## Existence and uniqueness of minimal blow-up solutions to an inhomogeneous mass critical NLS

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- by Pierre Raphaël and Jeremie Szeftel PDF
- J. Amer. Math. Soc.
**24**(2011), 471-546 Request permission

## Abstract:

We consider the 2-dimensional focusing mass critical NLS with an inhomogeneous nonlinearity: $i\partial _tu+\Delta u+k(x)|u|^{2}u=0$. From a standard argument, there exists a threshold $M_k>0$ such that $H^1$ solutions with $\|u\|_{L^2}<M_k$ are global in time while a finite time blow-up singularity formation may occur for $\|u\|_{L^2}>M_k$. In this paper, we consider the dynamics at threshold $\|u_0\|_{L^2}=M_k$ and give a necessary and sufficient condition on $k$ to ensure the existence of critical mass finite time blow-up elements. Moreover, we give a complete classification in the energy class of the minimal finite time blow-up elements at a nondegenerate point, hence extending the pioneering work by Merle who treated the pseudoconformal invariant case $k\equiv 1$.## References

- Valeria Banica,
*Remarks on the blow-up for the Schrödinger equation with critical mass on a plane domain*, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5)**3**(2004), no. 1, 139–170. MR**2064970** - Banica, V.; Carles, R.; Duyckaerts, T., Minimal blow-up solutions to the mass-critical inhomogeneous focusing NLS equation. To appear in Comm. Partial Diff. Eq.
- Adrien Blanchet, José A. Carrillo, and Nader Masmoudi,
*Infinite time aggregation for the critical Patlak-Keller-Segel model in $\Bbb R^2$*, Comm. Pure Appl. Math.**61**(2008), no. 10, 1449–1481. MR**2436186**, DOI 10.1002/cpa.20225 - Jean Bourgain and W. Wang,
*Construction of blowup solutions for the nonlinear Schrödinger equation with critical nonlinearity*, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)**25**(1997), no. 1-2, 197–215 (1998). Dedicated to Ennio De Giorgi. MR**1655515** - Burq, N.; Gérard, P.; Raphaël, P., Log-log blow-up solutions for the mass critical NLS on a manifold, in preparation.
- N. Burq, P. Gérard, and N. Tzvetkov,
*Two singular dynamics of the nonlinear Schrödinger equation on a plane domain*, Geom. Funct. Anal.**13**(2003), no. 1, 1–19. MR**1978490**, DOI 10.1007/s000390300000 - Shu-Ming Chang, Stephen Gustafson, Kenji Nakanishi, and Tai-Peng Tsai,
*Spectra of linearized operators for NLS solitary waves*, SIAM J. Math. Anal.**39**(2007/08), no. 4, 1070–1111. MR**2368894**, DOI 10.1137/050648389 - Thomas Duyckaerts and Frank Merle,
*Dynamic of threshold solutions for energy-critical NLS*, Geom. Funct. Anal.**18**(2009), no. 6, 1787–1840. MR**2491692**, DOI 10.1007/s00039-009-0707-x - Gadi Fibich, Frank Merle, and Pierre Raphaël,
*Proof of a spectral property related to the singularity formation for the $L^2$ critical nonlinear Schrödinger equation*, Phys. D**220**(2006), no. 1, 1–13. MR**2252148**, DOI 10.1016/j.physd.2006.06.010 - B. Gidas, Wei Ming Ni, and L. Nirenberg,
*Symmetry and related properties via the maximum principle*, Comm. Math. Phys.**68**(1979), no. 3, 209–243. MR**544879** - J. Ginibre and G. Velo,
*On a class of nonlinear Schrödinger equations. I. The Cauchy problem, general case*, J. Functional Analysis**32**(1979), no. 1, 1–32. MR**533218**, DOI 10.1016/0022-1236(79)90076-4 - Carlos E. Kenig and Frank Merle,
*Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case*, Invent. Math.**166**(2006), no. 3, 645–675. MR**2257393**, DOI 10.1007/s00222-006-0011-4 - Rowan Killip, Dong Li, Monica Visan, and Xiaoyi Zhang,
*Characterization of minimal-mass blowup solutions to the focusing mass-critical NLS*, SIAM J. Math. Anal.**41**(2009), no. 1, 219–236. MR**2505858**, DOI 10.1137/080720358 - Joachim Krieger, Enno Lenzmann, and Pierre Raphaël,
*On stability of pseudo-conformal blowup for $L^2$-critical Hartree NLS*, Ann. Henri Poincaré**10**(2009), no. 6, 1159–1205. MR**2557200**, DOI 10.1007/s00023-009-0010-2 - Joachim Krieger, Yvan Martel, and Pierre Raphaël,
*Two-soliton solutions to the three-dimensional gravitational Hartree equation*, Comm. Pure Appl. Math.**62**(2009), no. 11, 1501–1550. MR**2560043**, DOI 10.1002/cpa.20292 - J. Krieger and W. Schlag,
*Non-generic blow-up solutions for the critical focusing NLS in 1-D*, J. Eur. Math. Soc. (JEMS)**11**(2009), no. 1, 1–125. MR**2471133**, DOI 10.4171/JEMS/143 - Man Kam Kwong,
*Uniqueness of positive solutions of $\Delta u-u+u^p=0$ in $\textbf {R}^n$*, Arch. Rational Mech. Anal.**105**(1989), no. 3, 243–266. MR**969899**, DOI 10.1007/BF00251502 - Li, D.; Zhang, X., On the rigidity of solitary waves for the focusing mass-critical NLS in dimensions $d\geq 2$. Preprint, arXiv:0902.0802.
- P.-L. Lions,
*The concentration-compactness principle in the calculus of variations. The locally compact case. I*, Ann. Inst. H. Poincaré Anal. Non Linéaire**1**(1984), no. 2, 109–145 (English, with French summary). MR**778970** - Yvan Martel,
*Asymptotic $N$-soliton-like solutions of the subcritical and critical generalized Korteweg-de Vries equations*, Amer. J. Math.**127**(2005), no. 5, 1103–1140. MR**2170139** - Yvan Martel and Frank Merle,
*Stability of blow-up profile and lower bounds for blow-up rate for the critical generalized KdV equation*, Ann. of Math. (2)**155**(2002), no. 1, 235–280. MR**1888800**, DOI 10.2307/3062156 - Yvan Martel and Frank Merle,
*Nonexistence of blow-up solution with minimal $L^2$-mass for the critical gKdV equation*, Duke Math. J.**115**(2002), no. 2, 385–408. MR**1944576**, DOI 10.1215/S0012-7094-02-11526-9 - Yvan Martel and Frank Merle,
*Multi solitary waves for nonlinear Schrödinger equations*, Ann. Inst. H. Poincaré C Anal. Non Linéaire**23**(2006), no. 6, 849–864 (English, with English and French summaries). MR**2271697**, DOI 10.1016/j.anihpc.2006.01.001 - F. Merle,
*Determination of blow-up solutions with minimal mass for nonlinear Schrödinger equations with critical power*, Duke Math. J.**69**(1993), no. 2, 427–454. MR**1203233**, DOI 10.1215/S0012-7094-93-06919-0 - Frank Merle,
*Construction of solutions with exactly $k$ blow-up points for the Schrödinger equation with critical nonlinearity*, Comm. Math. Phys.**129**(1990), no. 2, 223–240. MR**1048692** - Franck Merle,
*Nonexistence of minimal blow-up solutions of equations $iu_t=-\Delta u-k(x)|u|^{4/N}u$ in $\textbf {R}^N$*, Ann. Inst. H. Poincaré Phys. Théor.**64**(1996), no. 1, 33–85 (English, with English and French summaries). MR**1378233** - Frank Merle and Pierre Raphael,
*The blow-up dynamic and upper bound on the blow-up rate for critical nonlinear Schrödinger equation*, Ann. of Math. (2)**161**(2005), no. 1, 157–222. MR**2150386**, DOI 10.4007/annals.2005.161.157 - F. Merle and P. Raphael,
*Sharp upper bound on the blow-up rate for the critical nonlinear Schrödinger equation*, Geom. Funct. Anal.**13**(2003), no. 3, 591–642. MR**1995801**, DOI 10.1007/s00039-003-0424-9 - Frank Merle and Pierre Raphael,
*On universality of blow-up profile for $L^2$ critical nonlinear Schrödinger equation*, Invent. Math.**156**(2004), no. 3, 565–672. MR**2061329**, DOI 10.1007/s00222-003-0346-z - Frank Merle and Pierre Raphael,
*On a sharp lower bound on the blow-up rate for the $L^2$ critical nonlinear Schrödinger equation*, J. Amer. Math. Soc.**19**(2006), no. 1, 37–90. MR**2169042**, DOI 10.1090/S0894-0347-05-00499-6 - Fabrice Planchon and Pierre Raphaël,
*Existence and stability of the log-log blow-up dynamics for the $L^2$-critical nonlinear Schrödinger equation in a domain*, Ann. Henri Poincaré**8**(2007), no. 6, 1177–1219. MR**2355345**, DOI 10.1007/s00023-007-0332-x - Pierre Raphael,
*Stability of the log-log bound for blow up solutions to the critical non linear Schrödinger equation*, Math. Ann.**331**(2005), no. 3, 577–609. MR**2122541**, DOI 10.1007/s00208-004-0596-0 - Raphaël, P.; Rodnianski, I., Stable blow-up dynamics for the critical corotational wave map and equivariant Yang Mills, submitted (2009).
- Michael I. Weinstein,
*Nonlinear Schrödinger equations and sharp interpolation estimates*, Comm. Math. Phys.**87**(1982/83), no. 4, 567–576. MR**691044** - Michael I. Weinstein,
*Lyapunov stability of ground states of nonlinear dispersive evolution equations*, Comm. Pure Appl. Math.**39**(1986), no. 1, 51–67. MR**820338**, DOI 10.1002/cpa.3160390103

## Additional Information

**Pierre Raphaël**- Affiliation: Institut de Mathématiques de Toulouse, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse, France
- Email: pierre.raphael@math.univ-toulouse.fr
**Jeremie Szeftel**- Affiliation: Département de Mathématiques et Applications, École Normale Supérieure, 45 rue d’Ulm, 75 005 Paris, France
- MR Author ID: 712495
- Email: jeremie.szeftel@ens.fr
- Received by editor(s): January 4, 2010
- Received by editor(s) in revised form: September 2, 2010
- Published electronically: December 2, 2010
- Additional Notes: The first author was supported by the French Agence Nationale de la Recherche, ANR jeunes chercheurs SWAP and by ANR OndeNonLin

The second author was supported by the French Agence Nationale de la Recherche, ANR jeunes chercheurs SWAP - © Copyright 2010 American Mathematical Society
- Journal: J. Amer. Math. Soc.
**24**(2011), 471-546 - MSC (2010): Primary 35B35, 35B44; Secondary 35Q41, 35Q55
- DOI: https://doi.org/10.1090/S0894-0347-2010-00688-1
- MathSciNet review: 2748399