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March  2020, 19(3): 1367-1386. doi: 10.3934/cpaa.2020067

Exponential stability for the locally damped defocusing Schrödinger equation on compact manifold

César Augusto Bortot 1,, , Wellington José Corrêa 2, , Ryuichi Fukuoka 3, and  Thales Maier Souza 1,

1. 

Department of Mobility Engineering, Federal University of Santa Catarina, Joinville-SC, 89219-600, Brazil
2. 

Academic Department of Mathematics, Federal Technological University of Paraná, Campo Mourão-PR, 87301-899, Brazil
3. 

Department of Mathematics, State University of Maringá, Maringá-PR, 87020-900, Brazil

* Corresponding author

Received  March 2019 Revised  September 2019 Published  November 2019

Fund Project: Research of Wellington J. Corrêa partially supported by the CNPq Grant 438807/2018-9.

In this paper we study the asymptotic dynamics for semilinear defocusing Schrödinger equation subject to a damping locally distributed on a n-dimentional compact Riemannian manifold $ M^n $ without boundary. The proofs are based on a result of unique continuation property, in the construction of a function $ f $ whose Hessian is positive definite and $ \Delta f = C_0 $ in some region contained in $ M $ and about the smoothing effect due to Aloui adapted to the present context.

Keywords: Schödinger equation, asymptotic stabilization, Riemannian manifolds.
Mathematics Subject Classification: Primary: 35L60; Secondary: 35B40.
Citation: César Augusto Bortot, Wellington José Corrêa, Ryuichi Fukuoka, Thales Maier Souza. Exponential stability for the locally damped defocusing Schrödinger equation on compact manifold. Communications on Pure & Applied Analysis, 2020, 19 (3) : 1367-1386. doi: 10.3934/cpaa.2020067
References:
[1]

L. Aloui, Smoothing effect for regularized Schrödinger equation on compact manifolds, Collect. Math., 59 (2008), 53-62.  doi: 10.1007/BF03191181.  Google Scholar

[2]

L. Aloui, Smoothing effect for regularized Schrödinger equation on bounded domains, Asymptot. Anal., 59 (2008), 179-193.   Google Scholar

[3]

R. Anton, Strichartz inequalities for Lipschitz metrics on manifolds and nonlinear Schrödinger equation on domains, Bull. Soc. Math. France, 136 (2008), 27-65.  doi: 10.24033/bsmf.2548.  Google Scholar

[4]

C. A. Bortot and M. M. Cavalcanti, Asymptotic stability for the damped Schrödinger equation on noncompact Riemannian manifolds and exterior domains, Communications in Partial Differential Equations, 39 (2014), 1791-1820.  doi: 10.1080/03605302.2014.908390.  Google Scholar

[5]

C. A. Bortot, M. M. Cavalcanti, W. J. Corrêa and V. N. Domingos Cavalcanti, Uniform decay rate estimates for Schrödinger and plate equations with nonlinear locally distributed damping, Journal of Differential Equations, 254 (2013), 3729-3764.  doi: 10.1016/j.jde.2013.01.040.  Google Scholar

[6]

C. A. Bortot and W. J. Corrêa, Exponential stability for the defocusing Schrödinger equation subject to strong damping locally distributed, Differential and Integral Equations, 31 (2018), 273-300.   Google Scholar

[7]

H. Brézis, Nonlinear Evolution Equations. Autumn Course on Semigroups, Theory and Applications, International Centre for Theoretical Physics. Trieste, 1984. Google Scholar

[8]

N. Burq, P. Gérard and N. Tzvetzkov, Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds, Am. J. Maths., 126 (2004), 569–605.  Google Scholar

[9]

N. Burq, P. Gérard and N. Tzvetkov, The Schrödinger equation on a compact manifold: Strichartz estimates and applications, Journées Équations aux Dérivées Partielles, (2001), 1–18. doi: 10.5802/jedp.589.  Google Scholar

[10]

N. Burq, P. Gérard and N. Tzvetkov, The Cauchy Problem for the Nonlinear Schrödinger Equation on a Compact Manifold, J. Nonlinear Math. Phys., 10 (2003), 12-27.  doi: 10.2991/jnmp.2003.10.s1.2.  Google Scholar

[11]

M. M. Cavalcanti, V. N. Domingos Cavalcanti, R. Fukuoka and J. A. Soriano, Uniform Stabilization of the wave equation on compact surfaces and locally distributed damping, Transactions of AMS, 361 (2009), 4561-4580.  doi: 10.1090/S0002-9947-09-04763-1.  Google Scholar

[12]

M. M. Cavalcanti, V. N. Domingos Cavalcanti, R. Fukuoka and J. A. Soriano, Uniform stabilization of the wave equation on compact manifolds and locally distributed damping - a sharp result, J. Math. Anal. Appl., 351 (2009), 661-674.  doi: 10.1016/j.jmaa.2008.11.008.  Google Scholar

[13]

M. M. Cavalcanti, V. N. Domingos Cavalcanti, R. Fukuoka and F. Natali, Exponential stability for the 2-D defocusing Schrödinger equation with locally distributed damping, Differential Integral Equations, 22 (2009), 617-636.   Google Scholar

[14]

M. M. Cavalcanti, V. N. Domingos Cavalcanti, R. Fukuoka and J. A. Soriano, Asymptotic stability of the wave equation on compact manifolds and locally distributed damping: A sharp result, Arch. Rational Mech. Anal., 197 (2010), 925-964.  doi: 10.1007/s00205-009-0284-z.  Google Scholar

[15]

M. M. Cavalcanti, V. N. Domingos Cavalcanti, J. A. Soriano and F. Natali, Qualitative aspects for the cubic nonlinear Schrödinger equations with localized damping: exponential and polynomial stabilization, J. Differential Equations, 248 (2010), 2955-2971.  doi: 10.1016/j.jde.2010.03.023.  Google Scholar

[16]

M. M. Cavalcanti, W. J. Corrêa, V. N. Domingos Cavalcanti and M. R. Astudillo et al., Z. Angew. Math. Phys., (2018) 69: 100. https://doi.org/10.1007/s00033-018-0985-y doi: 10.1007/s00033-018-0985-y.  Google Scholar

[17]

B. Dehman, P. Gérard and G. Lebeau, Stabilization and control for the nonlinear Schrödinger equation on a compact surface, Math. Z., 254 (2006), 729-749.  doi: 10.1007/s00209-006-0005-3.  Google Scholar

[18]

S. Demoulini, Global existence for a nonlinear Schroedinger-Chern-Simons system on a surface, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 207-225.  doi: 10.1016/j.anihpc.2006.01.004.  Google Scholar

[19]

R. Fukuoka, Mollifier smoothing of tensor fields on differentiable manifolds and applications to Riemannian Geometry, preprint, arXiv: math.DG/0608230. Google Scholar

[20]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Diferential Equations of Second Order, Springer-Verlag Berlin Heidelberg, 2001. doi: 10.1007/978-3-642-61798-0.  Google Scholar

[21]

I. Lasiecka and R. Triggiani, Optimal regularity, exact controllability and uniform stabilization of Schrödinger equations with Dirichlet control, Differential Integral Equations, 5 (1992), 521-535.   Google Scholar

[22]

I. Lasiecka and R. Triggiani, Well-posedness and sharp uniform decay rates at the $ L^2(\Omega) $ - level of the Schrödinger equation with nonlinear boundary dissipation, J. Evol. Equ., 6 (2006), 485-537.  doi: 10.1007/s00028-006-0267-6.  Google Scholar

[23]

C. Laurent, Global controlabilty and stabilzation for the nonlinear Schrödinger equation on some compact manifolds of dimension 3, SIAM J. Math. Anal., 42 (2010), 785-832.  doi: 10.1137/090749086.  Google Scholar

[24]

J. L. Lions and E. Magenes, Problèmes aux Limites non Homogènes, Aplications, Dunod, Paris, 1968.  Google Scholar

[25]

E. Machtyngier and E. Zuazua, Stabilization of the Schrödinger equation, Portugaliae Mathematica, 51 (1994), 243-256.   Google Scholar

[26]

F. Merle and P. Raphael, On universality of blow-up profile for $L^2$ critical nonlinear Schrödinger equation, Invent. Math., 156 (2004), 565-672.  doi: 10.1007/s00222-003-0346-z.  Google Scholar

[27]

C.E. Kenig and F. Merle, Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case, Invent. Math., 166 (2006), 645-675.  doi: 10.1007/s00222-006-0011-4.  Google Scholar

[28]

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

[29]

M. Spivak, A Comprehensive Introduction to Differential Geometry, Publish or Perish, INC., Houston, 1999.  Google Scholar

[30]

W. Strauss and C. Bu, An inhomogeneous boundary value problem for nonlinear Schrödinger equations, Journal of Differential Equations, 173 (2001), 79-91.  doi: 10.1006/jdeq.2000.3871.  Google Scholar

[31]

M. Taylor, Partial Differential Equations, Springer, Berlin, 1991. doi: 10.1007/978-1-4684-9320-7.  Google Scholar

[32]

L. Thomann, Instabilities for supercritical Schrödinger equations in analytic manifolds, Journal of Differential Equations, 245 (2008), 249-280.  doi: 10.1016/j.jde.2007.12.001.  Google Scholar

[33]

M. Tsutsumi, On global solutions to the initial boundary value problem for the damped nonlinear Schrödinger equations, J. Math. Anal. Appl., 145 (1990), 328-341.  doi: 10.1016/0022-247X(90)90403-3.  Google Scholar

[34]

F. W. Warner, Foundations of Differentiable Manifolds and Lie Groups, Foresman and Company, Scott, 1971.  Google Scholar

show all references

References:
[1]

L. Aloui, Smoothing effect for regularized Schrödinger equation on compact manifolds, Collect. Math., 59 (2008), 53-62.  doi: 10.1007/BF03191181.  Google Scholar

[2]

L. Aloui, Smoothing effect for regularized Schrödinger equation on bounded domains, Asymptot. Anal., 59 (2008), 179-193.   Google Scholar

[3]

R. Anton, Strichartz inequalities for Lipschitz metrics on manifolds and nonlinear Schrödinger equation on domains, Bull. Soc. Math. France, 136 (2008), 27-65.  doi: 10.24033/bsmf.2548.  Google Scholar

[4]

C. A. Bortot and M. M. Cavalcanti, Asymptotic stability for the damped Schrödinger equation on noncompact Riemannian manifolds and exterior domains, Communications in Partial Differential Equations, 39 (2014), 1791-1820.  doi: 10.1080/03605302.2014.908390.  Google Scholar

[5]

C. A. Bortot, M. M. Cavalcanti, W. J. Corrêa and V. N. Domingos Cavalcanti, Uniform decay rate estimates for Schrödinger and plate equations with nonlinear locally distributed damping, Journal of Differential Equations, 254 (2013), 3729-3764.  doi: 10.1016/j.jde.2013.01.040.  Google Scholar

[6]

C. A. Bortot and W. J. Corrêa, Exponential stability for the defocusing Schrödinger equation subject to strong damping locally distributed, Differential and Integral Equations, 31 (2018), 273-300.   Google Scholar

[7]

H. Brézis, Nonlinear Evolution Equations. Autumn Course on Semigroups, Theory and Applications, International Centre for Theoretical Physics. Trieste, 1984. Google Scholar

[8]

N. Burq, P. Gérard and N. Tzvetzkov, Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds, Am. J. Maths., 126 (2004), 569–605.  Google Scholar

[9]

N. Burq, P. Gérard and N. Tzvetkov, The Schrödinger equation on a compact manifold: Strichartz estimates and applications, Journées Équations aux Dérivées Partielles, (2001), 1–18. doi: 10.5802/jedp.589.  Google Scholar

[10]

N. Burq, P. Gérard and N. Tzvetkov, The Cauchy Problem for the Nonlinear Schrödinger Equation on a Compact Manifold, J. Nonlinear Math. Phys., 10 (2003), 12-27.  doi: 10.2991/jnmp.2003.10.s1.2.  Google Scholar

[11]

M. M. Cavalcanti, V. N. Domingos Cavalcanti, R. Fukuoka and J. A. Soriano, Uniform Stabilization of the wave equation on compact surfaces and locally distributed damping, Transactions of AMS, 361 (2009), 4561-4580.  doi: 10.1090/S0002-9947-09-04763-1.  Google Scholar

[12]

M. M. Cavalcanti, V. N. Domingos Cavalcanti, R. Fukuoka and J. A. Soriano, Uniform stabilization of the wave equation on compact manifolds and locally distributed damping - a sharp result, J. Math. Anal. Appl., 351 (2009), 661-674.  doi: 10.1016/j.jmaa.2008.11.008.  Google Scholar

[13]

M. M. Cavalcanti, V. N. Domingos Cavalcanti, R. Fukuoka and F. Natali, Exponential stability for the 2-D defocusing Schrödinger equation with locally distributed damping, Differential Integral Equations, 22 (2009), 617-636.   Google Scholar

[14]

M. M. Cavalcanti, V. N. Domingos Cavalcanti, R. Fukuoka and J. A. Soriano, Asymptotic stability of the wave equation on compact manifolds and locally distributed damping: A sharp result, Arch. Rational Mech. Anal., 197 (2010), 925-964.  doi: 10.1007/s00205-009-0284-z.  Google Scholar

[15]

M. M. Cavalcanti, V. N. Domingos Cavalcanti, J. A. Soriano and F. Natali, Qualitative aspects for the cubic nonlinear Schrödinger equations with localized damping: exponential and polynomial stabilization, J. Differential Equations, 248 (2010), 2955-2971.  doi: 10.1016/j.jde.2010.03.023.  Google Scholar

[16]

M. M. Cavalcanti, W. J. Corrêa, V. N. Domingos Cavalcanti and M. R. Astudillo et al., Z. Angew. Math. Phys., (2018) 69: 100. https://doi.org/10.1007/s00033-018-0985-y doi: 10.1007/s00033-018-0985-y.  Google Scholar

[17]

B. Dehman, P. Gérard and G. Lebeau, Stabilization and control for the nonlinear Schrödinger equation on a compact surface, Math. Z., 254 (2006), 729-749.  doi: 10.1007/s00209-006-0005-3.  Google Scholar

[18]

S. Demoulini, Global existence for a nonlinear Schroedinger-Chern-Simons system on a surface, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 207-225.  doi: 10.1016/j.anihpc.2006.01.004.  Google Scholar

[19]

R. Fukuoka, Mollifier smoothing of tensor fields on differentiable manifolds and applications to Riemannian Geometry, preprint, arXiv: math.DG/0608230. Google Scholar

[20]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Diferential Equations of Second Order, Springer-Verlag Berlin Heidelberg, 2001. doi: 10.1007/978-3-642-61798-0.  Google Scholar

[21]

I. Lasiecka and R. Triggiani, Optimal regularity, exact controllability and uniform stabilization of Schrödinger equations with Dirichlet control, Differential Integral Equations, 5 (1992), 521-535.   Google Scholar

[22]

I. Lasiecka and R. Triggiani, Well-posedness and sharp uniform decay rates at the $ L^2(\Omega) $ - level of the Schrödinger equation with nonlinear boundary dissipation, J. Evol. Equ., 6 (2006), 485-537.  doi: 10.1007/s00028-006-0267-6.  Google Scholar

[23]

C. Laurent, Global controlabilty and stabilzation for the nonlinear Schrödinger equation on some compact manifolds of dimension 3, SIAM J. Math. Anal., 42 (2010), 785-832.  doi: 10.1137/090749086.  Google Scholar

[24]

J. L. Lions and E. Magenes, Problèmes aux Limites non Homogènes, Aplications, Dunod, Paris, 1968.  Google Scholar

[25]

E. Machtyngier and E. Zuazua, Stabilization of the Schrödinger equation, Portugaliae Mathematica, 51 (1994), 243-256.   Google Scholar

[26]

F. Merle and P. Raphael, On universality of blow-up profile for $L^2$ critical nonlinear Schrödinger equation, Invent. Math., 156 (2004), 565-672.  doi: 10.1007/s00222-003-0346-z.  Google Scholar

[27]

C.E. Kenig and F. Merle, Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case, Invent. Math., 166 (2006), 645-675.  doi: 10.1007/s00222-006-0011-4.  Google Scholar

[28]

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

[29]

M. Spivak, A Comprehensive Introduction to Differential Geometry, Publish or Perish, INC., Houston, 1999.  Google Scholar

[30]

W. Strauss and C. Bu, An inhomogeneous boundary value problem for nonlinear Schrödinger equations, Journal of Differential Equations, 173 (2001), 79-91.  doi: 10.1006/jdeq.2000.3871.  Google Scholar

[31]

M. Taylor, Partial Differential Equations, Springer, Berlin, 1991. doi: 10.1007/978-1-4684-9320-7.  Google Scholar

[32]

L. Thomann, Instabilities for supercritical Schrödinger equations in analytic manifolds, Journal of Differential Equations, 245 (2008), 249-280.  doi: 10.1016/j.jde.2007.12.001.  Google Scholar

[33]

M. Tsutsumi, On global solutions to the initial boundary value problem for the damped nonlinear Schrödinger equations, J. Math. Anal. Appl., 145 (1990), 328-341.  doi: 10.1016/0022-247X(90)90403-3.  Google Scholar

[34]

F. W. Warner, Foundations of Differentiable Manifolds and Lie Groups, Foresman and Company, Scott, 1971.  Google Scholar

Figure 1.  $ \mathbb{V} $ is a non-dissipative area (in white) arbitrarily large while the demarcated area (in black) contains dissipative effects and can be considered arbitrarily small, both totally distributed on $ M $
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