Communications in Contemporary Mathematics

In this paper the isotopy classification of unions of an M-curve of degree 5 and two lines in ℝP² satisfying some conditions of maximality and general position is obtained. This classification consists of 20 isotopy types of such unions.

This paper is devoted to analyzing the long time behavior of solutions of the system of thermoelasticity of type III in a bounded domain of ℝⁿ (n = 1,2,3) and in the whole space ℝⁿ. For the first case, we introduce a decoupled system that allows to reduce the problem of the asymptotic behavior for the original system to a suitable observability inequality for the Lamé system. In this way most of the existing results for the classical system of thermoelasticity are shown to hold for this system too. In particular, we show that: (1) For most domains the energy of the system does not decay uniformly; (2) Under suitable conditions on the domain that may be described in terms of Geometric Optics, the energy of the system decays exponentially; and (3) For most domains in two space dimensions, the energy of smooth solutions decays polynomially. For the problem in the whole space ℝⁿ, first, based on Fourier analysis and Lyapunov's second method, we show that the energy of longitudinal and thermal waves of the system decays as that of the classical heat equation (while that of the transversal wave component is conservative). Then, by means of a careful spectral analysis, we give a sharp description on the decay rate of the high frequency longitudinal and thermal waves of the system.

This paper deals with semilinear elliptic problems of Dirichlet type, in star shaped domains. An abstract result is stated, which gives sufficient conditions for a positive solution of the problem to have strictly star shaped superlevels. Moreover, it turns out that the maximum point is the only critical point for these solutions.

Then we apply the result to the "single-peak" solutions of some widely studied problems. First a nonlinear and subcritical elliptic equation is considered, when the nonlinearity approaches the critical one. Then Schrödinger type problems are studied. Finally, the case when the potential is constant is also analyzed, on a bounded domain.

We consider a class of pencils (operator valued functions of a complex argument) in a separable Hilbert space. Bounds for the λ-nonlinear spectrum are suggested. Applications to differential operators, integral operators with delay and infinite matrix pencils are also discussed.

In this paper we study stable rank two bundles on a Calabi–Yau threefold. For hypersurfaces in a ℙ³-bundle over ℙ¹, we show that their moduli spaces have irreducible components which are birational to projective spaces.

We study the Cauchy problem for the nonlinear Landau–Ginzburg equation