Advanced Search
Article Contents
Article Contents

Null controllability for semilinear heat equation with dynamic boundary conditions

  • * Corresponding author: Omar Oukdach

    * Corresponding author: Omar Oukdach
Abstract Full Text(HTML) Related Papers Cited by
  • 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.

    Mathematics Subject Classification: Primary: 93Bxx; Secondary: 35K05.


    \begin{equation} \\ \end{equation}
  • 加载中
  • [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. WilliamsThe Mathematics of Diffusion, Oxford University Press, 1979. 
  • 加载中

Article Metrics

HTML views(2080) PDF downloads(180) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint