Monotonicity in Time and Stationary Solutions for a Quasilinear Heat Equation with Source

Sin resume

Nonlinear stability of self-similar solutions for semilinear wave equations

We prove nonlinear stability of the fundamental self--similar solution of the wave equation with a focusing power nonlinearity $\psi_{tt}-\Delta \psi=\psi^p$ for $p=3,5,7,...$ in the radial case. The proof is based on a semigroup formulation of the wave equation in similarity coordinates.Comment: References added and minor changes, accepted for publication in Communications in Partial Differential Equation

A particle system with explosions: law of large numbers for the density of particles and the blow-up time

Consider a system of independent random walks in the discrete torus with creation-annihilation of particles and possible explosion of the total number of particles in finite time. Rescaling space and rates for diffusion/creation/annihilation of particles, we obtain a stong law of large numbers for the density of particles in the supremum norm. The limiting object is a classical solution to the semilinear heat equation u_t =u_{xx} + f(u). If f(u)=u^p, 1<p \le 3, we also obtain a law of large numbers for the explosion time

Symmetries of a class of nonlinear fourth order partial differential equations

In this paper we study symmetry reductions of a class of nonlinear fourth order partial differential equations \be u_{tt} = \left(\kappa u + \gamma u^2\right)_{xx} + u u_{xxxx} +\mu u_{xxtt}+\alpha u_x u_{xxx} + \beta u_{xx}^2, \ee where $\alpha$, $\beta$, $\gamma$, $\kappa$ and $\mu$ are constants. This equation may be thought of as a fourth order analogue of a generalization of the Camassa-Holm equation, about which there has been considerable recent interest. Further equation (1) is a ``Boussinesq-type'' equation which arises as a model of vibrations of an anharmonic mass-spring chain and admits both ``compacton'' and conventional solitons. A catalogue of symmetry reductions for equation (1) is obtained using the classical Lie method and the nonclassical method due to Bluman and Cole. In particular we obtain several reductions using the nonclassical method which are no} obtainable through the classical method

Renormalizing Partial Differential Equations

In this review paper, we explain how to apply Renormalization Group ideas to the analysis of the long-time asymptotics of solutions of partial differential equations. We illustrate the method on several examples of nonlinear parabolic equations. We discuss many applications, including the stability of profiles and fronts in the Ginzburg-Landau equation, anomalous scaling laws in reaction-diffusion equations, and the shape of a solution near a blow-up point.Comment: 34 pages, Latex; [email protected]; [email protected]

Current Fluctuations and Electron-Electron Interactions in Coherent Conductors

We analyze current fluctuations in mesoscopic coherent conductors in the presence of electron-electron interactions. In a wide range of parameters we obtain explicit universal dependencies of the current noise on temperature, voltage and frequency. We demonstrate that Coulomb interaction decreases the Nyquist noise. In this case the interaction correction to the noise spectrum is governed by the combination $\sum_nT_n(T_n-1)$, where $T_n$ is the transmission of the $n$-th conducting mode. The effect of electron-electron interactions on the shot noise is more complicated. At sufficiently large voltages we recover two different interaction corrections entering with opposite signs. The net result is proportional to $\sum_nT_n(T_n-1)(1-2T_n)$, i.e. Coulomb interaction decreases the shot noise at low transmissions and increases it at high transmissions.Comment: 16 pages, 2 figure

Measurement of the Atmospheric Muon Spectrum from 20 to 3000 GeV

The absolute muon flux between 20 GeV and 3000 GeV is measured with the L3 magnetic muon spectrometer for zenith angles ranging from 0 degree to 58 degree. Due to the large exposure of about 150 m2 sr d, and the excellent momentum resolution of the L3 muon chambers, a precision of 2.3 % at 150 GeV in the vertical direction is achieved. The ratio of positive to negative muons is studied between 20 GeV and 500 GeV, and the average vertical muon charge ratio is found to be 1.285 +- 0.003 (stat.) +- 0.019 (syst.).Comment: Total 32 pages, 9Figure