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June  2022, 11(3): 749-779. doi: 10.3934/eect.2021024

Local null controllability for a parabolic equation with local and nonlocal nonlinearities in moving domains

1. 

Departamento de Análise Matemática, Universidade do Estado do Rio de Janeiro, Rio de Janeiro, RJ, 20550-900, Brazil

2. 

Departamento de Matemática Aplicada, Universidade Federal Fluminense, Niterói, RJ, 24020-140, Brazil

* Corresponding author: andre.lopes@ime.uerj.br

Received  July 2020 Revised  February 2021 Published  June 2022 Early access  May 2021

In this paper, we establish a local null controllability result for a nonlinear parabolic PDE with local and nonlocal nonlinearities in a domain whose boundary moves in time by a control force with a multiplicative part acting on a prescribed subdomain. We prove that, if the initial data is sufficiently small and the linearized system at zero satisfies an appropriate condition, the equation can be driven to zero.

Citation: André da Rocha Lopes, Juan Límaco. Local null controllability for a parabolic equation with local and nonlocal nonlinearities in moving domains. Evolution Equations and Control Theory, 2022, 11 (3) : 749-779. doi: 10.3934/eect.2021024
References:
[1]

V. M. Alekseev, V. M. Tikhomirov and S. V. Fomin, Optimal Control, Consultants Bureau, New York, 1987. doi: 10.1007/978-1-4615-7551-1.

[2]

M. L. BernardiG. Bonfanti and F. Lutteroti, Abstract Schrödinger type differential equations with variable domain, J. Math. Anal. and Appl, 211 (1997), 84-105.  doi: 10.1006/jmaa.1997.5422.

[3]

V. R. CabanillasS. B. De Menezes and E. Zuazua, Null controllability in unbounded domains for the semilinear heat equation with nonlinearities involving gradient terms, J. Optm. Theory Appl., 110 (2001), 245-264.  doi: 10.1023/A:1017515027783.

[4]

G.-Q. Chen and H. Frid, Divergence-measure fields and hyperbolic conservation laws, Archive Rat. Mech. Anal., 147 (1999), 89-118.  doi: 10.1007/s002050050146.

[5]

M. Chipot and B. Lovat, Existence and uniqueness results for a class of nonlocal elliptic and parabolic problems, Advances in quenching, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 8 (2001), 35-51. 

[6]

M. ChipotV. Valente and G. Caffarelli, Remarks on a nonlocal problem involving the Dirichlet energy, Rend. Semin. Mat. Univ. Padova, 110 (2003), 199-220. 

[7]

H. R. ClarkE. Fernández-CaraJ. Límaco and L. A Medeiros, Theoretical and numerical local null controllability for a parabolic system with local and nonlocal nonlinearities, Appl Math and Comp, 223 (2013), 483-505.  doi: 10.1016/j.amc.2013.08.035.

[8]

J. Cooper and C. Bardos, A nonlinear wave equation in a time dependent domain, J. Math. Anal. and Appl., 42 (1973), 29-60.  doi: 10.1016/0022-247X(73)90120-0.

[9]

J-M. Coron, Control and Nonlinearity, Mathematical Surveys and Monographs, 136. American Mathematical Society, Providence, RI, 2007. doi: 10.1090/surv/136.

[10]

S. B. De Menezes, J. Límaco and L. A. Medeiros, Remarks on null controllability for semilinear heat equation in moving domains, Eletronic J. of Qualitative Theory of Differential Equations, 16 (2003), No. 16, 32 pp.

[11]

S. B. De MenezesJ. Límaco and L. A. Medeiros, Finite approximate controllability for semilinear heat equations in non-cylindrical domains, Annals of the Brazilian Academy of Sciences, 76 (2004), 475-487.  doi: 10.1590/S0001-37652004000300002.

[12]

A. DoubovaE. Fernández-CaraM. González-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]

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

[14]

E. Fernández-Cara and S. Guerrero, Global Carleman inequalities for parabolic systems and applications to controllability, SIAM J. Control Optim., 45 (2006), 1395-1446.  doi: 10.1137/S0363012904439696.

[15]

E. Fernández-CaraJ. Límaco and S. B De Menezes, Null controllability for a parabolic equation with nonlocal nonlinearities, Systems and Control Letters, 61 (2012), 107-111.  doi: 10.1016/j.sysconle.2011.09.017.

[16]

E. Fernández-Cara and E. Zuazua, Null and approximate controllability of weakly blowing-up semilinear heat equations, Ann. Inst. Henri Poincaré, Analyse non Linéaire, 17 (2000), 583-616.  doi: 10.1016/S0294-1449(00)00117-7.

[17]

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

[18]

M. González-Burgos and R. Pérez-García, Controllabilty results for some nonlinear coupled parabolic systems by one control force, Asymptot. Anal., 46 (2006), 123-162. 

[19]

C. He and L. Hsiano, Two dimensional Euler equations in a time dependent domain, J. Diff. Equations, 163 (2000), 265-291.  doi: 10.1006/jdeq.1999.3702.

[20]

O. Yu Imanuvilov, Controllability of parabolic equations (in Russian), Mat. Sbornik. Novaya Seryia, 186 (1995), 109-132.  doi: 10.1070/SM1995v186n06ABEH000047.

[21]

O. Yu Imanuvilov, Remarks on exact controllability for the Navier-Stokes equations, ESAIM Control Optim. Calc. Var., 6 (2001), 39-72.  doi: 10.1051/cocv:2001103.

[22]

O. Yu Imanuvilov and M. Yamamoto, Carleman inequalities for parabolic equations in Sobolev spaces of negative order and exact controllability for semilinear parabolic equations, Publ. Res. Inst. Math. Sci., 39 (2003), 227-274.  doi: 10.2977/prims/1145476103.

[23]

A. Inoue, Sur $ \square u+u^{3} = f$ dans un domaine noncylindrique, J. Math. Anal. and Appl., 46 (1974), 777-819.  doi: 10.1016/0022-247X(74)90273-X.

[24]

J. LímacoM. ClarkA. MarinhoS. B. De Menezes and A. T. Louredo, Null controllability of some reaction-diffusion systems with only one control force in moving domains, Chin. Ann. Math. Ser. B, 37 (2016), 29-52.  doi: 10.1007/s11401-015-0959-8.

[25]

J. LímacoL. A. Medeiros and E. Zuazua, Existence, uniqueness and controllability for parabolic eqautions in non-cylindrical domains, Matemática Contemporânea, 23 (2002), 49-70.  doi: 10.1007/978-1-4615-7551-1.

[26]

J.-L. Lions, Quelques Méthodes de Résolution des Problèmes Aux Limites Nonlinéaires, Dunod, Paris, 1960.

[27]

J.-L. Lions, Une remarque sur les problèmes d'evolution nonlineaires dans le domaines noncylindriques (In french), Rev. Roumaine Math. Pures Appl., 9 (1964), 11-18. 

[28]

L. A. Medeiros, Nonlinear wave equations in domains with variable boundary, Arch. Rational Mech. Anal., 47 (1972), 47-58.  doi: 10.1007/BF00252188.

[29]

L. A MedeirosJ. Límaco and S. B. Menezes, Vibrations of elastic strings: Mathematical aspects - part one, J. Comput. Anal. Appl., 4 (2002), 91-127.  doi: 10.1023/A:1012934900316.

[30]

M. M. Miranda and J. L. Ferrel, The Navier-Stokes equation in noncylindrical domain, Comput. Appl. Math., 16 (1997), 247-265. 

[31]

M. M. Miranda and L. A. Medeiros, Contrôlabilité exacte de l'équation de Schrödinger dans des domaines noncylindriques, C. R. Acad. Sci. Paris, 319 (1994), 685-689. 

[32]

M. Nakao and T. Narazaki, Existence and decay of some nonlinear wave equations in noncylindrical domains, Math. Rep. Kyushu Univ., 11 (1978), 117-125. 

[33]

L. Prouvée and J. Límaco, Local null controllability for a parabolic-elliptic system with local and nonlocal nonlinearities, Eletronic J. of Qualitative Theory of Differential Equations, 74 (2019), 1-31.  doi: 10.14232/ejqtde.2019.1.74.

[34]

S. Zheng and M. Chipot, Asymptotic behavior of solutions to nonlinear parabolic equations with nonlocal terms, Asymptot. Anal., 45 (2005), 301-312. 

show all references

References:
[1]

V. M. Alekseev, V. M. Tikhomirov and S. V. Fomin, Optimal Control, Consultants Bureau, New York, 1987. doi: 10.1007/978-1-4615-7551-1.

[2]

M. L. BernardiG. Bonfanti and F. Lutteroti, Abstract Schrödinger type differential equations with variable domain, J. Math. Anal. and Appl, 211 (1997), 84-105.  doi: 10.1006/jmaa.1997.5422.

[3]

V. R. CabanillasS. B. De Menezes and E. Zuazua, Null controllability in unbounded domains for the semilinear heat equation with nonlinearities involving gradient terms, J. Optm. Theory Appl., 110 (2001), 245-264.  doi: 10.1023/A:1017515027783.

[4]

G.-Q. Chen and H. Frid, Divergence-measure fields and hyperbolic conservation laws, Archive Rat. Mech. Anal., 147 (1999), 89-118.  doi: 10.1007/s002050050146.

[5]

M. Chipot and B. Lovat, Existence and uniqueness results for a class of nonlocal elliptic and parabolic problems, Advances in quenching, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 8 (2001), 35-51. 

[6]

M. ChipotV. Valente and G. Caffarelli, Remarks on a nonlocal problem involving the Dirichlet energy, Rend. Semin. Mat. Univ. Padova, 110 (2003), 199-220. 

[7]

H. R. ClarkE. Fernández-CaraJ. Límaco and L. A Medeiros, Theoretical and numerical local null controllability for a parabolic system with local and nonlocal nonlinearities, Appl Math and Comp, 223 (2013), 483-505.  doi: 10.1016/j.amc.2013.08.035.

[8]

J. Cooper and C. Bardos, A nonlinear wave equation in a time dependent domain, J. Math. Anal. and Appl., 42 (1973), 29-60.  doi: 10.1016/0022-247X(73)90120-0.

[9]

J-M. Coron, Control and Nonlinearity, Mathematical Surveys and Monographs, 136. American Mathematical Society, Providence, RI, 2007. doi: 10.1090/surv/136.

[10]

S. B. De Menezes, J. Límaco and L. A. Medeiros, Remarks on null controllability for semilinear heat equation in moving domains, Eletronic J. of Qualitative Theory of Differential Equations, 16 (2003), No. 16, 32 pp.

[11]

S. B. De MenezesJ. Límaco and L. A. Medeiros, Finite approximate controllability for semilinear heat equations in non-cylindrical domains, Annals of the Brazilian Academy of Sciences, 76 (2004), 475-487.  doi: 10.1590/S0001-37652004000300002.

[12]

A. DoubovaE. Fernández-CaraM. González-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]

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

[14]

E. Fernández-Cara and S. Guerrero, Global Carleman inequalities for parabolic systems and applications to controllability, SIAM J. Control Optim., 45 (2006), 1395-1446.  doi: 10.1137/S0363012904439696.

[15]

E. Fernández-CaraJ. Límaco and S. B De Menezes, Null controllability for a parabolic equation with nonlocal nonlinearities, Systems and Control Letters, 61 (2012), 107-111.  doi: 10.1016/j.sysconle.2011.09.017.

[16]

E. Fernández-Cara and E. Zuazua, Null and approximate controllability of weakly blowing-up semilinear heat equations, Ann. Inst. Henri Poincaré, Analyse non Linéaire, 17 (2000), 583-616.  doi: 10.1016/S0294-1449(00)00117-7.

[17]

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

[18]

M. González-Burgos and R. Pérez-García, Controllabilty results for some nonlinear coupled parabolic systems by one control force, Asymptot. Anal., 46 (2006), 123-162. 

[19]

C. He and L. Hsiano, Two dimensional Euler equations in a time dependent domain, J. Diff. Equations, 163 (2000), 265-291.  doi: 10.1006/jdeq.1999.3702.

[20]

O. Yu Imanuvilov, Controllability of parabolic equations (in Russian), Mat. Sbornik. Novaya Seryia, 186 (1995), 109-132.  doi: 10.1070/SM1995v186n06ABEH000047.

[21]

O. Yu Imanuvilov, Remarks on exact controllability for the Navier-Stokes equations, ESAIM Control Optim. Calc. Var., 6 (2001), 39-72.  doi: 10.1051/cocv:2001103.

[22]

O. Yu Imanuvilov and M. Yamamoto, Carleman inequalities for parabolic equations in Sobolev spaces of negative order and exact controllability for semilinear parabolic equations, Publ. Res. Inst. Math. Sci., 39 (2003), 227-274.  doi: 10.2977/prims/1145476103.

[23]

A. Inoue, Sur $ \square u+u^{3} = f$ dans un domaine noncylindrique, J. Math. Anal. and Appl., 46 (1974), 777-819.  doi: 10.1016/0022-247X(74)90273-X.

[24]

J. LímacoM. ClarkA. MarinhoS. B. De Menezes and A. T. Louredo, Null controllability of some reaction-diffusion systems with only one control force in moving domains, Chin. Ann. Math. Ser. B, 37 (2016), 29-52.  doi: 10.1007/s11401-015-0959-8.

[25]

J. LímacoL. A. Medeiros and E. Zuazua, Existence, uniqueness and controllability for parabolic eqautions in non-cylindrical domains, Matemática Contemporânea, 23 (2002), 49-70.  doi: 10.1007/978-1-4615-7551-1.

[26]

J.-L. Lions, Quelques Méthodes de Résolution des Problèmes Aux Limites Nonlinéaires, Dunod, Paris, 1960.

[27]

J.-L. Lions, Une remarque sur les problèmes d'evolution nonlineaires dans le domaines noncylindriques (In french), Rev. Roumaine Math. Pures Appl., 9 (1964), 11-18. 

[28]

L. A. Medeiros, Nonlinear wave equations in domains with variable boundary, Arch. Rational Mech. Anal., 47 (1972), 47-58.  doi: 10.1007/BF00252188.

[29]

L. A MedeirosJ. Límaco and S. B. Menezes, Vibrations of elastic strings: Mathematical aspects - part one, J. Comput. Anal. Appl., 4 (2002), 91-127.  doi: 10.1023/A:1012934900316.

[30]

M. M. Miranda and J. L. Ferrel, The Navier-Stokes equation in noncylindrical domain, Comput. Appl. Math., 16 (1997), 247-265. 

[31]

M. M. Miranda and L. A. Medeiros, Contrôlabilité exacte de l'équation de Schrödinger dans des domaines noncylindriques, C. R. Acad. Sci. Paris, 319 (1994), 685-689. 

[32]

M. Nakao and T. Narazaki, Existence and decay of some nonlinear wave equations in noncylindrical domains, Math. Rep. Kyushu Univ., 11 (1978), 117-125. 

[33]

L. Prouvée and J. Límaco, Local null controllability for a parabolic-elliptic system with local and nonlocal nonlinearities, Eletronic J. of Qualitative Theory of Differential Equations, 74 (2019), 1-31.  doi: 10.14232/ejqtde.2019.1.74.

[34]

S. Zheng and M. Chipot, Asymptotic behavior of solutions to nonlinear parabolic equations with nonlocal terms, Asymptot. Anal., 45 (2005), 301-312. 

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