2009/11/20 by Justin Holmer, Holmer, Justin, Rodrigo Platte +3
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP
paper · pdf · doi:10.48550/arxiv.0911.3955
arxiv created 2009/11/20 · arxiv updated 2009/12/08
We consider solutions u to the 3d nonlinear Schrödinger equation i∂t u + Δu + |u|2u=0. In particular, we are interested in finding criteria on the initial data u0 that predict the asymptotic behavior of u(t), e.g., whether u(t) blows-up in finite time, exists globally in time but behaves like a linear solution for large times (scatters), or exists globally in time but does not scatter. This question has been resolved (at least for H1 data) if M[u]E[u]≤ M[Q]E[Q], where M[u] and E[u] denote the mass and energy of u, and Q denotes the ground state solution to -Q+ΔQ +|Q|2Q=0. Here, we prove a new sufficient condition for blow-up using an interpolation type inequality and the virial identity that is applicable to certain initial data satisfying M[u]E[u]>M[Q]E[Q]. Our condition is similar to one obtained by Lushnikov (1995) but our method allows for an adaptation to radial, infinite-variance initial data that can be stated conceptually: for real initial data, if a certain fraction of the mass is contained in the unit ball, then blow-up occurs. We also show analytically (if one takes the numerically computed value of ‖Q‖ H1/2) that there exist Gaussian initial data u0 with negative quadratic phase such that ‖u0‖ H1/2 < ‖Q‖ H1/2 but the solution u(t) blows-up. We conclude with several numerically computed examples.