Global existence and long-term behavior of a nonlinear Schrödinger-type equation. (English) Zbl 1373.35291
Ukr. Math. J. 67, No. 1, 74-97 (2015) and Ukr. Mat. Zh. 67, No. 1, 68-87 (2015).
This article is a revision of [“On nonlinear equation of Schrödinger type”, Preprint, arXiv:1208.2560] from the same author. The author considers the Cauchy problem for an NLS equation in a bounded domain \(\Omega\) with Dirichlet boundary conditions. The nonlinearity is of the form \(q|u|^{p-2}u\) with \(q\in W^{-1,p_0}\) and \(p,p_0\geq 2\).
Using an abstract Hahn-Banach based fix point theorem formerly proved by the author, the problem is shown to have a weak solution in \(L^2([0,T];H_0^1(\Omega))\cap L^m([0,T];H_0^1(\Omega))\) with \(m\geq 2^*\). If moreover the source term of the equation is bounded in \(L^2(\mathbb{R}^+;H^1(\Omega))\) then we can take \(T=\mathbb{R}^+\). This requires a smallness assumption on the coefficients.
Under the same smallness assumption the author shows that the \(L^2(\Omega)\) norm of \(\nabla u(t),\;\partial_t u(t)\) and \(u(t)\) is bounded by that of the source term at time \(t\).
Using an abstract Hahn-Banach based fix point theorem formerly proved by the author, the problem is shown to have a weak solution in \(L^2([0,T];H_0^1(\Omega))\cap L^m([0,T];H_0^1(\Omega))\) with \(m\geq 2^*\). If moreover the source term of the equation is bounded in \(L^2(\mathbb{R}^+;H^1(\Omega))\) then we can take \(T=\mathbb{R}^+\). This requires a smallness assumption on the coefficients.
Under the same smallness assumption the author shows that the \(L^2(\Omega)\) norm of \(\nabla u(t),\;\partial_t u(t)\) and \(u(t)\) is bounded by that of the source term at time \(t\).
Reviewer: Vincent Lescarret (Gif-sur-Yvette)
MSC:
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 35B40 | Asymptotic behavior of solutions to PDEs |
Citations:
Zbl 1227.47036References:
| [1] | M. Alves, M. Sepulveda, and O. Vera, “Smoothing properties for the higher-order nonlinear Schrödinger equation with constant coefficients,” J. Nonlin. Anal.: Theory, Methods, Appl., 71, 3-4 (2009). · Zbl 1238.94039 · doi:10.1016/j.na.2008.11.093 |
| [2] | A. Ambrosetti, M. Badiale, and S. Cingolani, “Semiclassical states of nonlinear Schrödinger equations,” Arch. Ration. Mech. Anal., 140, 285-300 (1997). · Zbl 0896.35042 · doi:10.1007/s002050050067 |
| [3] | A. Ambrosetti, A. Malchiodi, and D. Ruiz, “Bound states of nonlinear Schrödinger equations with potentials vanishing at infinity,” J. Anal. Math., 98, 317-348 (2006). · Zbl 1142.35082 · doi:10.1007/BF02790279 |
| [4] | T. Bartsch, A. Pankov, and Z. Q. Wang, “Nonlinear Schrödinger equations with steep potential well,” Comm. Contemp. Math., 3, 549-569 (2001). · Zbl 1076.35037 · doi:10.1142/S0219199701000494 |
| [5] | H. Brezis and A. C. Ponce, “Reduced measures on the boundary,” J. Funct. Anal., 229, No. 1 (2005). · Zbl 1081.31007 |
| [6] | Th. Cazenave, “Semilinear Schrödinger equations,” Courant Lect. Notes Math., 10 (2003). · Zbl 1055.35003 |
| [7] | S. Cingolani, “Semiclassical stationary states of nonlinear Schrödinger equations with an external magnetic field,” J. Different. Equat., 188, 52-79 (2003). · Zbl 1062.81056 · doi:10.1016/S0022-0396(02)00058-X |
| [8] | J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao, “Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in <Emphasis Type=”Italic“>R3,” Ann. Math., 167, No. 3, 767-865 (2008). · Zbl 1178.35345 · doi:10.4007/annals.2008.167.767 |
| [9] | J. Colliander, M. Grillakis, and N. Tzirakis, “Tensor products and correlation estimates with applications to nonlinear Schrödinger equations,” Comm. Pure Appl. Math., 62, No. 7, 920-968 (2009). · Zbl 1185.35250 · doi:10.1002/cpa.20278 |
| [10] | M. del Pino and P. Felmer, “Semiclassical states of nonlinear Schrödinger equations: A variational reduction method,” Math. Ann., 324, 1-32 (2002). · Zbl 1030.35031 · doi:10.1007/s002080200327 |
| [11] | A. Floer and A. Weinstein, “Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential,” J. Funct. Anal., 69, 397-408 (1986). · Zbl 0613.35076 · doi:10.1016/0022-1236(86)90096-0 |
| [12] | B. Gidas, W. M. Ni, and L. Nirenberg, “Symmetry of positive solutions of nonlinear elliptic equations in <Emphasis Type=”Italic“>R <Emphasis Type=”Italic“>N,” Math. Anal. Appl., Part A, Adv. in Math. Suppl. Stud.7, 369-402 (1981). · Zbl 0469.35052 |
| [13] | D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd edn., Springer-Verlag, Berlin-New York, 224 (1983). · Zbl 0562.35001 · doi:10.1007/978-3-642-61798-0 |
| [14] | B. Grebert and L. Thomann, “Resonant dynamics for the quintic nonlinear Schrödinger equation,” Ann. Inst. H. Poincaré. Non Lineare, 29, 455-477 (2012). · Zbl 1259.37045 · doi:10.1016/j.anihpc.2012.01.005 |
| [15] | M. Grossi, “On the number of single-peak solutions of the nonlinear Schrödinger equation,” Ann. Inst. H. Poincaré. Anal. Non Linéaire, 19, No. 3, 261-280 (2002). · Zbl 1034.35127 · doi:10.1016/S0294-1449(01)00089-0 |
| [16] | C. Gui, “Existence of multi-bump solutions for nonlinear Schrödinger equations via variational method,” Comm. Part. Different. Equat., 21, 787-820 (1996). · Zbl 0857.35116 · doi:10.1080/03605309608821208 |
| [17] | N. Hayashi, Ch. Li, and P. I. Naumkin, “Modified wave operator for a system of nonlinear Schrödinger equations in 2d,” Comm. Part. Different. Equat., 37, 947-968 (2012). · Zbl 1244.35135 · doi:10.1080/03605302.2012.668256 |
| [18] | C. Le Bris and P.-L. Lions, “From atoms to crystals: a mathematical journey,” Bull. Amer. Math. Soc. (N. S.), 42, No. 3, 291-363 (2005)(electronic). · Zbl 1136.81371 · doi:10.1090/S0273-0979-05-01059-1 |
| [19] | J. Lenells and A. S. Fokas, “On a novel integrable generalization of the nonlinear Schrodinger equation,” J. Nonlinearity, 22, No. 1, 11-27 (2009). · Zbl 1160.35536 · doi:10.1088/0951-7715/22/1/002 |
| [20] | F. Lin and P. Zhang, “Semiclassical limit of the Gross-Pitaevskii equation in an exterior domain,” Arch. Ration. Mech. Anal., 179, 79-107 (2006). · Zbl 1079.76016 · doi:10.1007/s00205-005-0383-4 |
| [21] | J.-L. Lions and E. Magenes, Nonhomogeneous Boundary-Value Problems and Applications, Springer-Verlag, Berlin, etc., 181 (1972). · Zbl 0227.35001 |
| [22] | V. G. Makhankov and V. K. Fedyanin, “Nonlinear effects in quasi-one-dimensional models of condensed matter theory,” Phys. Rep., 104, 1-86 (1984). · doi:10.1016/0370-1573(84)90106-6 |
| [23] | E. S. Noussair and C. A. Swanson, “Oscillation theory for semilinear Schrödinger equations and inequalities,” Proc. Roy. Soc. Edinburgh A, 75, 67-81 (1975/76). · Zbl 0372.35004 |
| [24] | Y. G. Oh, “Existence of semiclassical bound states of nonlinear Schrödinger equations with potentials of the class (<Emphasis Type=”Italic“>V ) <Emphasis Type=”Italic“>α,” Comm. Part. Different. Equat., 13, 1499-1519 (1988). · Zbl 0702.35228 · doi:10.1080/03605308808820585 |
| [25] | P. H. Rabinowitz, “On a class of nonlinear Schrödinger equations,” Z. Angew. Math. Phys., 43, 270-291 (1992). · Zbl 0763.35087 · doi:10.1007/BF00946631 |
| [26] | P. H. Rabinowitz, “Mixed states for an Allen-Cahn type equation,” Comm. Pure Appl. Math., 56, No. 8, 1078-1134 (2003). · Zbl 1274.35122 · doi:10.1002/cpa.10087 |
| [27] | K. N. Soltanov and M. Akhmedov, “On nonlinear parabolic equation in nondivergent form with implicit degeneration and embedding theorems,” arXiv:1207.7063 (2012), 25 p. · Zbl 0896.35042 |
| [28] | K. N. Soltanov, “On a nonlinear equation with coefficients which are generalized functions,” Novi Sad J. Math., 41, No. 1, 43-52 (2011). · Zbl 1289.35112 |
| [29] | K. N. Soltanov, “On semi-continuous mappings, equations, and inclusions in the Banach space,” Hacettepe J. Math. Statist., 37, No. 1 (2008). · Zbl 1165.47043 |
| [30] | K. N. Soltanov, “Some nonlinear equations of the nonstable filtration type and embedding theorems,” J. Nonlinear Anal.: Theory, Methods, Appl., 65, 2103-2134 (2006). · Zbl 1107.35093 · doi:10.1016/j.na.2005.11.053 |
| [31] | K. N. Soltanov, “Perturbation of the mapping and solvability theorems in the Banach space,” Nonlinear Anal.: Theory, Methods, Appl., 72, No. 1 (2010). · Zbl 1227.47036 |
| [32] | K. N. Soltanov, “On nonlinear equation of Schrödinger type,” arXiv:1208.2560v1 (2012), 16 p. · Zbl 1289.35112 |
| [33] | C. A. Stuart, “Lectures on the orbital stability of standing waves and application to the nonlinear Schrödinger equation,” Milan J. Math., 76, No. 1, 329-399 (2008). · Zbl 1179.37101 · doi:10.1007/s00032-008-0089-9 |
| [34] | X. Wang and B. Zeng, “On concentration of positive bound states of nonlinear Schrödinger equations with competing potential functions,” SIAM J. Math. Anal., 28, 633-655 (1997). · Zbl 0879.35053 · doi:10.1137/S0036141095290240 |
| [35] | J. Wang, J. X. Xu, and F. B. Zhang, “Existence and multiplicity of semiclassical solutions for a Schrodinger equation,” J. Math. Anal. Appl., 357, No. 2 (2009). · Zbl 1173.35695 |
| [36] | H. Yin and P. Zhang, “Bound states of nonlinear Schrödinger equations with potentials tending to zero at infinity,” J. Different. Equat., 247, No. 2, 618-647 (2009). · Zbl 1178.35353 · doi:10.1016/j.jde.2009.03.002 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
