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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

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

[19]

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

[20]

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

[21]

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

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

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

[19]

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

[20]

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

[21]

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

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