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June  2019, 9(2): 351-384. doi: 10.3934/mcrf.2019017

## Nonlinear Schrödinger equations on a finite interval with point dissipation

 Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, VA USA

* Corresponding author: Shu-Ming Sun

Received  July 2017 Revised  July 2018 Published  December 2018

Fund Project: The research was partially supported by the National Science Foundation under grant No. DMS-1210979.

The paper considers the initial value problem of a general type of nonlinear Schrödinger equations
 $iu_t+u_{xx}+f(u) = 0 , \;\;\;\; u ( x, 0 ) = w_0 (x)$
posed on a finite domain
 $x\in [0, L]$
with an
 $L^2$
-stabilizing feedback control law
 $u(0, t) = \beta u(L, t), \beta u_x(0, t)-u_x(L, t) = i\alpha u(0, t),$
where
 $L>0$
,
 $\alpha, \beta$
are real constants with
 $\alpha\beta<0$
and
 $\beta\neq \pm 1$
, and
 $f(u)$
is a smooth function from
 $\mathbb{C}$
to
 $\mathbb{C}$
satisfying some growth conditions. It is shown that for
 $s \in \left ( \frac12, 1\right ]$
and
 $w_0 (x) \in H^s(0, L )$
with the boundary conditions described above, the problem is locally well-posed for
 $u \in C([0, T]; H^s (0, L ))$
. Moreover, the solution with small initial condition exists globally and approaches to 0 as
 $t \rightarrow + \infty$
.
Citation: Jing Cui, Shu-Ming Sun. Nonlinear Schrödinger equations on a finite interval with point dissipation. Mathematical Control and Related Fields, 2019, 9 (2) : 351-384. doi: 10.3934/mcrf.2019017
##### References:
 [1] J. L. Bona, S. M. Sun and B.-Y. Zhang, Nonhomogeneous boundary-value problems for one-dimensional nonlinear Schrödinger equations, J. Math. Pures Appl., 109 (2018), 1-66.  doi: 10.1016/j.matpur.2017.11.001. [2] J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to non-linear evolution equations, part Ⅰ: Schrödinger equations, Geom. Funct. Anal., 3 (1993), 107-156.  doi: 10.1007/BF01896020. [3] J. Bourgain, Global Solutions of Nonlinear Schrödinger Equations, Colloqium Publication, Vol. 46, American Mathematical Society, Providence, RI, 1999. doi: 10.1090/coll/046. [4] H. Brézis and T. Gallouet, Nonlinear Schrödinger evolution equation, Nonlinear Anal. TMA, 4 (1980), 677-681.  doi: 10.1016/0362-546X(80)90068-1. [5] C. Bu, An initial-boundary value problem of the nonlinear Schrödinger equation, Appl. Anal., 53 (1994), 241-254.  doi: 10.1080/00036819408840260. [6] C. Bu, Nonlinear Schrödinger equation on the semi-infinite line, Chinese Annals of Math., 21 (2000), 209-222. [7] C. Bu, K. Tsutaya and C Zhang, Nonlinear Schrödinger equation with inhomogebeous Dirichlet boundary data, J. Math. Phys., 46 (2005), 083504, 6pp. doi: 10.1063/1.1914730. [8] T. Cazenave, Semilinear Schrödinger Equations, American Math. Soc., Providence, RI, 2003. doi: 10.1090/cln/010. [9] T. Cazenave, D. Fang and Z. Han, Continuous dependence for NLS in fractional order spaces, Ann. Inst. H. Poincaré, Anal. Non Linéaire, 28 (2011), 135-147.  doi: 10.1016/j.anihpc.2010.11.005. [10] T. Cazenave and F. B. Weissler, The Cauchy problem for the critical nonlinear Schrödinger equation in $H^s$, Nonlinear Anal. TMA, 14 (1990), 807-836.  doi: 10.1016/0362-546X(90)90023-A. [11] A. Chabchoub, N. Hoffmann and N. Akhmediev, Rogue wave observation in a water wave tank, Phys. Rev. Lett., 106 (2011), 204502. doi: 10.1103/PhysRevLett.106.204502. [12] N. Dunford and J. T. Schwartz, Linear Operators, Part III, Wiley-Interscience, New York, 1971. [13] M. Fujii and R. Nakamoto, Simultaneous Extensions of Selberg inequality and Heinz-Kato-Furuta inequality, Nihonkai Math., 9 (1998), 219-225. [14] G. Gao and S. M. Sun, A Korteweg-de Vries type of fifth-order equations on a finite domain with point dissipation, J. Math. Anal. Appl., 438 (2016), 200-239.  doi: 10.1016/j.jmaa.2016.01.050. [15] J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equations. Ⅰ. The Cauchy problem, general case, J. Functional Anal., 32 (1979), 1-32.  doi: 10.1016/0022-1236(79)90076-4. [16] J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equations. Ⅱ. Scattering theory, general case, J. Functinal Anal., 32 (1979), 33-71.  doi: 10.1016/0022-1236(79)90077-6. [17] L. F. Ho and D. L. Russell, Admissible input elements for systems in Hillbert space and Carleson measure criterion, SIAM J. Control. Optim., 21 (1983), 614-640.  doi: 10.1137/0321037. [18] J. Holmer, The initial-boundary value problem for the $1$-$d$ nonlinear Schrödinger equation on the half-line, Diff. Integral Equations, 18 (2005), 647-668. [19] F.-L. Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces, Ann. Differential Equations, 1 (1985), 43-56. [20] R. Illner, H. Lange and H. Teismann, A note on the exact internal control of nonlinear Schrödinger equations, CRM Proc. Lecture Notes, 33 (2003), 127-137. [21] R. Illner, H. Lange and H. Teismann, Limitations on the control of Schrödinger equations, ESAIM Control Optim. Calc. Var., 12 (2006), 615-635.  doi: 10.1051/cocv:2006014. [22] T. Kato, On nonlinear Schrödinger equations, Ann. Inst. H. Poincaré, Phys. Theor., 46 (1987), 113--129. [23] T. Kato, On nonlinear Scrhödinger equations. Ⅱ. $H^s$-solutions and unconditional well-posedness, J. d'Analyse Math., 67 (1995), 281-306.  doi: 10.1007/BF02787794. [24] S. Kamvissis, Semiclassical nonlinear Schrödinger on the half line, J. Math. Phys., 44 (2003), 5849--5868. doi: 10.1063/1.1624091. [25] V. Komornik, A generalization of Ingham's inequality, in Colloq. Math. Soc. $J\grave{a}nos$ Bolyai, Differential Equations Applications, 62 (1991), 213--217. [26] H. Lange and H. Teismann, Controllability of the nonlinear Schrödinger equation in the vicinity of the ground state, Math. Methods Appl. Sci., 30 (2007), 1483-1505.  doi: 10.1002/mma.849. [27] G. Lumer and R. S. Phillips, Dissipative operators in a Banach space, Pacific J. Math., 11 (1961), 679-698.  doi: 10.2140/pjm.1961.11.679. [28] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. [29] D. Peregrine, Water waves, nonlinear Schrödinger equations and their solutions, J. Austral. Math. Soc. B, 25 (1983), 16-43.  doi: 10.1017/S0334270000003891. [30] L. Rosier and B.-Y. Zhang, Local exact controllability and stabilizability of the nonlinear Schrödinger equation on a bounded interval, SIAM J. Control Optim., 48 (2009), 972-992.  doi: 10.1137/070709578. [31] D. L. Russell, Controllability and stabilizability theory for linear partial differential equations: Recent progress and open questions, SIAM Rev., 20 (1978), 639-739.  doi: 10.1137/1020095. [32] D. L. Russell and B. Y. Zhang, Controllability and stabilizability of the third-order linear dispersion equation on a periodic domain, SIAM J. Control Optim., 31 (1993), 659-676.  doi: 10.1137/0331030. [33] D. L. Russell and B. Y. Zhang, Smoothing and decay properties of solutions of the Korteweg-de Vries equation on a periodic domain with point dissipation, J. Math. Anal. Appl., 190 (1995), 449-488.  doi: 10.1006/jmaa.1995.1087. [34] W. Strauss and C. Bu, Inhomogeneous boundary value problem for a nonlinear Schrödinger equation, J. Diff. Equations, 173 (2001), 79-91.  doi: 10.1006/jdeq.2000.3871. [35] S. M. Sun, The Korteweg-de Vries equation on a periodic domain with singular-point dissipation, SIAM J. Control and Optimization, 34 (1996), 892-912.  doi: 10.1137/S0363012994269491. [36] Y. Tsutsumi, $L^2$-solutions for nonlinear Schrödinger equations and nonlinear groups, Funk. Ekva., 30 (1987), 115-125. [37] V. E. Zakharov and S. V. Manakov, On the complete integrability of a nonlinear Schrödinger equation, J. Theore. and Math. Phys., 19 (1974), 551-559. [38] V. E. Zakharov and A. B. Shabat, Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media, J. Experi. and Theore. Phys., 34 (1972), 62-69.

show all references

##### References:
 [1] J. L. Bona, S. M. Sun and B.-Y. Zhang, Nonhomogeneous boundary-value problems for one-dimensional nonlinear Schrödinger equations, J. Math. Pures Appl., 109 (2018), 1-66.  doi: 10.1016/j.matpur.2017.11.001. [2] J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to non-linear evolution equations, part Ⅰ: Schrödinger equations, Geom. Funct. Anal., 3 (1993), 107-156.  doi: 10.1007/BF01896020. [3] J. Bourgain, Global Solutions of Nonlinear Schrödinger Equations, Colloqium Publication, Vol. 46, American Mathematical Society, Providence, RI, 1999. doi: 10.1090/coll/046. [4] H. Brézis and T. Gallouet, Nonlinear Schrödinger evolution equation, Nonlinear Anal. TMA, 4 (1980), 677-681.  doi: 10.1016/0362-546X(80)90068-1. [5] C. Bu, An initial-boundary value problem of the nonlinear Schrödinger equation, Appl. Anal., 53 (1994), 241-254.  doi: 10.1080/00036819408840260. [6] C. Bu, Nonlinear Schrödinger equation on the semi-infinite line, Chinese Annals of Math., 21 (2000), 209-222. [7] C. Bu, K. Tsutaya and C Zhang, Nonlinear Schrödinger equation with inhomogebeous Dirichlet boundary data, J. Math. Phys., 46 (2005), 083504, 6pp. doi: 10.1063/1.1914730. [8] T. Cazenave, Semilinear Schrödinger Equations, American Math. Soc., Providence, RI, 2003. doi: 10.1090/cln/010. [9] T. Cazenave, D. Fang and Z. Han, Continuous dependence for NLS in fractional order spaces, Ann. Inst. H. Poincaré, Anal. Non Linéaire, 28 (2011), 135-147.  doi: 10.1016/j.anihpc.2010.11.005. [10] T. Cazenave and F. B. Weissler, The Cauchy problem for the critical nonlinear Schrödinger equation in $H^s$, Nonlinear Anal. TMA, 14 (1990), 807-836.  doi: 10.1016/0362-546X(90)90023-A. [11] A. Chabchoub, N. Hoffmann and N. Akhmediev, Rogue wave observation in a water wave tank, Phys. Rev. Lett., 106 (2011), 204502. doi: 10.1103/PhysRevLett.106.204502. [12] N. Dunford and J. T. Schwartz, Linear Operators, Part III, Wiley-Interscience, New York, 1971. [13] M. Fujii and R. Nakamoto, Simultaneous Extensions of Selberg inequality and Heinz-Kato-Furuta inequality, Nihonkai Math., 9 (1998), 219-225. [14] G. Gao and S. M. Sun, A Korteweg-de Vries type of fifth-order equations on a finite domain with point dissipation, J. Math. Anal. Appl., 438 (2016), 200-239.  doi: 10.1016/j.jmaa.2016.01.050. [15] J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equations. Ⅰ. The Cauchy problem, general case, J. Functional Anal., 32 (1979), 1-32.  doi: 10.1016/0022-1236(79)90076-4. [16] J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equations. Ⅱ. Scattering theory, general case, J. Functinal Anal., 32 (1979), 33-71.  doi: 10.1016/0022-1236(79)90077-6. [17] L. F. Ho and D. L. Russell, Admissible input elements for systems in Hillbert space and Carleson measure criterion, SIAM J. Control. Optim., 21 (1983), 614-640.  doi: 10.1137/0321037. [18] J. Holmer, The initial-boundary value problem for the $1$-$d$ nonlinear Schrödinger equation on the half-line, Diff. Integral Equations, 18 (2005), 647-668. [19] F.-L. Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces, Ann. Differential Equations, 1 (1985), 43-56. [20] R. Illner, H. Lange and H. Teismann, A note on the exact internal control of nonlinear Schrödinger equations, CRM Proc. Lecture Notes, 33 (2003), 127-137. [21] R. Illner, H. Lange and H. Teismann, Limitations on the control of Schrödinger equations, ESAIM Control Optim. Calc. Var., 12 (2006), 615-635.  doi: 10.1051/cocv:2006014. [22] T. Kato, On nonlinear Schrödinger equations, Ann. Inst. H. Poincaré, Phys. Theor., 46 (1987), 113--129. [23] T. Kato, On nonlinear Scrhödinger equations. Ⅱ. $H^s$-solutions and unconditional well-posedness, J. d'Analyse Math., 67 (1995), 281-306.  doi: 10.1007/BF02787794. [24] S. Kamvissis, Semiclassical nonlinear Schrödinger on the half line, J. Math. Phys., 44 (2003), 5849--5868. doi: 10.1063/1.1624091. [25] V. Komornik, A generalization of Ingham's inequality, in Colloq. Math. Soc. $J\grave{a}nos$ Bolyai, Differential Equations Applications, 62 (1991), 213--217. [26] H. Lange and H. Teismann, Controllability of the nonlinear Schrödinger equation in the vicinity of the ground state, Math. Methods Appl. Sci., 30 (2007), 1483-1505.  doi: 10.1002/mma.849. [27] G. Lumer and R. S. Phillips, Dissipative operators in a Banach space, Pacific J. Math., 11 (1961), 679-698.  doi: 10.2140/pjm.1961.11.679. [28] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. [29] D. Peregrine, Water waves, nonlinear Schrödinger equations and their solutions, J. Austral. Math. Soc. B, 25 (1983), 16-43.  doi: 10.1017/S0334270000003891. [30] L. Rosier and B.-Y. Zhang, Local exact controllability and stabilizability of the nonlinear Schrödinger equation on a bounded interval, SIAM J. Control Optim., 48 (2009), 972-992.  doi: 10.1137/070709578. [31] D. L. Russell, Controllability and stabilizability theory for linear partial differential equations: Recent progress and open questions, SIAM Rev., 20 (1978), 639-739.  doi: 10.1137/1020095. [32] D. L. Russell and B. Y. Zhang, Controllability and stabilizability of the third-order linear dispersion equation on a periodic domain, SIAM J. Control Optim., 31 (1993), 659-676.  doi: 10.1137/0331030. [33] D. L. Russell and B. Y. Zhang, Smoothing and decay properties of solutions of the Korteweg-de Vries equation on a periodic domain with point dissipation, J. Math. Anal. Appl., 190 (1995), 449-488.  doi: 10.1006/jmaa.1995.1087. [34] W. Strauss and C. Bu, Inhomogeneous boundary value problem for a nonlinear Schrödinger equation, J. Diff. Equations, 173 (2001), 79-91.  doi: 10.1006/jdeq.2000.3871. [35] S. M. Sun, The Korteweg-de Vries equation on a periodic domain with singular-point dissipation, SIAM J. Control and Optimization, 34 (1996), 892-912.  doi: 10.1137/S0363012994269491. [36] Y. Tsutsumi, $L^2$-solutions for nonlinear Schrödinger equations and nonlinear groups, Funk. Ekva., 30 (1987), 115-125. [37] V. E. Zakharov and S. V. Manakov, On the complete integrability of a nonlinear Schrödinger equation, J. Theore. and Math. Phys., 19 (1974), 551-559. [38] V. E. Zakharov and A. B. Shabat, Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media, J. Experi. and Theore. Phys., 34 (1972), 62-69.

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