We study the nonlinear Schrödinger equation with an inversesquare potential in dimensions 3 ≤ d ≤ 6. We consider both focusing and defocusing nonlinearities in the mass-supercritical and energy-subcritical regime. In the focusing case, we prove a scattering/blowup dichotomy below the ground state. In the defocusing case, we prove scattering in H 1 for arbitrary data.
In this paper, we consider the Cauchy problem for the generalized Davey-Stewartson systemWe first use a variational approach to give a dichotomy of blow-up and scattering for the solution of mass supercritical equation with the initial data satisfying J(u 0 ) < J(R), where J stands for the Lagrange functional. The basic strategy is the concentration-compactness arguments from Kenig and Merle [17]. We overcome the main difficulties coming from the lack of scaling invariance and the asymmetrical structure of nonlinearity (in particular, the nonlinearity is non-local). Furthermore, we adapt the standard method from [9] to obtain the blow up criterion.
To explore the effects
of end groups on the confined crystallization
of an alkyl chain, 3-pentadecylphenol (PDP) was infiltrated into the
anodic aluminum oxide template (AAO) to investigate the melting and
crystallization behaviors of PDP in a nanoconfined environment. Wide-angle
X-ray diffraction (WAXD) found that the solid–solid phase transition
of PDP occurred under confined conditions, and the absence of the
(00L) reflections indicated that the stacking of the end groups of
the alkyl chain layered structure was seriously disturbed. Thermal
analysis (TG) showed that the thermal stability of the confined samples
decreased due to the confinement effect, and the introduction of end
groups made the confinement effect more obvious. Differential scanning
calorimeter (DSC) results well reflected the space–time equivalence
in the PDP crystallization processes, i.e., the solid–solid
phase transition can be achieved by reducing the cooling rate or confining
PDP in the nanometer space. Compared with C15, the introduction
of the end groups with a phenol ring led to the disappearance of the
solid–solid phase transition of an alkyl chain at high cooling
rates. In the confined environment, the introduction of the end groups
with a phenol ring caused the melting double peaks of the alkyl chain
to become a single melting peak, and it also caused the disappearance
of the surface freezing monolayer for alkyl chains. Through the analysis
of crystallinity, it was found that AAO-PDP was more sensitive to
AAO pore size changes than AAO-C15, the X
c of AAO-PDP had a good linear relationship with the pore
size d, but the X
c of
the AAO-C15 had a nonlinear relationship with the pore
size d. Attenuated total reflection (ATR)-IR proved
that in the confined environment, the order of the alkyl chain decreased
and the degree of chain distortion increased.
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