On the stability of critical chemotactic aggregation

We consider the two dimensional parabolic-elliptic Patlak-Keller-Segel model of chemotactic aggregation for radially symmetric initial data. We show the existence of a stable mechanism of singularity formation and obtain a complete description of the associated aggregation process.Comment: 80 page

Uniform Decay of Local Energy and the Semi-Linear Wave Equation on Schwarzchild Space

We provide a uniform decay estimate of Morawetz type for the local energy of general solutions to the inhomogeneous wave equation on a Schwarzchild background. This estimate is both uniform in space and time, so in particular it implies a uniform bound on the sup norm of solutions which can be given in terms of certain inverse powers of the radial and advanced/retarded time coordinate variables. As a model application, we show these estimates give a very simple proof small amplitude scattering for nonlinear scalar fields with higher than cubic interactions.Comment: 24 page

Morrey spaces and classification of global solutions for a supercritical semilinear heat equation in RnR^n

We prove the boundedness of global classical solutions for the semilinear heat equation $u_t-\Delta u= |u|^{p-1}u$ in the whole space $R^n$, with $n\ge 3$ and supercritical power $p>(n+2)/(n-2)$. This is proved {\rmb without any radial symmetry or sign assumptions}, unlike in all the previously known results for the Cauchy problem, and under spatial decay assumptions on the initial data that are essentially optimal in view of the known counter-examples. Moreover, we show that any global classical solution has to decay in time faster than $t^{-1/(p-1)}$, which is also optimal and in contrast with the subcritical case. The proof relies on nontrivial modifications of techniques developed by Chou, Du and Zheng [Calc. Var. PDE 2007] and by Blatt and Struwe [IMRN, 2015] for the case of convex bounded domains. They are based on weighted energy estimates of Giga-Kohn type, combined with an analysis of the equation in a suitable Morrey space. We in particular simplify the approach of Blatt and Struwe by establishing and using a result on global existence and decay for small initial data in the critical Morrey space $M^{2,4/(p-1)}(R^n)$, rather than \eps-regularity in a parabolic Morrey space. This method actually works for any convex, bounded or unbounded, smooth domain, but at the same time captures some of the specific behaviors associated with the case of the whole space $R^n$. As a consequence we also prove that the set of initial data producing global solutions is open in suitable topologies, and we show that the so-called "borderline" global weak solutions blow up in finite time and then become classical again and decay as $t\to\infty$. All these results put into light the key role played by the Morrey space $M^{2,4/(p-1)}$ in the understanding of the structure of the set of global solutions for $p>p_S$.Comment: 29 page

Inhomogeneities in 3 dimensional oscillatory media

We consider localized perturbations to spatially homogeneous oscillations in dimension 3 using the complex Ginzburg-Landau equation as a prototype. In particular, we will focus on heterogeneities that locally change the phase of the oscillations. In the usual translation invariant spaces and at $\varepsilon=0$ the linearization about these spatially homogeneous solutions result in an operator with zero eigenvalue embedded in the essential spectrum. In contrast, we show that when considered as an operator between Kondratiev spaces, the linearization is a Fredholm operator. These spaces consist of functions with algebraical localization that increases with each derivative. We use this result to construct solutions close to the equilibrium via the Implicit Function Theorem and derive asymptotics for wavenumbers in the far field.Comment: 3 figures, 15 pages. More accurate numerical results. Added a figure illustrating the decay of Amplitude of solution

Optical and electrical activity of defects in rare earth implanted Si

A common technique for introducing rare earth atoms into Si and related materials for photonic applications is ion implantation. It is compatible with standard Si processing, and also allows high, non-equilibrium concentrations of rare earths to be introduced. However, the high energies often employed mean that there are collision cascades and potentially severe end-of-range damage. This paper reports on studies of this damage, and the competition it may present to the optical activity of the rare earths. Er-, Si, and Yb-implanted Si samples have been investigated, before and after anneals designed to restore the sample crystallinity. The electrical activity of defects in as-implanted Er, Si, and Yb doped Si has been studied by Deep Level Transient Spectroscopy (DTLS) and the related, high resolution technique, Laplace DLTS (LDLTS), as a function of annealing. Er-implanted Si, regrown by solid phase epitaxy at 600degrees C and then subject to a rapid thermal anneal, has also been studied by time-resolved photoluminescence (PL). The LDLTS studies reveal that there are clear differences in the defect population as a function of depth from the surface, and this is attributed to different defects in the vacancy-rich and interstitial-rich regions. Defects in the interstitial-rich region have electrical characteristics typical of small extended defects, and these may provide the precursors for larger structural defects in annealed layers. The time-resolved PL of the annealed layers, in combination with electron microscopy, shows that the Er emission at 1.54microns contains a fast component attributed to non-radiative recombination at deep states due to small dislocations. It is concluded that there can be measurable competition to the radiative efficiency in rare-earth implanted Si that is due to the implantation and is not specific to Er.</p