A complete classification of simultaneous blow-up rates

We study the simultaneous blow-up rates of a system of two heat equations coupled through the boundary in a nonlinear way. We complete the previous known results by covering the whole range of possible parameters.Fil: Brändle, Cristina. Universidad Autónoma de Madrid; EspañaFil: Quirós, Fernando. Universidad Autónoma de Madrid; EspañaFil: Rossi, Julio Daniel. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentin

Blowup solutions and their blowup rates for parabolic equations with non-standard growth conditions

This paper concerns classical solutions for homogeneous Dirichlet problem of parabolic equations coupled via exponential sources involving variable exponents. We first establish blowup criteria for positive solutions. And then, for radial solutions, we obtain optimal classification for simultaneous and non-simultaneous blowup, which is represented by using the maxima of the involved variable exponents. At last, all kinds of blowup rates are determined for both simultaneous and non-simultaneous blowup solutions

Transient Information Flow in a Network of Excitatory and Inhibitory Model Neurons: Role of Noise and Signal Autocorrelation

We investigate the performance of sparsely-connected networks of integrate-and-fire neurons for ultra-short term information processing. We exploit the fact that the population activity of networks with balanced excitation and inhibition can switch from an oscillatory firing regime to a state of asynchronous irregular firing or quiescence depending on the rate of external background spikes. We find that in terms of information buffering the network performs best for a moderate, non-zero, amount of noise. Analogous to the phenomenon of stochastic resonance the performance decreases for higher and lower noise levels. The optimal amount of noise corresponds to the transition zone between a quiescent state and a regime of stochastic dynamics. This provides a potential explanation on the role of non-oscillatory population activity in a simplified model of cortical micro-circuits.Comment: 27 pages, 7 figures, to appear in J. Physiology (Paris) Vol. 9

Lipschitz geometry of complex surfaces: analytic invariants and equisingularity

We prove that the outer Lipschitz geometry of a germ $(X,0)$ of a normal complex surface singularity determines a large amount of its analytic structure. In particular, it follows that any analytic family of normal surface singularities with constant Lipschitz geometry is Zariski equisingular. We also prove a strong converse for families of normal complex hypersurface singularities in $\mathbb C^3$: Zariski equisingularity implies Lipschitz triviality. So for such a family Lipschitz triviality, constant Lipschitz geometry and Zariski equisingularity are equivalent to each other.Comment: Added a new section 10 to correct a minor gap and simplify some argument

The thick-thin decomposition and the bilipschitz classification of normal surface singularities

We describe a natural decomposition of a normal complex surface singularity $(X,0)$ into its "thick" and "thin" parts. The former is essentially metrically conical, while the latter shrinks rapidly in thickness as it approaches the origin. The thin part is empty if and only if the singularity is metrically conical; the link of the singularity is then Seifert fibered. In general the thin part will not be empty, in which case it always carries essential topology. Our decomposition has some analogy with the Margulis thick-thin decomposition for a negatively curved manifold. However, the geometric behavior is very different; for example, often most of the topology of a normal surface singularity is concentrated in the thin parts. By refining the thick-thin decomposition, we then give a complete description of the intrinsic bilipschitz geometry of $(X,0)$ in terms of its topology and a finite list of numerical bilipschitz invariants.Comment: Minor corrections. To appear in Acta Mathematic

A two-parameter criterion for classifying the explodability of massive stars by the neutrino-driven mechanism

Thus far, judging the fate of a massive star (either a neutron star (NS) or a black hole) solely by its structure prior to core collapse has been ambiguous. Our work and previous attempts find a non-monotonic variation of successful and failed supernovae with zero-age main-sequence mass, for which no single structural parameter can serve as a good predictive measure. However, we identify two parameters computed from the pre-collapse structure of the progenitor, which in combination allow for a clear separation of exploding and non-exploding cases with only few exceptions (~1-2.5%) in our set of 621 investigated stellar models. One parameter is M4, defining the normalized enclosed mass for a dimensionless entropy per nucleon of s=4, and the other is mu4 = d(m/M_sun)/d(r/1000 km) at s=4, being the normalized mass-derivative at this location. The two parameters mu4 and M4*mu4 can be directly linked to the mass-infall rate, Mdot, of the collapsing star and the electron-type neutrino luminosity of the accreting proto-NS, L_nue ~ M_ns*Mdot, which play a crucial role in the "critical luminosity" concept for the theoretical description of neutrino-driven explosions as runaway phenomenon of the stalled accretion shock. All models were evolved employing the approach of Ugliano et al. for simulating neutrino-driven explosions in spherical symmetry. The neutrino emission of the accretion layer is approximated by a gray transport solver, while the uncertain neutrino emission of the 1.1 M_sun proto-NS core is parametrized by an analytic model. The free parameters connected to the core-boundary prescription are calibrated to reproduce the observables of Supernova 1987A for five different progenitor models.Comment: 23 pages, 12 figures; accepted by ApJ; revised version considerably enlarged (Fig. 7 and Sect.3.6 added

Non-simultaneous quenching in a system of heat equations coupled at the boundary

We study the solutions of a parabolic system of heat equations coupled at the boundary through a nonlinear flux. We characterize in terms of the parameters involved when nonsimultaneous quenching may appear. Moreover, if quenching is non-simultaneous we find the quenching rate, which surprisingly depends on the flux associated to the other component.Fil: Ferreira, Raúl. Universidad Carlos III de Madrid; EspañaFil: de Pablo, Arturo. Universidad Carlos III de Madrid; EspañaFil: Quirós, Fernando. Universidad Autónoma de Madrid; EspañaFil: Rossi, Julio Daniel. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentin