We consider the blow-up results of the solution in W-2,W-2(R-N) for the following quasilinear Schrodinger equation {iu(t) + Delta u + 2uh '(|u|(2))Delta h(|u|(2)) + uf(|u|(2)) = 0, x is an element of R-N, u(x, 0) = u(0)(x), x is an element of R-N, where h and f are real functions which related to various physical models. We prove that the W-2,W-2(R-N) solutions must blow up if |x|u(0) is an element of L-2(R-N)(finite variance), and we give the upper bound of the blow-up time. We also show that without the finite variance assumption, the radial symmetric solutions in W-2,W-2(R-N) must blow up in finite time for the whole class of initial data with strictly negative energy.
We develop a scalar interface between numerical detection and a posteriori certification of T-periodic solutions in time-periodic ODEs. Starting from the time-T map, we introduce the periodicity degree, a bounded indicator of one-period recurrence, and establish its basic analytical properties on compact sets. The periodicity degree serves as an optimizationfriendly detector of near-fixed points of the time-T map. To convert such numerical evidence into a mathematically valid existence statement, we provide a contraction-based a posteriori criterion that upgrades a detected near-fixed point to the existence of an exact fixed point with an explicit error bound. Numerical illustrations for two forced benchmarks clarify the distinction between detection and certification, while a Lorenz double-scroll example shows how recurrence-guided lag selection and local Poincare/Newton refinement can be combined to obtain and interpret a high-accuracy periodic-orbit candidate in a chaotic regime.
