A Note on a Damped Focusing Inhomogeneous Choquard Equation
DOI:
https://doi.org/10.15407/mag17.03.295Анотація
Стаття присвячена фокусувальному неоднорiдному рiвнянню Шокарда з лiнiйним демпфуванням:$$ i\dot{u}+\triangle u+ia u=-|x|^{-\gamma}(I_{\alpha}\ast|u|^p)|u|^{p-2}u \quad \text{on} \ \mathbb R^N, $$
де $a\geq 0$ i $0<\gamma<\inf(N,2+\alpha)$. Глобальне iснування i розсiювання доведенi для вiдносно великого демпфування. Для довiльного демпфування одержано глобальне iснування, коли початковi данi належать до певних iнварiантних множин.
Mathematics Subject Classification: 35Q55
Ключові слова:
демпфоване рiвняння Шокарда, велике демпфування, глобальне iснування, розсiювання, iнварiантнi множиниПосилання
G.D. Akrivis, V.A. Dougalis, O.A. Karakashian, and W.R. Mckinney, Numerical approximation of singular solution of the damped nonlinear Schrödinger equation, In: ENUMATH 97 (Heidelberg), World Scientific River Edge, NJ, 1998, 117–124.
I.V. Barashenkov, N.V. Alexeeva, and E.V. Zemlianaya, Two and three dimensional oscillons in nonlinear Faraday resonance, Phys. Rev. Lett. 89 (2002), 104101. https://doi.org/10.1103/PhysRevLett.89.104101
C. Bonanno, P. d’Avenia, M. Ghimenti, and M. Squassina, Soliton dynamics for the generalized Choquard equation, J. Math. Anal. Appl. 417 (2014), 180–199. https://doi.org/10.1016/j.jmaa.2014.02.063
T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes In Mathematics, 10, Courant Institute of Mathematical Sciences, New York University New York, N.Y., Amer. Math. Soc. Providence, RI, 2003.
J. Chen and B. Guo, Strong instability of standing waves for a nonlocal Schrödinger equation, Physica D 227 (2007), 142–148. https://doi.org/10.1016/j.physd.2007.01.004
M. Darwich and L. Molinet, Some remarks on the nonlinear Schrödinger equation with fractional dissipation, J. Math. Phys. 57 (2015), 101–502. https://doi.org/10.1063/1.4965225
M. Darwich, On the Cauchy problem for the nonlinear Schrödinger equation including fractional dissipation with variable coefficient, Math. Methods Appl. Sci. 41 (2018), 2930–2938. https://doi.org/10.1002/mma.4792
B. Feng and X. Yuan, On the Cauchy problem for the Schrödinger-Hartree equation, Evol. Equ. Control Theory 4 (2015), 431–445. https://doi.org/10.3934/eect.2015.4.431
G. Fibich, Self-focusing in the damped nonlinear Schrödinger equation, SIAM J. Appl. Math. 61(5) (2001), 1680–1705. https://doi.org/10.1137/S0036139999362609
H. Genev and G. Venkov, Soliton and blow-up solutions to the time-dependent Schrödinger Hartree equation, Discrete Contin. Dyn. Syst. Ser. S 5 (2012), 903– 923. https://doi.org/10.3934/dcdss.2012.5.903
M.V. Goldman, K. Rypdal, and B. Hafizi, Dimensionality and dissipation in Langmuir collapse, Phys. Fluids 23 (1980), 945–955. https://doi.org/10.1063/1.862982
E.P. Gross, Dynamics of interacting bosons, In: Physics of Many-Particle Systems (Ed. E. Meeron), 1, Gordon & Breach, New York, NY, 1966, 231–406.
M. Lewin and N. Rougerie, Derivation of Pekar´s polarons from a microscopic model of quantum crystal, SIAM J. Math. Anal. 45 (2013), 1267–1301. https://doi.org/10.1137/110846312
E.H. Lieb and M. Loss, Analysis, 2nd ed., Graduate Studies in Mathematics, 14, Amer. Math. Soc, RI, 2001. https://doi.org/10.1090/gsm/014
P. L. Lions, The Choquard equation and related questions, Nonlinear Anal. 4 (1980), 1063–1073. https://doi.org/10.1016/0362-546X(80)90016-4
P. L. Lions, Symetrie et compacité dans les espaces de Sobolev, J. Funct. Anal. 49 (1982), 315–334. https://doi.org/10.1016/0022-1236(82)90072-6
V. Moroz and J.V. Schaftingen, Ground states of nonlinear Choquard equations: existence, qualitative properties and decay asymptotics, J. Funct. Anal. 265 (2013), 153–184. https://doi.org/10.1016/j.jfa.2013.04.007
M. Ohta and G. Todorova, Remarks on global existence and blowup for damped non-linear Schrödinger equations, Discrete Contin. Dyn. Syst. 23 (2009), 1313–1325. https://doi.org/10.3934/dcds.2009.23.1313
L.E. Payne and D.H. Sattinger, Saddle points and instability of non-linear hyperbolic equations, Israel J. Math. 22 (1976), 273–303. https://doi.org/10.1007/BF02761595
R. Penrose, Quantum computation, entanglement and state reduction, Philos. Trans. Roy. Soc. 356 (1998), 1927–1939. https://doi.org/10.1098/rsta.1998.0256
T. Saanouni, Sharp threshold of global well-posedness vs finite time blow-up for a class of inhomogeneous Choquard equations, J. Math. Phys. 60 (2019), id.081514. https://doi.org/10.1063/1.5111550
T. Saanouni, Scattering threshold for the focusing Choquard equation, Nonlinear Differ. Equ. Appl. 26 (2019), 1–32. https://doi.org/10.1007/s00030-018-0548-0
T. Tao, Non-linear dispersive equations: local and global analysis, CBMS regional series in mathematics, (2006). Available from: https://www.math.ucla.edu/~tao/ preprints/chapter.pdf. https://doi.org/10.1090/cbms/106
T. Saanouni, Remarks on damped fractional Schrödinger equation with pure power nonlinearity, J. Math. Phys. 56 (2015), 061502. https://doi.org/10.1063/1.4922114
T. Saanouni, Damped non-linear coupled Schrödinger equation, Complex Anal. Oper. Theory 13 (2019), 1093–1110. https://doi.org/10.1007/s11785-018-0841-2
M. Tsutsumi, Nonexistence of global solutions to the Cauchy problem for the damped nonlinear Schrödinger equations, SIAM J. Math. Anal. 15 (1984), 357– 366. https://doi.org/10.1137/0515028
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. https://doi.org/10.1016/0022-247X(90)90403-3
