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November  2017, 16(6): 2269-2297. doi: 10.3934/cpaa.2017112

The focusing NLS on exterior domains in three dimensions

Department of Mathematics, University of Iowa, Iowa city, IA 52242, USA

Received  May 2016 Revised  November 2016 Published  July 2017

We consider the Dirichlet problem of the focusing energy subcritical NLS outside a smooth compact strictly convex obstacle in dimension three. The critical space of our problem is $\dot{H}^s$ with $0<s<1$. In this paper, we proved that if the initial data $u_{0}$ satisfy $\Vert u_{0}\Vert _{2}^{1-s}\Vert \nabla u_{0}\Vert _{2}^{s}<\Vert \nabla Q\Vert _{2}^{s}\Vert Q\Vert _{2}^{1-s}$ and $ M(u_{0})^{1-s}E(u_{0})^{s}<M(Q)^{1-s}E(Q)^{s},$ then there exists a unique global solution which scatters in both time directions, where $Q$ denotes the ground state solution in the whole space case.

Citation: Kai Yang. The focusing NLS on exterior domains in three dimensions. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2269-2297. doi: 10.3934/cpaa.2017112
References:
[1]

R. Anton, Global existence for defocusing cubic NLS and Gross--Pitaveskii equations in exterior domains, J. Math. Pures Appl., 89 (2008), 335-354. doi: 10.1016/j.matpur.2007.12.006.

[2]

M. D. BlairH. F. Smith and C. D. Sogge, Strichartz estimates and nonlinear Schrödinger equation on manifolds with boundary, Math. Ann., 354 (2012), 1397-1430. doi: 10.1007/s00208-011-0772-y.

[3]

J. Bourgain, Global well-posedness of defocusing 3D critical NLS in the radial case, JAMS, 12 (1999), 145-171. doi: 10.1090/S0894-0347-99-00283-0.

[4]

H. Brézis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490. doi: 10.2307/2044999.

[5]

T. Cazenave, Semilinear Schrödinger equations Courant Lecture Notes in Mathematics 10 New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003. doi: 10.1090/cln/010.

[6]

J. CollianderM. KeelG. StaffilaniH. Takaoka and T. Tao, Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in $\mathbb{R}^{3}$, Ann. of Math., 167 (2008), 767-865. doi: 10.4007/annals.2008.167.767.

[7]

E. B. Davies, Spectral theory and differential operators Cambridge Studies in Advanced Mathematics42 Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511623721.

[8]

T. Duyckaerts J. Holmer and S. Roudenko, Scattering for the non-radial 3D cubic nonlinear Schrödinger equation, Math. Res. Lett., 15 (2008), 1233-1250. doi: 10.4310/MRL.2008.v15.n6.a13.

[9]

D. FangJ. Xie and T. Cazenave, Scattering for the focusing energy-subcritical nonlinear Schrödinger equation, Sci. China Math., 54 (2011), 2037-2062. doi: 10.1007/s11425-011-4283-9.

[10]

J. Holmer and S. Roudenko, A sharp condition for scattering of the radial 3D cubic nonlinear Schrödinger equation, Communications in Mathematical Physics, 282 (2008), 435-467. doi: 10.1007/s00220-008-0529-y.

[11]

O. Ivanovici, On the Schrödinger equation outside strictly convex obstacles, Anal. PDE, 3 (2010), 261-293. doi: 10.2140/apde.2010.3.261.

[12]

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., 116 (2006), 645-675. doi: 10.1007/s00222-006-0011-4.

[13]

S. Keraani, On the defect of compactness for the Strichartz estimates of the Schrödinger equations, J. Differential Equations, 175 (2001), 353-392. doi: 10.1006/jdeq.2000.3951.

[14]

R. KillipM. Visan and X. Zhang, The focusing cubic NLS on exterior domains in three dimensions, Applied Mathematics Research eXpress, 1 (2016), 146-180. doi: 10.1093/amrx/abv012.

[15]

R. KillipM. Visan and X. Zhang, Riesz transforms outside a convex obstacle, Int. Math. Res. Not., 19 (2016), 5875-5921. doi: 10.1093/imrn/rnv338.

[16]

R. KillipM. Visan and X. Zhang, Quintic NLS in the exterior of a strictly convex obstacle, American Journal of Mathematics, 138 (2016), 1193-1346. doi: 10.1353/ajm.2016.0039.

[17]

M. K. Kwong, Uniqueness of positive solutions of $Δ u-u+u^p=0$ in $\mathbb{R}^n$, Arch. Rational Mech. Anal., 105 (1989), 243-266. doi: 10.1007/BF00251502.

[18]

H. Bahouri and P. Gérard, High frequency approximation of solutions to critical nonlinear wave equations, Amer. J. Math., 121 (1999), 131-175.

[19]

W. A. Strauss, Existence of solitary waves in higher dimensions, Commun. Math. Phys., 55 (1977), 149-162.

[20]

T. Tao and M. Visan, Stability of energy-critical nonlinear Schrödinger equations in high dimensions, Electron. J. Differential Equations 118 (2005), 28pp. (electronic).

[21]

M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Commun. Math. Phys., 87 (1983), 567-576.

show all references

References:
[1]

R. Anton, Global existence for defocusing cubic NLS and Gross--Pitaveskii equations in exterior domains, J. Math. Pures Appl., 89 (2008), 335-354. doi: 10.1016/j.matpur.2007.12.006.

[2]

M. D. BlairH. F. Smith and C. D. Sogge, Strichartz estimates and nonlinear Schrödinger equation on manifolds with boundary, Math. Ann., 354 (2012), 1397-1430. doi: 10.1007/s00208-011-0772-y.

[3]

J. Bourgain, Global well-posedness of defocusing 3D critical NLS in the radial case, JAMS, 12 (1999), 145-171. doi: 10.1090/S0894-0347-99-00283-0.

[4]

H. Brézis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490. doi: 10.2307/2044999.

[5]

T. Cazenave, Semilinear Schrödinger equations Courant Lecture Notes in Mathematics 10 New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003. doi: 10.1090/cln/010.

[6]

J. CollianderM. KeelG. StaffilaniH. Takaoka and T. Tao, Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in $\mathbb{R}^{3}$, Ann. of Math., 167 (2008), 767-865. doi: 10.4007/annals.2008.167.767.

[7]

E. B. Davies, Spectral theory and differential operators Cambridge Studies in Advanced Mathematics42 Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511623721.

[8]

T. Duyckaerts J. Holmer and S. Roudenko, Scattering for the non-radial 3D cubic nonlinear Schrödinger equation, Math. Res. Lett., 15 (2008), 1233-1250. doi: 10.4310/MRL.2008.v15.n6.a13.

[9]

D. FangJ. Xie and T. Cazenave, Scattering for the focusing energy-subcritical nonlinear Schrödinger equation, Sci. China Math., 54 (2011), 2037-2062. doi: 10.1007/s11425-011-4283-9.

[10]

J. Holmer and S. Roudenko, A sharp condition for scattering of the radial 3D cubic nonlinear Schrödinger equation, Communications in Mathematical Physics, 282 (2008), 435-467. doi: 10.1007/s00220-008-0529-y.

[11]

O. Ivanovici, On the Schrödinger equation outside strictly convex obstacles, Anal. PDE, 3 (2010), 261-293. doi: 10.2140/apde.2010.3.261.

[12]

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., 116 (2006), 645-675. doi: 10.1007/s00222-006-0011-4.

[13]

S. Keraani, On the defect of compactness for the Strichartz estimates of the Schrödinger equations, J. Differential Equations, 175 (2001), 353-392. doi: 10.1006/jdeq.2000.3951.

[14]

R. KillipM. Visan and X. Zhang, The focusing cubic NLS on exterior domains in three dimensions, Applied Mathematics Research eXpress, 1 (2016), 146-180. doi: 10.1093/amrx/abv012.

[15]

R. KillipM. Visan and X. Zhang, Riesz transforms outside a convex obstacle, Int. Math. Res. Not., 19 (2016), 5875-5921. doi: 10.1093/imrn/rnv338.

[16]

R. KillipM. Visan and X. Zhang, Quintic NLS in the exterior of a strictly convex obstacle, American Journal of Mathematics, 138 (2016), 1193-1346. doi: 10.1353/ajm.2016.0039.

[17]

M. K. Kwong, Uniqueness of positive solutions of $Δ u-u+u^p=0$ in $\mathbb{R}^n$, Arch. Rational Mech. Anal., 105 (1989), 243-266. doi: 10.1007/BF00251502.

[18]

H. Bahouri and P. Gérard, High frequency approximation of solutions to critical nonlinear wave equations, Amer. J. Math., 121 (1999), 131-175.

[19]

W. A. Strauss, Existence of solitary waves in higher dimensions, Commun. Math. Phys., 55 (1977), 149-162.

[20]

T. Tao and M. Visan, Stability of energy-critical nonlinear Schrödinger equations in high dimensions, Electron. J. Differential Equations 118 (2005), 28pp. (electronic).

[21]

M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Commun. Math. Phys., 87 (1983), 567-576.

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