Existence of standing waves for dirac fields with singular nonlinearities

We prove the existence of stationary states for nonlinear Dirac equations of the form (E) $$i\sum\limits_{\mu = 0}^3 {\gamma ^\mu \partial _\mu \psi - M\psi + F\left( {\bar \psi \psi } \right)\psi = 0,} $$ whereM>0 andF is a singular self-interaction. In particular, in the model case whereF(s)=?s ^?α, for some 0<α<1, and for every ω>M, there exists a solution of (E) of the form ψ(t, x)=e ^iωt?(x), wherex ₀=t andx=(x ₁,x ₂,x ₃), such that ? has compact support. IF 0<α<1/3, then ? is of classC ¹. If 1/3<α<1, then ? is continuously differentiable, except on some sphere {|x|=R}, where |??| is infinite.     

A general approach toward the synthesis of C-nucleoside pyrazolo[1,5-a]-1,3,5-triazines and their 3',5'-bisphosphate C-nucleotide analogues as the first reported in vivo stable P2Y(1)-receptor antagonists

In our effort to identify potent purinergic P2Y(1) receptor antagonists as potent platelet aggregation inhibitors with enhanced metabolic stability, we developed an efficient route for the large-scale preparation of 2'-deoxy-C-nucleosides of pyrazolo[1,5-a]-1,3,5-triazine. The key strategic elements of this novel synthetic approach involved the following: (i) the use of a novel activating group, the N-methyl-N-phenylamino group, which was easily generated in high yield by treatment of the pyrazolo[1,5-a]-1,3,5-triazin-4-one (5) with phosphorus oxychloride and dimethylaniline under high pressure, (ii) a regio- and stereospecific palladium-mediated coupling reaction of the readily available unprotected glycal 1,4-anhydro-2-deoxy-D-erythro-pent-1-enitol (4b) and the 8-iodo derivative (16), and (iii) the stereoselective reduction of the ketone group of the furanosyl ring followed by the subsequent displacement of the N-methyl-N-phenylamino group upon treatment with methylamine. The beta configuration at the anomeric C-1' position of the glycal moieties was perfectly retained throughout this conversion. This procedure afforded 8-(2'-deoxy-beta-D-ribofuranosyl)-2-methyl-4-(N-methylamino)pyrazolo[1,5-a]-1,3,5-triazine (21) and 8-(2'-deoxy-beta-D-xylofuranosyl)-2-methyl-4-(N-methylamino)pyrazolo[1,5-a]-1,3,5-triazine (24) with an overall yield of 50% and 39%, respectively. Finally, the conversion of nucleosides 21 and 24 to the pyrazolotriazine C-nucleotides 3',5'-bisphosphate 2 and 3',5'-cyclophosphate 26 is also described herein and represents the first reported nucleotide derivatives within the pyrazolo[1,5-a]-1,3,5-triazine series. Preliminary biological testing has shown that compound 2 strongly inhibits ADP-induced human platelet aggregation and shape change and possesses significant efficacies 30 min after injection in rat, highlighting a strong P2Y(1)-receptor antagonist activity in vitro combined with a prolonged duration of action in vivo.     

Rapidly decaying solutions of the nonlinear Schrödinger equation

We consider global solutions of the nonlinear Schrödinger equation

Finite-time blowup for a complex Ginzburg–Landau equation with linear driving

In this paper, we consider the complex Ginzburg–Landau equation ${u_t = e^{i\theta} [\Delta u + |u|^\alpha u] + \gamma u}$ on ${\mathbb{R}^N}$ , where ${\alpha > 0,\,\gamma \in \mathbb{R}}$ and ${-\pi /2 < \theta < \pi /2}$ . By convexity arguments, we prove that, under certain conditions on ${\alpha,\theta,\gamma}$ , a class of solutions with negative initial energy blows up in finite time.     

Nonparabolic Asymptotic Limits of Solutions of the Heat Equation on $${\mathbb{R}}^N$$

In this paper, we construct solutions u(t,x) of the heat equation on such that has nontrivial limit points in as t → ∞ for certain values of μ > 0 and β > 1/2. We also show the existence of solutions of this type for nonlinear heat equations.      

Continuous dependence for NLS in fractional order spaces

For the nonlinear Schrödinger equation iut+Δu+λα|u|u=0 in RN, local existence of solutions in Hs is well known in the Hs-subcritical and critical cases 0<α?4/(N−2s), where 0<s<min{N/2,1}. However, even though the solution is constructed by a fixed-point technique, continuous dependence in Hs does not follow from the contraction mapping argument. In this paper, we show that the solution depends continuously on the initial value in the sense that the local flow is continuous Hs→Hs. If, in addition, α?1 then the flow is locally Lipschitz.     

More self-similar solutions of the nonlinear Schrödinger equation

We prove the existence of self-similar solutions of the Schr?dinger equation in with the power nonlinearity . Existence is obtained for a range of which differs from, but overlaps with, the range of powers considered in [4]. We also obtain regularity results for the self-similar solutions constructed in [4]. Received May 16, 1997     

Uniform estimates for solutions of nonlinear Klein-Gordon equations

Uniform estimates in $\text{H}{}_0{}^1\text{(Ω)}$ of global solutions to nonlinear Klein-Gordon equations of the form $\text{u}{}_{\mathrm{tt}}\text{? Δu + mu = g(u)}\text{in}\text{Ω, u = 0}\text{in}\text{?Ω}$, where Ω is an open subset of $\text{R}$^N, m > 0, and g satisfies some growth conditions are established.