Volume 2, Issue 1

The Dirichlet problem to nonlinear degenerate diagonal parabolic system with some special right–hand sides but still satisfying the maximal growth conditions is considered. Applying the idea of Stampacchia a L ∞ – estimate in terms of data for a priori bounded weak solutions is found.

We show that the slice convergence of the convex parents of saddle functions implies the epi/hypo–convergence of these saddle functions and hence the convergence of their saddle points. We also obtain conditions for the slice convergence of sums of convex functions. We then apply these results to problems in convex programming, optimal control and Chebyshev approximations.

Starting with the classical Sierpiński partition [Bull. Internat. Acad. Sci. Cracovie Ser. A: 1–3, 1919], we consider a question concerning the existence of a nonmeasurable set in the product measure space. Also, we discuss an analogous question on the existence of a set without the Baire property in the product topological space. In particular, it is shown that the situations of measure and category are essentially different.

This paper is an extension of joint work with A. Milani [J. Math. Anal. Appl. 190: 77–100, 1995]. The so–called magnetohydrodynamic limit case of Maxwell's equations with a monotone and Lipschitz continuous material relation is considered in a bounded domain of arbitrary topological genus. The solution theory is presented in a space–time Hilbert space setting. Existence, uniqueness and continuous dependence results are obtained.

In the paper the existence theorem for a initial – boundary problem with the Neumann type boundary condition to equations of the motion of a generalized newtonian incompressible fluid is proved. Moreover, for a special case of a non–newtonian fluid uniqueness of a solution is proved, as well.

Some oscillation criteria are given for the second order linear differential equation with damping [ r ( t ) x ′( t )]′+ p ( t ) x ′( t )+ q ( t ) x ( t ) = 0, t ≥ t 0 , where p ( t ) and q ( t ) are allowed to change sign on [ t 0 , ∞), and r ( t ) > 0. These results generalize and improve some known results for the differential equations x″ ( t ) + q ( t ) x ( t ) = 0, and x″ ( t ) + p ( t ) x ′( t ) + q ( t ) x ( t ) = 0.