×

zbMATH — the first resource for mathematics

Blow-up of \(H^ 1\) solution for the nonlinear Schrödinger equation. (English) Zbl 0739.35093
This paper considers the blow-up of the solution in \(H^ 1\) for the following nonlinear Schrödinger equation: \[ i{\partial\over\partial t}u+\Delta u=-| u|^{p-1}u,\quad x\in\mathbb{R}^ n,\quad t\geq 0,\quad u(0,x)=u_ 0(x),\quad x\in \mathbb{R}^ n,\quad t=0, (*) \] where \(n\geq 2\) and \(1+4/n\leq p<\min\{(n+2)/(n-2),5\}\). It is proved that if the initial data \(u_ 0\) in \(H^ 1\) are radially symmetric and have negative energy, then the solution of (*) in \(H^ 1\) blows up in finite time. It is not assumed that \(xu_ 0\in L^ 2\), and therefore the result is the generalization of the results of R. T. Glassey [e.g.: J. Math. Phys. 18, 1794-1797 (1977; Zbl 0372.35009)] and M. Tsutsumi [e.g.: with H. Nawa, Funkc. Ekvacioj, Ser. Int. 32, No. 3, 417-428 (1989; Zbl 0703.35018)] for the radially symmetric case.

MSC:
35Q55 NLS equations (nonlinear Schrödinger equations)
35K55 Nonlinear parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Cazenave, T; Weisslar, F.B, The Cauchy problem for the nonlinear Schrödinger equation in H1, Manuscripta math., 61, 477-498, (1988)
[2] {\scT. Cazenave and F. B. Weisslar}, The structure of solutions to the pseudo-conformally invariant nonlinear Schrödinger equation, Proc. Roy Soc. Edinburgh, in press.
[3] Ginibre, J; Velo, G; Ginibre, J; Velo, G, On a class of nonlinear Schrödinger equations. II. scattering theory general case, J. funct. anal., J. funct. anal., 32, 33-71, (1979) · Zbl 0396.35029
[4] Glassey, R.T, On the blowing-up of solutions to the Cauchy problem for the nonlinear Schrödinger equation, J. math. phys., 18, 1794-1797, (1977) · Zbl 0372.35009
[5] Kato, T, On nonlinear Schrödinger equations, Ann. inst. H. Poincaré phys. théor., 46, 113-129, (1987) · Zbl 0632.35038
[6] Kato, T, Nonlinear Schrödinger equations, (), 218-263
[7] Kavian, O, A remark on the blow-up of solutions to the Cauchy problem for nonlinear Schrödinger equations, Trans. amer. math. soc., 299, 193-203, (1987) · Zbl 0638.35043
[8] LeMesurier, B; Papanjcolau, G; Sulem, C; Sulem, P.L, The focusing singularity of the nonlinear Schrödinger equation, (), 159-201
[9] Lin, J.E; Strauss, W.A, Decay and scattering of solutions of a nonlinear Schrödinger equation, J. funct. anal., 30, 245-263, (1978) · Zbl 0395.35070
[10] Merle, F, Limit of the solution of the nonlinear Schrödinger equation at the blow-up time, J. funct. anal., 84, 201-214, (1989) · Zbl 0681.35078
[11] Merle, F, Construction of solutions with exactly k blow-up points for the Schrödinger equation with critical nonlinearity, Comm. math. phys., 129, 223-240, (1990) · Zbl 0707.35021
[12] Merle, F; Tsutsumi, Y, L2 concentration of blow-up solutions for the nonlinear Schrödinger equation with critical power nonlinearity, J. differential equations, 84, 205-214, (1990) · Zbl 0722.35047
[13] {\scH. Nawa}, “Mass concentration” phenomenon for the nonlinear Schrödinger equation with the critical power nonlinearity 2, Kodai Math. J., in press.
[14] Nawa, H; Tsutsumi, M, On blow-up for the pseudo-conformally invariant nonlinear Schrödinger equation, Funk. ekvac., 32, 417-428, (1989) · Zbl 0703.35018
[15] Shatah, J, Unstable ground state of nonlinear Klein-Gordon equations, Trans. amer. math. soc., 290, 701-710, (1985) · Zbl 0617.35072
[16] Strauss, W.A, Existence of solitary waves in higher dimensions, Comm. math. phys., 55, 149-162, (1977) · Zbl 0356.35028
[17] Strauss, W.A, Nonlinear wave equations, () · Zbl 0869.35029
[18] Tsutsumi, M, Nonexistence of global solutions to the Cauchy problem for the damped nonlinear Schrödinger equation, SIAM J. math. anal., 15, 357-366, (1984) · Zbl 0539.35022
[19] Tsutsumi, Y, Rate of L2 concentration of blow-up solutions for the nonlinear Schrödinger equation with the critical power nonlinearity, Nonlinear anal. TMA, 15, 719-724, (1990) · Zbl 0726.35124
[20] Weinstein, M.I, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. math. phys., 87, 567-576, (1983) · Zbl 0527.35023
[21] Weinstein, M.I, On the structure and formulation of singularities in solutions to non-linear dispersive evolution equations, Comm. partial differential equations, 11, 545-565, (1986) · Zbl 0596.35022
[22] Weinstein, M.I, The nonlinear Schrödinger equation—singularity formation, stability and dispertion, Contemp. math., 99, 213-232, (1989)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.