Aggregative movement and front propagation for bi-stable population models

Front propagation for the aggregation-diffusion-reaction equation is investigated, where f is a bi-stable reaction-term and D(v) is a diffusion coefficient with changing sign, modeling aggregating-diffusing processes. We provide necessary and sufficient conditions for the existence of traveling wave solutions and classify them according to how or if they attain their equilibria at finite times. We also show that the dynamics can exhibit the phenomena of finite speed of propagation and/or finite speed of saturation

Diffusion-aggregation processes with mono-stable reaction terms

This paper analyses front propagation of the equation $u_\tau=[D(u)v_x]_x +f(v) \;\;\; \tau < 0, x \in \mathbb{R}$ where $f$ is a monostable (ie Fisher-type) nonlinear reaction term and $D(v)$ changes its sign once, from positive to negative values,in the interval $v \in[0,1]$ where the process is studied. This model equation accounts for simultaneous diffusive and aggregative behaviors of a population dynamic depending on the population density $v$ at time $\tau$ and position $x$. The existence of infinitely many travelling wave solutions is proven. These fronts are parametrized by their wave speed and monotonically connect the stationary states u = 0 and v = 1. In the degenerate case, i.e. when D(0) and/or D(1) = 0, sharp profiles appear, corresponding to the minimum wave speed. They also have new behaviors, in addition to those already observed in diffusive models, since they can be right compactly supported, left compactly supported, or both. The dynamics can exhibit, respectively, the phenomena of finite speed of propagation, finite speed of saturation, or both

Heteroclinic Orbits in Plane Dynamical Systems

We consider general second order boundary value problems on the whole line of the type u''=h(t, u, u'), u(-∞)=0, u(+∞)=1, for which we provide existence, non-existence, multiplicity results. The solutions we find can be reviewed as heteroclinic orbits in the (u, u') plane dynamical system

Existence and multiplicity of heteroclinic solutions for non-autonomous boundary eigenvalue problem

In this paper we investigate the boundary eigenvalue problem x''-b(c,t,x)x'+g(t,x)=0, x(-∞)=0, x(+∞)=1 depending on the real parameter c. We take the function b continuous and positive and assume that g is bounded and becomes active and positive only when it exceeds a threshold value theta in (0,1). At the point theta we allow g to have a jump. Additional monotonicity properties are required, when needed. Our main discussion deals with the non-autonomous case. In this context we prove the existence of a continuum of values for which this problem is solvable and we estimate the interval of such admissible values. In the autonomous case, we show its solvability for at most one c*. In the special case when b=c+h(x) with h continuous, we also give a non-existence result, for any real c. Our methods combine comparison-type arguments, both for first and second order dynamics, with a shooting technique. Some applications of the obtained results are included

The X-ray spectra of Compton-thick Seyfert 2 galaxies as seen by BeppoSAX

Results from BeppoSAX observations of Compton--thick Seyfert 2 galaxies are summarized and reviewed, and their general properties derived and discussed. In five out of the seven observed sources, the nucleus is directly visible at high X-ray energies, where the photons penetrate absorbers with column densities in the range 1.1-4.3x10^{24} cm-2 (in the other two sources, NGC 1068 and NGC 7674, the nucleus is instead totally obscured at all energies, implying even larger column densities). In most sources there is unambiguous evidence of a reflection component from optically thick, cold matter, while in two (or maybe four) cases there is also evidence of reflection from ionized matter. For the sources with a measured X-ray luminosity, a comparison with the infrared luminosity is made; while in two cases (the Circinus galaxy and NGC 4945) the IR emission appears to be dominated by starburst activity, in the other three sources (NGC 6240, Mrk 3 and TOL 0109-383) it is likely to be dominated by reprocessing of the UV and X-ray photons emitted by an AGN.Comment: Accepted for publication in MNRA