June  2022, 15(6): 1525-1546. doi: 10.3934/dcdss.2022087

Null controllability for semilinear heat equation with dynamic boundary conditions

Cadi Ayyad University, Faculty of Sciences Semlalia, LMDP, UMMISCO (IRD-UPMC), B.P. 2390, Marrakesh, Morocco

* Corresponding author: Omar Oukdach

Received  December 2021 Revised  March 2022 Published  June 2022 Early access  April 2022

This paper deals with the null controllability of the semilinear heat equation with dynamic boundary conditions of surface diffusion type, with nonlinearities involving drift terms. First, we prove a negative result for some function $ F $ that behaves at infinity like $ |s| \ln ^{p}(1+|s|), $ with $ p > 2 $. Then, by a careful analysis of the linearized system and a fixed point method, a null controllability result is proved for nonlinearties $ F(s, \xi) $ and $ G(s, \xi) $ growing slower than $ |s| \ln ^{3 / 2}(1+|s|+\|\xi\|)+\|\xi\| \ln^{1 / 2}(1+|s|+\|\xi\|) $ at infinity.

Citation: Abdelaziz Khoutaibi, Lahcen Maniar, Omar Oukdach. Null controllability for semilinear heat equation with dynamic boundary conditions. Discrete and Continuous Dynamical Systems - S, 2022, 15 (6) : 1525-1546. doi: 10.3934/dcdss.2022087
References:
[1]

E. M. Ait Ben HassiS.-E. ChorfiL. Maniar and O. Oukdach, Lipschitz stability for an inverse source problem in anisotropic parabolic equations with dynamic boundary conditions, Evol. Eq. Control Theory, 10 (2021), 837-859.  doi: 10.3934/eect.2020094.

[2]

S. Anita and V. Barbu, Null controllability of nonlinear convective heat equation, ESAIM Control Optim. Calc. Var., 5 (2000), 157-173.  doi: 10.1051/cocv:2000105.

[3]

W. Arendt, Semigroups and evolution equations: Functional calculus, regularity and kernel estimates, Evolutionary equations, Handb. Differ. Equ., North-Holland, Amsterdam, 1 (2004), 1-85. 

[4]

J.-P. Aubin, L'Analyse non Linéaire et ses Motivations Économiques, Masson, Paris, 1984.

[5]

I. BoutaayamouS. E. ChorfiL. Maniar and O. Oukdach, The cost of approximate controllability of heat equation with general dynamical boundary conditions, Portugal. Math., 78 (2021), 65-99.  doi: 10.4171/PM/2061.

[6]

L. Calatroni and P. Colli, Global solution to the Allen-Cahn equation with singular potentials and dynamic boundary conditions, Nonlinear Anal., 79 (2013), 12-27.  doi: 10.1016/j.na.2012.11.010.

[7]

D. ChaeO. Y. Imaniviliv and S. M. Kim, Exact controllability for semilinear parabolic equations with neumann boundary conditions, J. Dyn. Control Syst., 2 (1996), 449-483.  doi: 10.1007/BF02254698.

[8]

L. Cherfils and A. Miranville, On the Caginalp system with dynamic boundary conditions and singular potentials, Appl. Math., 54 (2009), 89-115.  doi: 10.1007/s10492-009-0008-6.

[9]

P. Colli and J. Sprekels, Optimal control of an Allen-Cahn equation with singular potentials and dynamic boundary condition, SIAM J. Control Optim., 53 (2015), 213-234.  doi: 10.1137/120902422.

[10]

R. Courant and D. Hilbert, Methoden der Mathematischen Physik. I., Heidelberger Taschenbücher, Band 30. Springer-Verlag, Berlin-New York, 1968.

[11]

M. C. Delfour and J. P. Zoleosio, Shape analysis via oriented distance functions, J. Functional Anal., 123 (1994), 129-201.  doi: 10.1006/jfan.1994.1086.

[12]

A. DoubvaE. Fernández-CaraM. Gonzalez-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.

[13]

J. Escher, Quasilinear parabolic systems with dynamical boundary conditions, Comm. Partial Differential Equations, 18 (1993), 1309-1364.  doi: 10.1080/03605309308820976.

[14]

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

[15]

E. Fernández-CaraM. Gonzalez-BurgosS. Guerrero and J.-P. Puel, Null controllability of the heat equation with boundary fourier conditions: The linear case, ESAIM Control Optim. Calc. Var., 12 (2006), 442-465.  doi: 10.1051/cocv:2006010.

[16]

E. Fernández-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.

[17]

A. V. Fursikov and O. Y. Imanuviliv, Controllability of Evolution equations, Lecture Notes Series, 34. Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1996.

[18]

C. G. Gal and M. Grasselli, The non-isothermal Allen-Cahn equation with dynamic boundary conditions, Discrete Contin. Dyn. Syst. Ser. A, 22 (2008), 1009-1040.  doi: 10.3934/dcds.2008.22.1009.

[19]

C. G. Gal and L. Tebou, Carleman inequalities for wave equations with oscillatory boundary conditions and application, SIAM J. Control. Optim., 55 (2017), 324-364.  doi: 10.1137/15M1032211.

[20]

G. R. Goldstein, Derivation and physical interpretation of general boundary conditions, Adv. Differential. Equ., 11 (2006), 457-480. 

[21]

G. R. GoldsteinJ. A. GoldsteinD. Guidetti and S. Romanelli, Maximal regularity, analytic semigroups, and dynamic and general wentzell boundary conditions with a diffusion term on the boundary, Ann. Mat. Pura Appl., 199 (2020), 127-146.  doi: 10.1007/s10231-019-00868-3.

[22]

M. GrasselliA. Miranville and G. Schimperna, The Caginalp phase-field system with coupled dynamic boundary conditions and singular potentials, Discrete Contin. Dyn. Syst., 28 (2010), 67-98.  doi: 10.3934/dcds.2010.28.67.

[23]

O. Y. Imanuvilov, Controllability of parabolic equations, Sb. Math., 186 (1995), 879-900.  doi: 10.1070/SM1995v186n06ABEH000047.

[24]

A. Khoutaibi and L. Maniar, Null controllability for a heat equation with dynamic boundary conditions and drift terms, Evol. Equ. Control Theory, 9 (2019), 535-559.  doi: 10.3934/eect.2020023.

[25]

A. Khoutaibi, L. Maniar, D. Mugnolo and A. Rhandi, Parabolic equations with dynamic boundary conditions and drift terms, Math. Nachr..

[26]

R. E. Langer, A problem in diffusion or in the flow of heat for a solid in contact with a fluid, Tohoku Math. J., First Series, 35 (1932), 260-275. 

[27]

I. LasieckaR. Triggiani and P.-F. Yao, Inverse/observability estimates for second-order hyperbolic equations with variable coefficients, J. Math. Anal. Appl., 235 (1999), 13-57.  doi: 10.1006/jmaa.1999.6348.

[28]

K. Le Balc'H, Global null-controllability and nonnegative-controllability of slightly superlinear heat equations, J. Math. Pures Appl., 135 (2020), 103-139.  doi: 10.1016/j.matpur.2019.10.009.

[29]

M. Liero, Passing from bulk to bulk-surface evolution in the Allen-Cahn equation, NoDEA Nonlinear Differential Equations Appl., 20 (2013), 919-942.  doi: 10.1007/s00030-012-0189-7.

[30]

L. ManiarM. Meyries and R. Schnaubelt, Null controllability for parabolic equations with dynamic boundary conditions of reactive-diffusion type, Evol. Equ. Control Theory, 6 (2017), 381-407.  doi: 10.3934/eect.2017020.

[31]

D. S. Mitrinović, J. E. Pe(v)carić and A. M. Fink, Classical and New Inequalities in Analysis Kluwer Academic Publishers, 1993.

[32]

A. Miranville and S. Zelik, Exponential attractors for the Cahn-Hilliard equation with dynamic boundary conditions, Math. Methods Appl., 28 (2005), 709-735.  doi: 10.1002/mma.590.

[33]

D. Mugnolo and S. Romanelli, Dirichlet forms for general wentzell boundary conditions, analytic semigroups, and cosine operator functions, Electron. J. Differential Equations, (2006), 20 pp.

[34]

N. Sauer, Dynamic boundary conditions and the Carslaw-Jaeger constitutive relation in heat transfer, Partial Differ. Equ. Appl., 1 (2020), Paper No. 48, 20 pp. doi: 10.1007/s42985-020-00050-y.

[35]

J. Simon, Compact sets in the spacel $L^p (0, T; B)$, Ann. Mat. Pura Appl., 146 (1986), 65-96.  doi: 10.1007/BF01762360.

[36]

M. E. Taylor, Partial Differential Equations I: Basic Theory, Applied Mathematical Sciences, Second edition, Applied Mathematical Sciences, 115. Springer, New York, 2011. doi: 10.1007/978-1-4419-7055-8.

[37]

L. Vázquez and E. Vitillaro, Heat equation with dynamical boundary conditions of reactive type, Commun. Part. Diff. Eq., 33 (2008), 561-612.  doi: 10.1080/03605300801970960.

[38]

R. Vold and M. Vold, Colloid and Interface Chemistry, Addision-wesley, Reading-Mass, 1983.

[39] M. M. R. Williams, The Mathematics of Diffusion, Oxford University Press, 1979. 

show all references

References:
[1]

E. M. Ait Ben HassiS.-E. ChorfiL. Maniar and O. Oukdach, Lipschitz stability for an inverse source problem in anisotropic parabolic equations with dynamic boundary conditions, Evol. Eq. Control Theory, 10 (2021), 837-859.  doi: 10.3934/eect.2020094.

[2]

S. Anita and V. Barbu, Null controllability of nonlinear convective heat equation, ESAIM Control Optim. Calc. Var., 5 (2000), 157-173.  doi: 10.1051/cocv:2000105.

[3]

W. Arendt, Semigroups and evolution equations: Functional calculus, regularity and kernel estimates, Evolutionary equations, Handb. Differ. Equ., North-Holland, Amsterdam, 1 (2004), 1-85. 

[4]

J.-P. Aubin, L'Analyse non Linéaire et ses Motivations Économiques, Masson, Paris, 1984.

[5]

I. BoutaayamouS. E. ChorfiL. Maniar and O. Oukdach, The cost of approximate controllability of heat equation with general dynamical boundary conditions, Portugal. Math., 78 (2021), 65-99.  doi: 10.4171/PM/2061.

[6]

L. Calatroni and P. Colli, Global solution to the Allen-Cahn equation with singular potentials and dynamic boundary conditions, Nonlinear Anal., 79 (2013), 12-27.  doi: 10.1016/j.na.2012.11.010.

[7]

D. ChaeO. Y. Imaniviliv and S. M. Kim, Exact controllability for semilinear parabolic equations with neumann boundary conditions, J. Dyn. Control Syst., 2 (1996), 449-483.  doi: 10.1007/BF02254698.

[8]

L. Cherfils and A. Miranville, On the Caginalp system with dynamic boundary conditions and singular potentials, Appl. Math., 54 (2009), 89-115.  doi: 10.1007/s10492-009-0008-6.

[9]

P. Colli and J. Sprekels, Optimal control of an Allen-Cahn equation with singular potentials and dynamic boundary condition, SIAM J. Control Optim., 53 (2015), 213-234.  doi: 10.1137/120902422.

[10]

R. Courant and D. Hilbert, Methoden der Mathematischen Physik. I., Heidelberger Taschenbücher, Band 30. Springer-Verlag, Berlin-New York, 1968.

[11]

M. C. Delfour and J. P. Zoleosio, Shape analysis via oriented distance functions, J. Functional Anal., 123 (1994), 129-201.  doi: 10.1006/jfan.1994.1086.

[12]

A. DoubvaE. Fernández-CaraM. Gonzalez-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.

[13]

J. Escher, Quasilinear parabolic systems with dynamical boundary conditions, Comm. Partial Differential Equations, 18 (1993), 1309-1364.  doi: 10.1080/03605309308820976.

[14]

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

[15]

E. Fernández-CaraM. Gonzalez-BurgosS. Guerrero and J.-P. Puel, Null controllability of the heat equation with boundary fourier conditions: The linear case, ESAIM Control Optim. Calc. Var., 12 (2006), 442-465.  doi: 10.1051/cocv:2006010.

[16]

E. Fernández-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.

[17]

A. V. Fursikov and O. Y. Imanuviliv, Controllability of Evolution equations, Lecture Notes Series, 34. Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1996.

[18]

C. G. Gal and M. Grasselli, The non-isothermal Allen-Cahn equation with dynamic boundary conditions, Discrete Contin. Dyn. Syst. Ser. A, 22 (2008), 1009-1040.  doi: 10.3934/dcds.2008.22.1009.

[19]

C. G. Gal and L. Tebou, Carleman inequalities for wave equations with oscillatory boundary conditions and application, SIAM J. Control. Optim., 55 (2017), 324-364.  doi: 10.1137/15M1032211.

[20]

G. R. Goldstein, Derivation and physical interpretation of general boundary conditions, Adv. Differential. Equ., 11 (2006), 457-480. 

[21]

G. R. GoldsteinJ. A. GoldsteinD. Guidetti and S. Romanelli, Maximal regularity, analytic semigroups, and dynamic and general wentzell boundary conditions with a diffusion term on the boundary, Ann. Mat. Pura Appl., 199 (2020), 127-146.  doi: 10.1007/s10231-019-00868-3.

[22]

M. GrasselliA. Miranville and G. Schimperna, The Caginalp phase-field system with coupled dynamic boundary conditions and singular potentials, Discrete Contin. Dyn. Syst., 28 (2010), 67-98.  doi: 10.3934/dcds.2010.28.67.

[23]

O. Y. Imanuvilov, Controllability of parabolic equations, Sb. Math., 186 (1995), 879-900.  doi: 10.1070/SM1995v186n06ABEH000047.

[24]

A. Khoutaibi and L. Maniar, Null controllability for a heat equation with dynamic boundary conditions and drift terms, Evol. Equ. Control Theory, 9 (2019), 535-559.  doi: 10.3934/eect.2020023.

[25]

A. Khoutaibi, L. Maniar, D. Mugnolo and A. Rhandi, Parabolic equations with dynamic boundary conditions and drift terms, Math. Nachr..

[26]

R. E. Langer, A problem in diffusion or in the flow of heat for a solid in contact with a fluid, Tohoku Math. J., First Series, 35 (1932), 260-275. 

[27]

I. LasieckaR. Triggiani and P.-F. Yao, Inverse/observability estimates for second-order hyperbolic equations with variable coefficients, J. Math. Anal. Appl., 235 (1999), 13-57.  doi: 10.1006/jmaa.1999.6348.

[28]

K. Le Balc'H, Global null-controllability and nonnegative-controllability of slightly superlinear heat equations, J. Math. Pures Appl., 135 (2020), 103-139.  doi: 10.1016/j.matpur.2019.10.009.

[29]

M. Liero, Passing from bulk to bulk-surface evolution in the Allen-Cahn equation, NoDEA Nonlinear Differential Equations Appl., 20 (2013), 919-942.  doi: 10.1007/s00030-012-0189-7.

[30]

L. ManiarM. Meyries and R. Schnaubelt, Null controllability for parabolic equations with dynamic boundary conditions of reactive-diffusion type, Evol. Equ. Control Theory, 6 (2017), 381-407.  doi: 10.3934/eect.2017020.

[31]

D. S. Mitrinović, J. E. Pe(v)carić and A. M. Fink, Classical and New Inequalities in Analysis Kluwer Academic Publishers, 1993.

[32]

A. Miranville and S. Zelik, Exponential attractors for the Cahn-Hilliard equation with dynamic boundary conditions, Math. Methods Appl., 28 (2005), 709-735.  doi: 10.1002/mma.590.

[33]

D. Mugnolo and S. Romanelli, Dirichlet forms for general wentzell boundary conditions, analytic semigroups, and cosine operator functions, Electron. J. Differential Equations, (2006), 20 pp.

[34]

N. Sauer, Dynamic boundary conditions and the Carslaw-Jaeger constitutive relation in heat transfer, Partial Differ. Equ. Appl., 1 (2020), Paper No. 48, 20 pp. doi: 10.1007/s42985-020-00050-y.

[35]

J. Simon, Compact sets in the spacel $L^p (0, T; B)$, Ann. Mat. Pura Appl., 146 (1986), 65-96.  doi: 10.1007/BF01762360.

[36]

M. E. Taylor, Partial Differential Equations I: Basic Theory, Applied Mathematical Sciences, Second edition, Applied Mathematical Sciences, 115. Springer, New York, 2011. doi: 10.1007/978-1-4419-7055-8.

[37]

L. Vázquez and E. Vitillaro, Heat equation with dynamical boundary conditions of reactive type, Commun. Part. Diff. Eq., 33 (2008), 561-612.  doi: 10.1080/03605300801970960.

[38]

R. Vold and M. Vold, Colloid and Interface Chemistry, Addision-wesley, Reading-Mass, 1983.

[39] M. M. R. Williams, The Mathematics of Diffusion, Oxford University Press, 1979. 
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