Global behavior of finite energy solutions to the focusing nonlinear Schrödinger Equation in d dimension 0)$ and energy-subcritical $(s<1).$ For solutions with $ME[u_0]<1$ ($ME[u_0]$ stands for an invariant and conserved quantity in terms of the mass and energy of $u_0$), a sharp threshold for scattering and blowup is given. Namely, if the renormalized gradient $g_u$ of a solution $u$ to NLS is initially less than 1, i.e., $g_u(0)<1,$ then the solution exists globally in time and scatters in $H^1$ (approaches some linear Schr"odinger evolution as $ttopminfty$); if the renormalized gradient $g_u(0)>1,$ then the solution exhibits a blowup behavior, that is, either a finite time blowup occurs, or there is a divergence of $H^1$ norm in infinite time. This work generalizes the results for the 3d cubic NLS obtained in a series of papers by Holmer-Roudenko and Duyckaerts-Holmer-Roudenko with the key ingredients, the concentration compactness and localized variance, developed in the context of the energy-critical NLS and Nonlinear Wave equations by Kenig and Merle. One of the difficulties is fractional powers of nonlinearities which are overcome by considering Besov-Strichartz estimates and various fractional differentiation rules.]]>autGuevara, Cristi DarleythsRoudenko, SvetlanadgcCastillo_Chavez, CarlosdgcJones, DonalddgcMahalov, AlexdgcSuslov, SergeipblArizona State UniversityengPartial requirement for: Ph.D., Arizona State University, 2011Includes bibliographical references (p. 101-108)Field of study: Mathematicsby Cristi Darley Guevarahttps://hdl.handle.net/2286/R.I.902600Doctoral DissertationAcademic thesesviii, 108 p. : ill113131823131630349659149730adminIn CopyrightAll Rights Reserved2011TextMathematicsBlowupConcentration CompactnessDivergenceProfile decompositionScatteringSchrödingerSchrödinger equationGross-Pitaevskii equations