413-6.

[31]

K. Kime, Simultaneous control of a rod equation and a simple Schr$\ddot{o}$dinger equation, Systems & Control Letters, 24 (1995), 301-306.  doi: 10.1016/0167-6911(94)00022-N.

[32]

G. Lebeau and L. Robbiano, Contrôle exacte de l'équation de la chaleur, Comm. PDE, 20 (1995), 335-356.  doi: 10.1080/03605309508821097.

[33]

P. LinZ. Zhou and H. Gao, Exact controllability of the parabolic system with bilinear control, Applied Mathematics Letters, 19 (2006), 568-575.  doi: 10.1016/j.aml.2005.05.016.

[34]

G. G. Lorentz, Bernstein Polynomials, Chelsea Publishing Co., New York, 1986.

[35]

V. Nersesyan, Growth of Sobolev norms and controllability of the Schr$\ddot{o}$dinger equation, Comm. Math. Phys., 290 (2009) 371–387. doi: 10.1007/s00220-009-0842-0.

[36]

M. Ouzahra, Controllability of the wave equation with bilinear controls, European Journal of Control, 20 (2014), 57-63.  doi: 10.1016/j.ejcon.2013.10.007.

[37]

M. OuzahraA. Tsouli and A. Boutoulout, Exact controllability of the heat equation with bilinear control, Mathematical Methods in the Applied Sciences, 38 (2015), 5074-5084.  doi: 10.1002/mma.3428.

[38]

M. Ouzahra, Approximate and exact controllability of a reaction-diffusion equation governed by bilinear control, European Journal of Control, 32 (2016), 32-38.  doi: 10.1016/j.ejcon.2016.05.004.

[39]

M. Ouzahra, Controllability of the semilinear wave equation governed by a multiplicative control, Evolution Equations & Control Theory, 8 (2019), 669-686.  doi: 10.3934/eect.2019039.

[40]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[41]

B. Perthame, Some mathematical aspects of tumor growth and therapy, Proceedings of the International Congress of Mathematicians—Seoul, Vol. 1, Kyung Moon Sa, Seoul, 2014,529–545.

[42]

T. Roose, S. J. Chapman and P. K. Maini, Mathematical models of avascular tumor growth, SIAM Rev., 49 (2007) 179–208. doi: 10.1137/S0036144504446291.

[43]

S. Salsa, Partial Differential Equations in Action: From Modelling to Theory, Springer-Verlag Italia, Milan, 2008.

[44]

A. Tosin, Initial/boundary-value problems of tumor growth within a host tissue, J. Math.Biol., 66 (2013) 163–202. doi: 10.1007/s00285-012-0505-1.

[45]

H. Ye, J. Gao and Y. Ding, A generalized Gronwall inequality and its application to a fractional differential equation, J. Math. Anal. Appl., 328 (2007) 1075–1081. doi: 10.1016/j.jmaa.2006.05.061.

[46]

E. Zuazua, Approximate controllability for semilinear heat equations with globally Lipschitz nonlinearities, Control Cybern., 28 (1999), 665-683. 

[1]

Hugo Leiva, Nelson Merentes, José L. Sánchez. Approximate controllability of semilinear reaction diffusion equations. Mathematical Control and Related Fields, 2012, 2 (2) : 171-182. doi: 10.3934/mcrf.2012.2.171

[2]

Lianwen Wang. Approximate controllability and approximate null controllability of semilinear systems. Communications on Pure and Applied Analysis, 2006, 5 (4) : 953-962. doi: 10.3934/cpaa.2006.5.953

[3]

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

[4]

Wilhelm Stannat, Lukas Wessels. Deterministic control of stochastic reaction-diffusion equations. Evolution Equations and Control Theory, 2021, 10 (4) : 701-722. doi: 10.3934/eect.2020087

[5]

Mohamed Ouzahra. Controllability of the semilinear wave equation governed by a multiplicative control. Evolution Equations and Control Theory, 2019, 8 (4) : 669-686. doi: 10.3934/eect.2019039

[6]

Filipa Caetano, Martin J. Gander, Laurence Halpern, Jérémie Szeftel. Schwarz waveform relaxation algorithms for semilinear reaction-diffusion equations. Networks and Heterogeneous Media, 2010, 5 (3) : 487-505. doi: 10.3934/nhm.2010.5.487

[7]

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

[8]

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

[9]

Razvan Gabriel Iagar, Ariel Sánchez. Eternal solutions for a reaction-diffusion equation with weighted reaction. Discrete and Continuous Dynamical Systems, 2022, 42 (3) : 1465-1491. doi: 10.3934/dcds.2021160

[10]

Hugo Leiva, Jahnett Uzcategui. Approximate controllability of discrete semilinear systems and applications. Discrete and Continuous Dynamical Systems - B, 2016, 21 (6) : 1803-1812. doi: 10.3934/dcdsb.2016023

[11]

Perla El Kettani, Danielle Hilhorst, Kai Lee. A stochastic mass conserved reaction-diffusion equation with nonlinear diffusion. Discrete and Continuous Dynamical Systems, 2018, 38 (11) : 5615-5648. doi: 10.3934/dcds.2018246

[12]

Heather Finotti, Suzanne Lenhart, Tuoc Van Phan. Optimal control of advective direction in reaction-diffusion population models. Evolution Equations and Control Theory, 2012, 1 (1) : 81-107. doi: 10.3934/eect.2012.1.81

[13]

Henri Berestycki, Nancy Rodríguez. A non-local bistable reaction-diffusion equation with a gap. Discrete and Continuous Dynamical Systems, 2017, 37 (2) : 685-723. doi: 10.3934/dcds.2017029

[14]

Maho Endo, Yuki Kaneko, Yoshio Yamada. Free boundary problem for a reaction-diffusion equation with positive bistable nonlinearity. Discrete and Continuous Dynamical Systems, 2020, 40 (6) : 3375-3394. doi: 10.3934/dcds.2020033

[15]

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

[16]

Tomás Caraballo, José A. Langa, James C. Robinson. Stability and random attractors for a reaction-diffusion equation with multiplicative noise. Discrete and Continuous Dynamical Systems, 2000, 6 (4) : 875-892. doi: 10.3934/dcds.2000.6.875

[17]

Elena Beretta, Cecilia Cavaterra. Identifying a space dependent coefficient in a reaction-diffusion equation. Inverse Problems and Imaging, 2011, 5 (2) : 285-296. doi: 10.3934/ipi.2011.5.285

[18]

Michio Urano, Kimie Nakashima, Yoshio Yamada. Transition layers and spikes for a reaction-diffusion equation with bistable nonlinearity. Conference Publications, 2005, 2005 (Special) : 868-877. doi: 10.3934/proc.2005.2005.868

[19]

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

[20]

Takanori Ide, Kazuhiro Kurata, Kazunaga Tanaka. Multiple stable patterns for some reaction-diffusion equation in disrupted environments. Discrete and Continuous Dynamical Systems, 2006, 14 (1) : 93-116. doi: 10.3934/dcds.2006.14.93

2021 Impact Factor: 1.497

Metrics

  • PDF downloads (280)
  • HTML views (378)
  • Cited by (0)

Other articles
by authors

[Back to Top]