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Asymptotic profiles of blow-up solutions of the nonlinear Schrödinger equation with critical power nonlinearity
Hayato NAWA
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Hayato NAWA
Department of Mathematics Faculty of Science Tokyo institute of Technology
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1994 Volume 46 Issue 4 Pages 557-586
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- Published: 1994 Received: June 29, 1992 Available on J-STAGE: October 20, 2006 Accepted: - Advance online publication: - Revised: -
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Correction information
Date of correction: October 20, 2006 Reason for correction: - Correction: SUBTITLE Details: Wrong : Dedicated to Professor R. Iino on his 70th birthday
Date of correction: October 20, 2006 Reason for correction: - Correction: CITATION Details: Wrong : 1) S.A. Akhmanov, A.P. Sukhorukov and R.V. Khokhlov, Self-focusing and self-trapping of intense light beams in a nonlinear medium, Soviet Phys. JETP, 23 (1966), 1025-1033.
2) S.A. Akhmanov, A.P. Sukhorukov and R.V. Khokhlov, Self-focusing and diffraction of light in a nonlinear medium, Soviet Phys. Uspekhi, 93 (1968), 609-633.
3) H. Berestycki and P.L. Lions, Nonlinear scalar field equations. I, II, Arch. Rational Mech. Anal., 82 (1983), 313-376.
4) H. Brézis and J.M. Coron, Convergence of H-system or how to blow bubbles, Arch. Rational Mech. Anal., 89 (1985), 21-56.
5) H. Brézis and E.H. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 485-489.
6) H. Brézis and E. H. Lieb, Minimum action solutions of some vector field equations, Comm. Math. Phys, 96 (1983), 97-113.
7) T. Cazenave and F.B. Weissler, The structure of solution to the pseudoconformaly invariant nonlinear Schrödinger equations, Preprint.
8) X.-Y. Chen and H. Matano, Convergence, asymptotic periodicity, and finite-point blow-up in one-dimensional semilinear heat equations, J. Differential Equations, 78 (1989), 160-190.
9) J. Fröhlich, E.H. Lieb and M. Loss, Stability of coulomb systems with magnetic fields I, The one-electron atom, Comm. Math. Phys., 104 (1986), 251-270.
10) Y. Giga and R.V. Kohn, Nondegeneracy of blowup for semilinear heat equations, Comm. Pure Appl. Math., 17 (1989), 845-884.
11) J. Ginibre and G. Velo, On a class of nonlinear Schrodinger equations, I, II, J. Funct. Anal., 32 (1979), 1-71.
12) J. Ginibre and G. Velo, The global Cauchy problem for the nonlinear Schrödinger equation revisited, Ann. Inst. H. Poincaré Anal. Non Linéaire, 2 (1985), 309-327.
13) R.T. Glassey, On the blowing up solution to the Cauchy problem for nonlinear Schrödinger equations, J. Math. Phys, 18 (1979), 1794-1797.
14) T. Kato, On nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Phys. Théor., 46 (1987), 113-129.
15) B. LeMesurier, C. Papanicolau, C. Sulem and P.L. Sulem, The focusing singularity of the nonlinear Schrodinger equation, Direction in Partial Differential Equations, (eds. M.G. Crandall, P.H. Rabinswitz and R.E. Turner), Academic Press, New York, 1987, pp. 159-201.
16) E.H. Lieb, On the lowest eigenvalue of Laplacian for the intersection of two domains, Invent. Math., 74 (1983), 441-448.
17) P.L. Lions, The concentration compactness principle in the calculus of variations. The locally compact case. I, II, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 109-145, 223-283.
18) P.L. Lions, The concentration compactness principle in the calculus of variations. The limit case. I, II., Rev. Mat. Iberoamericana, 1 (1985), 45-121, 145-201.
19) F. Merle, Limit of the solution of the nonlinear Schrodinger equation at the blow-up time, J. Funct. Anal., 84 (1989), 201-214.
20) F. Merle, Construction of solutions with exactly k blow-up points for the Schrödinger equation with the critical power nonlinearity, Comm. Math. Phys, 129 (1990), 223-240.
21) F. Merle and Y. Tsutsumi, L2 concentration of blow-up solutions for the nonlinear Schrödinger equation with the critical power nonlinearity, J. Differential Equations, 84 (1990), 205-214.
22) K. Nagasaki and T. Suzuki, Asymptotic analysis for two-dimensional ellipticeigenvalue problems with exponentially-dominated nonlinearities, Asymptotic Analysis, 3 (1990), 173-188.
23) H. Nawa, Blow-up solutions for the nonlinear Schrödinger equations, unpublished.
24) H. Nawa, “Mass concentration” phenomenon for the nonlinear Schrödinger equation with the critical power nonlinearity, Funkcial. Ekvac., 35 (1992), 1-18.
25) H. Nawa, “Mass concentration” phenomenon for the nonlinear Schrödinger equation with the critical power nonlinearity. II, Kodai Math. J., 13 (1990), 333-348.
26) H. Nawa, Convergence theorems for the pseudo-conformally invariant nonlinear Schrödinger equation, Proc. Japan Acad., 66(A) (1990), 214-216; RIMS Kokuroku, 755 (1991), 73-92.
27) H. Nawa, Formation of singularities in solutions of the nonlinear Schrödinger equation, Proc. Japan Acad., 67(A) (1991), 29-34.
28) H. Nawa, Asymptotic profiles of blow-up solutions of the nonlinear Schrödinger equation, Singularities in fluids, plasmas and optics (R.E. Caflisch), NATO ASI series (to appear), Kluwer, Dordrecht, 1993; its revised manuscript, T.I.T, preprint series, 1993 (to appear).
29) H. Nawa and M. Tsutsumi, On blow-up for the pseudo-conformally invariant nonlinear Schrödinger equation, Funkcial. Ekvac., 32 (1989), 417-428.
30) T. Ogawa and Y. Tsutsumi, Blow-up of H1-solution for the nonlinear Schrödinger equation, J. Differential Equations, 92 (1991), 317-330.
31) T. Ogawa and Y. Tsutsumi, Blow-up of H1-solution for the one dimensional nonlinear Schrödinger equation with critical power nonlinearity, Proc. Amer. Math. Soc., 111 (1991), 487-496.
32) W.A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162.
33) M. Struwe, A global compactness result for elliptic boundary value problem involving limiting nonlinearities, Math. Z., 187 (1983), 567-576.
34) S. Takakuwa, Behavior of minimizing sequences for the Yamabe problem, Tokyo Metropolitan University Mathematics Preprint Series, 7 (1990).
35) Y. Tsutsumi, Rate of L2 concentration of blow-up solutions for the nonlinear Schrödinger equation with critical power nonlinearity, Nonlinear Anal. T.M.A., 15 (1990), 719-724.
36) M.I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys., 87 (1983), 511-517.
37) M.I. Weinstein, On the structure and formation singularitres in solutions to nonlinear dispersive evolution equations, Comm. Partial Differential Equations, 11 (1986), 545-565.
38) M.I. Weinstein, The nonlinear Schrödinger equation-Singularity formation, Stability and dispersion, The Connection between infinite and Finite Dimensional Dynamical Systems, Contemp. Math., 99, 1989, pp. 213-232.
39) K. Yajima, Existence of solutions for Schrödinger evolution equations, Comm. Math. Phys., 110 (1987), 415-426.
40) V.E. Zakharov and V.S. Synakh, The nature of self-focusing singularity, Soviet Phys. JETP, 41 (1976), 441-448.
Right : [1] S. A. Akhmanov, A. P. Sukhorukov and R. V. Khokhlov, Self-focusing and self-trapping of intense light beams in a nonlinear medium, Soviet Phys. JETP, 23 (1966), 1025-1033.
[2] S. A. Akhmanov, A. P. Sukhorukov and R. V. Khokhlov, Self-focusing and diffraction of light in a nonlinear medium, Soviet Phys. Uspekhi, 93 (1968), 609-633.
[3] H. Berestycki and P. L. Lions, Nonlinear scalar field equations. I, II, Arch. Rational Mech. Anal., 82 (1983), 313-376.
[4] H. Brézis and J. M. Coron, Convergence of H-system or how to blow bubbles, Arch. Rational Mech. Anal., 89 (1985), 21-56.
[5] H. Brézis and E.H. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 485-489.
[6] H. Brézis and E. H. Lieb, Minimum action solutions of some vector field equations, Comm. Math. Phys, 96 (1983), 97-113.
[7] T. Cazenave and F.B. Weissler, The structure of solution to the pseudoconformaly invariant nonlinear Schrödinger equations, Preprint.
[8] X. -Y. Chen and H. Matano, Convergence, asymptotic periodicity, and finite-point blow-up in one-dimensional semilinear heat equations, J. Differential Equations, 78 (1989), 160-190.
[9] J. Fröhlich, E. H. Lieb and M. Loss, Stability of coulomb systems with magnetic fields I, The one-electron atom, Comm. Math. Phys., 104 (1986), 251-270.
[10] Y. Giga and R. V. Kohn, Nondegeneracy of blowup for semilinear heat equations, Comm. Pure Appl. Math., 17 (1989), 845-884.
[11] J. Ginibre and G. Velo, On a class of nonlinear Schrodinger equations, I, II, J. Funct. Anal., 32 (1979), 1-71.
[12] J. Ginibre and G. Velo, The global Cauchy problem for the nonlinear Schrödinger equation revisited, Ann. Inst. H. Poincaré Anal. Non Linéaire, 2 (1985), 309-327.
[13] R. T. Glassey, On the blowing up solution to the Cauchy problem for nonlinear Schrödinger equations, J. Math. Phys, 18 (1979), 1794-1797.
[14] T. Kato, On nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Phys. Théor., 46 (1987), 113-129.
[15] B. LeMesurier, C. Papanicolau, C. Sulem and P. L. Sulem, The focusing singularity of the nonlinear Schrodinger equation, Direction in Partial Differential Equations, (eds. M. G. Crandall, P.H. Rabinswitz and R. E. Turner), Academic Press, New York, 1987, pp. 159-201.
[16] E. H. Lieb, On the lowest eigenvalue of Laplacian for the intersection of two domains, Invent. Math., 74 (1983), 441-448.
[17] P. L. Lions, The concentration compactness principle in the calculus of variations. The locally compact case. I, II, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 109-145, 223-283.
[18] P. L. Lions, The concentration compactness principle in the calculus of variations. The limit case. I, II., Rev. Mat. Iberoamericana, 1 (1985), 45-121, 145-201.
[19] F. Merle, Limit of the solution of the nonlinear Schrodinger equation at the blow-up time, J. Funct. Anal., 84 (1989), 201-214.
[20] F. Merle, Construction of solutions with exactly k blow-up points for the Schrödinger equation with the critical power nonlinearity, Comm. Math. Phys, 129 (1990), 223-240.
[21] F. Merle and Y. Tsutsumi, L2 concentration of blow-up solutions for the nonlinear Schrödinger equation with the critical power nonlinearity, J. Differential Equations, 84 (1990), 205-214.
[22] K. Nagasaki and T. Suzuki, Asymptotic analysis for two-dimensional ellipticeigenvalue problems with exponentially-dominated nonlinearities, Asymptotic Analysis, 3 (1990), 173-188.
[23] H. Nawa, Blow-up solutions for the nonlinear Schrödinger equations, unpublished.
[24] H. Nawa, “Mass concentration” phenomenon for the nonlinear Schrödinger equation with the critical power nonlinearity, Funkcial. Ekvac., 35 (1992), 1-18.
[25] H. Nawa, “Mass concentration” phenomenon for the nonlinear Schrödinger equation with the critical power nonlinearity. II, Kodai Math. J., 13 (1990), 333-348.
[26] H. Nawa, Convergence theorems for the pseudo-conformally invariant nonlinear Schrödinger equation, Proc. Japan Acad., 66 (A) (1990), 214-216; RIMS Kokuroku, 755 (1991), 73-92.
[27] H. Nawa, Formation of singularities in solutions of the nonlinear Schrödinger equation, Proc. Japan Acad., 67(A) (1991), 29-34.
[28] H. Nawa, Asymptotic profiles of blow-up solutions of the nonlinear Schrödinger equation, Singularities in fluids, plasmas and optics (R. E. Caflisch), NATO ASI series (to appear), Kluwer, Dordrecht, 1993; its revised manuscript, T. I. T, preprint series, 1993 (to appear).
[29] H. Nawa and M. Tsutsumi, On blow-up for the pseudo-conformally invariant nonlinear Schrödinger equation, Funkcial. Ekvac., 32 (1989), 417-428.
[30] T. Ogawa and Y. Tsutsumi, Blow-up of H1-solution for the nonlinear Schrödinger equation, J. Differential Equations, 92 (1991), 317-330.
[31] T. Ogawa and Y. Tsutsumi, Blow-up of H1-solution for the one dimensional nonlinear Schrödinger equation with critical power nonlinearity, Proc. Amer. Math. Soc., 111 (1991), 487-496.
[32] W. A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162.
[33] M. Struwe, A global compactness result for elliptic boundary value problem involving limiting nonlinearities, Math. Z., 187 (1983), 567-576.
[34] S. Takakuwa, Behavior of minimizing sequences for the Yamabe problem, Tokyo Metropolitan University Mathematics Preprint Series, 7 (1990).
[35] Y. Tsutsumi, Rate of L2 concentration of blow-up solutions for the nonlinear Schrödinger equation with critical power nonlinearity, Nonlinear Anal. T. M. A., 15 (1990), 719-724.
[36] M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys., 87 (1983), 511-517.
[37] M. I. Weinstein, On the structure and formation singularitres in solutions to nonlinear dispersive evolution equations, Comm. Partial Differential Equations, 11 (1986), 545-565.
[38] M. I. Weinstein, The nonlinear Schrödinger equation-Singularity formation, Stability and dispersion, The Connection between infinite and Finite Dimensional Dynamical Systems, Contemp. Math., 99, 1989, pp. 213-232.
[39] K. Yajima, Existence of solutions for Schrödinger evolution equations, Comm. Math. Phys., 110 (1987), 415-426.
[40] V. E. Zakharov and V. S. Synakh, The nature of self-focusing singularity, Soviet Phys. JETP, 41 (1976), 441-448.
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