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A positive solution of asymptotically periodic Choquard equations with locally defined nonlinearities
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 |
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.
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. |

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