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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

 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.

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 and 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. [2] L. Aloui, Smoothing effect for regularized Schrödinger equation on bounded domains, Asymptot. Anal., 59 (2008), 179-193. [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. [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. [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. [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. [7] H. Brézis, Nonlinear Evolution Equations. Autumn Course on Semigroups, Theory and Applications, International Centre for Theoretical Physics. Trieste, 1984. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [19] R. Fukuoka, Mollifier smoothing of tensor fields on differentiable manifolds and applications to Riemannian Geometry, preprint, arXiv: math.DG/0608230. [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. [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. [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. [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. [24] J. L. Lions and E. Magenes, Problèmes aux Limites non Homogènes, Aplications, Dunod, Paris, 1968. [25] E. Machtyngier and E. Zuazua, Stabilization of the Schrödinger equation, Portugaliae Mathematica, 51 (1994), 243-256. [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. [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. [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. [29] M. Spivak, A Comprehensive Introduction to Differential Geometry, Publish or Perish, INC., Houston, 1999. [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. [31] M. Taylor, Partial Differential Equations, Springer, Berlin, 1991. doi: 10.1007/978-1-4684-9320-7. [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. [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. [34] F. W. Warner, Foundations of Differentiable Manifolds and Lie Groups, Foresman and Company, Scott, 1971.

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. [2] L. Aloui, Smoothing effect for regularized Schrödinger equation on bounded domains, Asymptot. Anal., 59 (2008), 179-193. [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. [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. [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. [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. [7] H. Brézis, Nonlinear Evolution Equations. Autumn Course on Semigroups, Theory and Applications, International Centre for Theoretical Physics. Trieste, 1984. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [19] R. Fukuoka, Mollifier smoothing of tensor fields on differentiable manifolds and applications to Riemannian Geometry, preprint, arXiv: math.DG/0608230. [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. [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. [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. [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. [24] J. L. Lions and E. Magenes, Problèmes aux Limites non Homogènes, Aplications, Dunod, Paris, 1968. [25] E. Machtyngier and E. Zuazua, Stabilization of the Schrödinger equation, Portugaliae Mathematica, 51 (1994), 243-256. [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. [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. [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. [29] M. Spivak, A Comprehensive Introduction to Differential Geometry, Publish or Perish, INC., Houston, 1999. [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. [31] M. Taylor, Partial Differential Equations, Springer, Berlin, 1991. doi: 10.1007/978-1-4684-9320-7. [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. [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. [34] F. W. Warner, Foundations of Differentiable Manifolds and Lie Groups, Foresman and Company, Scott, 1971.
$\mathbb{V}$ 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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