ASYMPTOTIC AND LIMITING PROFILES OF BLOWUP SOLUTIONS OF THE NONLINEAR SCHRODINGER EQUATION WITH CRITICAL POWER

This paper is a sequel to previous ones 38, 39, 41. We continue the study of the blowup problem for the nonlinear Schrodinger equation with critical power nonlinearity (NSC). We introduce a new idea to prove the existence of a blowup solution in H1(ℝN) without any weight condition and reduce the problem to a kind of variational problem. Our new method refines the previous results concerning the asymptotic and limiting profiles of blowup solutions: For a certain class of initial data, the blowup solution behaves like a finite superposition of zero-energy, H1-bounded, global-in-time solutions of (NSC); these singularities stay in a bounded region in ℝN, and one can see that the so-called shoulder emerges outside these singularities as suggested by some numerical computations (see, e.g., [26]). We investigate the asymptotic behavior of zero-energy, global-in-time solutions of (NSC) and find that such a solution behaves like a “multisoliton.” However, it is not an assemblage of free “particles”; the “solitons” interact with each other. © 1999 John Wiley & Sons, Inc.

[1]  H. Nawa.  ASYMPTOTIC PROFILES OF BLOW-UP SOLUTIONS OF THE NONLINEAR SCHRODINGER EQUATION , 1993 .

[2]  G. Papanicolaou,et al.  The Focusing Singularity of the Nonlinear Schrödinger Equation , 1987 .

[3]  Pierre-Louis Lions,et al.  The concentration-compactness principle in the Calculus of Variations , 1985 .

[4]  Michael I. Weinstein,et al.  On the structure and formation of singularities in solutions to nonlinear dispersive evolution equations , 1986 .

[5]  Takayoshi Ogawa,et al.  Blow-up of H1 solution for the nonlinear Schrödinger equation , 1991 .

[6]  Formation of singularities in solutions of the nonlinear Schrödinger equation , 1991 .

[7]  On blowup for the pseudo‐conformally invariant nonlinear Schrödinger equation II , 1998 .

[8]  M. Kwong.  Uniqueness of positive solutions of Δu−u+up=0 in Rn , 1989 .

[9]  P. Lions.  The concentration-compactness principle in the Calculus of Variations , 1984 .

[10]  Frank Merle,et al.  LOWER BOUNDS FOR THE BLOWUP RATE OF SOLUTIONS OF THE ZAKHAROV EQUATION IN DIMENSION TWO , 1996 .

[11]  U. Niederer,et al.  The maximal kinematical invariance group of the free Schrodinger equation. , 1972 .

[12]  E. Lieb,et al.  Stability of Coulomb Systems with Magnetic Fields , 1997 .

[13]  F. Merle,et al.  Determination of blow-up solutions with minimal mass for nonlinear Schrödinger equations with critical power , 1993 .

[14]  R. Glassey.  On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equations , 1977 .

[15]  J. Ginibre,et al.  On a class of nonlinear Schrödinger equations. I. The Cauchy problem, general case , 1979 .

[16]  Jean Bourgain,et al.  Construction of blowup solutions for the nonlinear Schr ? odinger equation with critical nonlineari , 1997 .

[17]  Pierre-Louis Lions,et al.  Nonlinear scalar field equations, I existence of a ground state , 1983 .

[18]  Takayoshi Ogawa,et al.  Blow-up of $H\sp 1$ solutions for the one-dimensional nonlinear Schrödinger equation with critical power nonlinearity , 1991 .

[19]  Frank Merle,et al.  Construction of solutions with exactly k blow-up points for the Schrödinger equation with critical nonlinearity , 1990 .

[20]  Masayoshi Tsutsumi,et al.  Nonexistence of Global Solutions to the Cauchy Problem for the Damped Nonlinear Schrödinger Equations , 1984 .

[21]  E. Lieb,et al.  Stability of coulomb systems with magnetic fields , 1986 .

[22]  H. Nawa.  Limiting profiles of blow-up solutions of the nonlinear schrödinger equation with critical power nonlinearity , 1997 .

[23]  Hayato Nawa.  Asymptotic profiles of blow-up solutions of the nonlinear Schrodinger equation with critical power nonlinearity , 1994 .

[24]  Hiroshi Matano,et al.  Convergence, asymptotic periodicity, and finite-point blow-up in one-dimensional semilinear heat equations , 1989 .

[25]  Frank Merle,et al.  On uniqueness and continuation properties after blow‐up time of self‐similar solutions of nonlinear schrödinger equation with critical exponent and critical mass , 1992 .

[26]  Elliott H. Lieb,et al.  On the lowest eigenvalue of the Laplacian for the intersection of two domains , 1983 .

[27]  M. Weinstein.  Nonlinear Schrödinger equations and sharp interpolation estimates , 1983 .

[28]  R. Khokhlov,et al.  Reviews of Topical Problems: Self-Focusing and Diffraction of Light in a Nonlinear Medium , 1968 .

[29]  H. Nawa.  “Mass concentration” phenomenon for the nonlinear Schrödinger equation with the critical power nonlinearity II , 1990 .

[30]  Yoshio Tsutsumi,et al.  Rate of L 2 concentration of blow-up solutions for the nonlinear Schro¨dinger equation with critical power , 1990 .

[31]  H. Nawa.  Two Points Blow-up in Solutions of the Nonlinear Schrödinger Equation with Quartic Potential on $$\mathbb{R} $$ , 1998 .

[32]  Jean-Michel Coron,et al.  Convergence of solutions of H-systems or how to blow bubbles , 1985 .

[33]  Thierry Cazenave,et al.  The Cauchy problem for the critical nonlinear Schro¨dinger equation in H s , 1990 .

[34]  Michael Struwe,et al.  A global compactness result for elliptic boundary value problems involving limiting nonlinearities , 1984 .

[35]  Yoshikazu Giga,et al.  Nondegeneracy of blowup for semilinear heat equations , 1989 .

[36]  Elliott H. Lieb,et al.  Minimum action solutions of some vector field equations , 1984 .

[37]  J. Sacks,et al.  The Existence of Minimal Immersions of 2-Spheres , 1981 .

[38]  Takashi Suzuki,et al.  Asymptotic analysis for two-dimensional elliptic eigenvalue problems with exponentially dominated nonlinearities , 1990 .

[39]  Walter A. Strauss,et al.  Existence of solitary waves in higher dimensions , 1977 .

[40]  J. Ginibre,et al.  The global Cauchy problem for the nonlinear Schrodinger equation revisited , 1985 .

[41]  Papanicolaou,et al.  Focusing singularity of the cubic Schrödinger equation. , 1986, Physical review. A, General physics.

[42]  Yoshio Tsutsumi,et al.  L2 concentration of blow-up solutions for the nonlinear Schrödinger equation with critical power nonlinearity , 1990 .

[43]  George Papanicolaou,et al.  Stability of isotropic singularities for the nonlinear schro¨dinger equation , 1991 .

[44]  K. Yajima.  Existence of solutions for Schrödinger evolution equations , 1987 .

[45]  Tosio Kato,et al.  On nonlinear Schrödinger equations , 1987 .

[46]  T. Cazenave,et al.  The structure of solutions to the pseudo-conformally invariant nonlinear Schrödinger equation , 1991, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.