Abstract:
It is well known that the Kirchhoff equation admits infinitely many simple modes, i.e., time periodic solutions with only one Fourier component in the space variable(s). We prove that for some form of the nonlinear term these simple modes are stable provided that their energy is large enough. Here stable means orbitally stable as solutions of the two-modes system obtained considering initial data with two Fourier components.

Abstract:
We consider mildly degenerate Kirchhoff equations with a small parameter and a weak dissipation term. We prove the existence of global solutions when the parameter is small with respect to the size of initial data. Then we provide global-in-time error estimates on the difference between the solution of our problem and the solution of the corresponding first order problem.

Abstract:
We study the asymptotic behavior of the solutions of the mildly degenerate Kirchhoff equation with a dissipative term. We obtain a new estimate on second-in-time derivative of the solution. Moreover we renormalize the solution in such a way that the renormalization as a no zero limit as t goes to infinity. Finally we calculate explicitly the norm of this limit and we prove that it does not depend on the initial data.

Abstract:
the choice among different materials, products and systems employed in the construction sector may take into consideration economic, social, cultural and environmental criteria. in the case of plumbing fixtures, economic evaluation and water consumption comparisons are the most frequently used criteria. this paper aims to propose a method for quantifying the energy consumption of plumbing fixtures based on the concept of life cycle assessment. such estimate allows a comparison between the performances of different plumbing fixtures in terms of energy consumption. the proposed method was applied in a case study to quantify energy consumption in the life cycle of two types of taps. the total energy consumption in the life cycle of conventional and water-saving taps used in the study was, respectively, 151.66 mj and 127.58 mj. the use phase accounts for 64.6% of energy consumption for conventional taps and 56.5% for self-closing taps. the proposed method has shown to be adequate for quantifying energy consumption in the life cycle and subsequent choice among different plumbing fixtures that perform the same function.

Abstract:
the aim of this article is to present results from a study realized in a public organizationin the state of santa catarina, brazil. the main objective was to verify if organizationalculture and power dependences acted as sources of resistance or acceptation to changeof the structural framework, which took place during the 1991-92 period. as a case study,data analysis was carried out through descriptive-qualitative procedures taking intoconsideration the configuration of organizational structure before and after the event ofmajor change at the year of 1991, and minor modifications that follows. organizationalculture was approached through identification of the values shared by managers at differentlevels and technical staff; dependences of power through the interests of both managersand technical staff. the results revealed that organizational culture acted as an acceptationsource of the structural change, while power dependences worked out as source ofresistance.

Abstract:
In this paper we consider the Cauchy boundary value problem for the abstract Kirchhoff equation with a continuous nonlinearity m : [0,+\infty) --> [0,+\infty). It is well known that a local solution exists provided that the initial data are regular enough. The required regularity depends on the continuity modulus of m. In this paper we present some counterexamples in order to show that the regularity required in the existence results is sharp, at least if we want solutions with the same space regularity of initial data. In these examples we construct indeed local solutions which are regular at t = 0, but exhibit an instantaneous (often infinite) derivative loss in the space variables.

Abstract:
We consider the hyperbolic-parabolic singular perturbation problem for a nondegenerate quasilinear equation of Kirchhoff type with weak dissipation. This means that the dissipative term is multiplied by a coefficient b(t) which tends to 0 as t tends to +infinity. The case where b(t) behaves like (1+t)^{-p} with p<1 has recently been considered. The result is that the hyperbolic problem has a unique global solution, and the difference between solutions of the hyperbolic problem and the corresponding solutions of the parabolic problem converges to zero both as t tends to +infinity and as epsilon goes to 0. In this paper we show that these results cannot be true for p>1, but they remain true in the critical case p=1.

Abstract:
In this note we present some recent results for Kirchhoff equations in generalized Gevrey spaces. We show that these spaces are the natural framework where classical results can be unified and extended. In particular we focus on existence and uniqueness results for initial data whose regularity depends on the continuity modulus of the nonlinear term, both in the strictly hyperbolic case, and in the degenerate hyperbolic case.

Abstract:
We consider Kirchhoff equations with a small parameter epsilon in front of the second-order time-derivative, and a dissipative term whose coefficient may tend to 0 as t -> + infinity (weak dissipation). In this note we present some recent results concerning existence of global solutions, and their asymptotic behavior both as t -> + infinity and as epsilon -> 0. Since the limit equation is of parabolic type, this is usually referred to as a hyperbolic-parabolic singular perturbation problem. We show in particular that the equation exhibits hyperbolic or parabolic behavior depending on the values of the parameters.

Abstract:
We consider the Cauchy problem for the Perona-Malik equation in an open subset of R^{n}, with Neumann boundary conditions. It is well known that in the one-dimensional case this problem does not admit any global C^{1} solution if the initial condition is transcritical, namely when the norm of the gradient of the initial condition is smaller than 1 in some region, and larger than 1 in some other region In this paper we show that this result cannot be extended to higher dimension. We show indeed that for n >= 2 the problem admits radial solutions of class C^{2,1} with a transcritical initial condition.