Global and exploding solutions in a model of self-gravitating systems

We study asymptotic properties of solutions to an extension to arbitrary dimensions of the astrophysical model proposed by Chavanis et al. to explain phenomena of gravitational collapse in clouds of self-gravitating particles. In particular, we show that in the two-dimensional case the solutions can be continued to global ones, while in three space dimensions large data of negative energy blow up in a finite time. Relations between isothermal, Streater's energy-transport and the present models are also studied.     

Stochastic interpretation of potential scattering in quantum mechanics

We generalize Shucker's result on the asymptotically-classical behaviour of the quantum system with zero potential in Nelson's stochastic interpretation of quantum mechanics to the case of the one-dimensional potential scattering.     

Coordination compounds of nickel with trithiocyanuric acid. Part IV. Structure of [Ni(pmdien)(ttcH)] (pmdien = N,N,N′,N′,N′′-pentamethyldiethylenetriamine,ttcH3 = trithiocyanuric acid)

Ni^II mixed-ligand complexes of compositions [Ni(pmdien)(ttcH)] (1), [Ni(baphen)₂(ttcH)] · 4H₂O (2), [Ni-(dpa)(ttcH)(H₂O)] (3), [Ni(cyclam)(ttcH)] · 2H₂O (4), [Ni(hexaa)](ttcH) (5) and [Ni(hexab)(ttcH)] · 2H₂O (6), (baphen = 4,7-diphenyl-1,10-phenanthroline, dpa = 2,2-dipyridylamine, cyclam = 1,4,8,11-tetraazacyclotetradecane, hexaa = 1,3,6,9,11,14-hexaazatricyclo[12.2.1.1^6,9]-octadecane, hexab = 1,8-dimethyl-1,3,6,8,10,13-hexaazacyclotetradecane) have been prepared and characterized by means of i.r., u.v.–vis. spectroscopies and magnetochemical measurements. The redox properties of the complexes were studied by cyclic voltammetry. The crystal and molecular structure of [Ni(pmdien)(ttcH)] was determined. The nickel atom is penta-coordinated by three N atoms of pmdien, and by S and N atoms of trithiocyanurate(2–) anion.     

Xanthate complexes of nickel with nitrogen donor ligands. Part V

Nickel(II) xanthate complexes of general formula, [Ni(Rxa)₂(L)] [R = i-Pr, i-Am, xa = OCS^– ₂, L = 4,7-diphenyl-1,10-phenanthroline(baphen), 4,4-dimethyl-2,2-bipyridyl (me₂bpy), 2-9-dimethyl-1,10-phenanthroline(neo) or trans-1,2-diaminocyclohexane (dch)] and [Ni(Rxa)(cyclam)](Rxa) (= 1,4,8,11-tetraazacyclotetradecane) have been synthesized and characterized by elemental analyses, i.r. and electronic spectroscopy, magnetic and conductivity measurements. The complexes with cyclam were studied by cyclic voltammetry. Values of magnetic moments at room temperature lie within the 3.25–3.51 B.M. range and thus indicate the presence of two unpaired electrons, except for the complexes with cyclam, where the anomalous values of 1.76 and 1.74 B.M. were obtained in respect to the nickel(II) complexes. Magnetic susceptibility data for [Ni(i-Prxa)(cyclam)](i-Prxa) were measured over the 79–298 K range and indicate no exchange interaction between paramagnetic nickel(II) centres.     

Partition of energy in strongly damped generalized wave equations

Dissipative perturbations of hyperbolic equations such as u_tt + Bu_t + A²u = 0 with positive operators A, B are considered. The rates of decay and partition of energy theorems are established for solutions of these equations.     

Time decay of solutions of semilinear strongly damped generalized wave equations

We study the asymptotic behaviour in time of the solutions of dissipative perturbations of wave-type equations in ?^N, u_tt + Bu_t + Au + G(u) = 0, with commuting positive operators A, B and a power like non-linearity G(u). First we give some (pseudo) conformal invariants of the linear operator in the equation. This allows us to derive optimal decay rates for the solutions of the linearized problems. We then prove some decay estimates for the non-linear problems using the tools of scattering theory and the aforementioned conformal invariants.     

Blowup of solutions to generalized Keller–Segel model

The existence and nonexistence of global in time solutions is studied for a class of equations generalizing the chemotaxis model of Keller and Segel. These equations involve Lévy diffusion operators and general potential type nonlinear terms.     

Scattering of small solutions to generalized benjamin-bona-mahony equation in several space dimensions

We show that the supremum norm of solutions with small initial data of the generalized Benjamin-Bona-Mahony equation u_t-△u_t=(b,▽u)+u^p(a,▽u)in x?Rⁿ,n≥2, with integer p≥3 , decays to zero like t^-2/3 if n=2 and like t^-1+6, for any δ0, if n≥3, when t tends to infinity. The proofs of these results are based on an analysis of the linear equation u_t-△=(b,▽u)) and the associated oscillatory integral which may have nonisolated stationary points of the phase function.     

Two‐dimensional chemotaxis models with fractional diffusion

In this paper we study the Cauchy problem for the fractional diffusion equation u_t + (?Δ)^α/2u=?·(u?(Δ^?1u)), generalizing the Keller–Segel model of chemotaxis, for the initial data u₀ in critical Besov spaces ?(?²) with r∈[1, ∞], where 1<α<2. Making use of some estimates of the linear dissipative equation in the frame of mixed time–space spaces, the Chemin ‘mono‐norm method,’ Fourier localization technique and the Littlewood–Paley theory, we obtain a local well‐posedness result. We also consider analogous ‘doubly parabolic’ models. Copyright © 2008 John Wiley & Sons, Ltd.     

Nonexistence of solutions of the heat diffusion problem on a punctured disc

This note presents an elementary approach to the nonexistence of solutions of a linear initial-boundary value problem for the heat diffusion equation on a punctured disc.