doi: 10.3934/eect.2021024
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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 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 & Control Theory, 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[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.   Google Scholar

[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.  Google Scholar

[20]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

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

[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.   Google Scholar

[28]

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

[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.  Google Scholar

[30]

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

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[34]

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

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[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.   Google Scholar

[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.  Google Scholar

[20]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

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

[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.   Google Scholar

[28]

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

[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.  Google Scholar

[30]

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

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[34]

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

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