Five types of blow-up in a semilinear fourth-order reaction-diffusion equation: an analytic-numerical approach

Five types of blow-up patterns that can occur for the 4th-order semilinear parabolic equation of reaction-diffusion type u_t= -\Delta^2 u + |u|^{p-1} u \quad {in} \quad \ren \times (0,T), p>1, \quad \lim_{t \to T^-}\sup_{x \in \ren} |u(x,t)|= +\iy, are discussed. For the semilinear heat equation $u_t= \Delta u+ u^p$, various blow-up patterns were under scrutiny since 1980s, while the case of higher-order diffusion was studied much less, regardless a wide range of its application.Comment: 41 pages, 27 figure

Measurement of Cross-sections and Leptonic Forward-backward Asymmetries At the Z-pole and Determination of Electroweak Parameters

We report on the measurement of the leptonic and hadronic cross sections and leptonic forward-backward asymmetries at the Z peak with the L3 detector at LEP. The total luminosity of 40.8 pb-1 collected in the years 1990, 1991 and 1992 corresponds to 1.09 . 10(6) hadronic and 0.98 . 10(5) leptonic Z decays observed. These data allow us to determine the electroweak parameters. From the cross sections we derive the properties of the Z boson: M(Z) = 91 195 +/- 9 MeV GAMMA(Z) = 2494 +/- 10 MeV GAMMA(had) = 1748 +/- 10 MeV GAMMA(l) = 83.49 +/- 0.46 MeV, assuming lepton universality. We obtain an invisible width of GAMMA(inv) = 496.5 +/- 7.9 MeV which, in the Standard Model, corresponds to a number of light neutrino species of N(v) = 2.981 +/- 0.050. Using also the three leptonic forward-backward asymmetries and the average tau polarization, we determine the effective vector and axial-vector coupling constants of the neutral weak current to charged leptons to be: g(V)-l = -0.0378(+0.0045/-0.0042) g(A)-l = -0.4998 +/- 0.0014 Within the framework of the Standard Model, and including our measurements of the Z --> bbBAR forward-backward asymmetry and partial decay width, we derive an effective electroweak mixing angle of sin2 theta(W)BAR = 0.2326 +/- 0.0012. We obtain an estimate for the strong coupling constant, alpha(s) = 0.142 +/- 0.013, and for the top-quark mass, m(t) = 158(+32/-40) +/- 19(Higgs) GeV, where the second error arises due to the uncertainty in the Higgs-boson mass