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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.

35Q55 NLS equations (nonlinear Schrödinger equations)
35K55 Nonlinear parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
Full Text: DOI
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