论文信息 - On uniqueness and continuation properties after blow‐up time of self‐similar solutions of nonlinear schrödinger equation with critical exponent and critical mass

On uniqueness and continuation properties after blow‐up time of self‐similar solutions of nonlinear schrödinger equation with critical exponent and critical mass

We consider the nonlinear Schrodinger equation with critical power where u: (0, T) × ℝN C and o ϕ H1 ∪ {o;|x|o ϵ L2}. With this nonlinear term, the equation (1)-(1′) has a conformal invariance. Thus one yields explicit “self-similar” solutions which have the following property: they have minimum L2 norm among blow-up solutions. In this paper, we first focus on uniqueness properties of these self-similar solutions in a certain class. We then look at the possible continuations of these solutions after the blow-up time.

Frank Merle | F. Merle

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

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

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

[4] P. Lions,et al. Orbital stability of standing waves for some nonlinear Schrödinger equations , 1982 .

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

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

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

[8] Thierry Cazenave,et al. The Cauchy problem for the nonlinear Schrödinger equation in H1 , 1988 .

[9] Limit of the solution of a nonlinear Schrödinger equation at blow-up time , 1989 .

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

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