We obtain the $L_p$ estimate of solutions to the resolvent problem for Stokes system with Neumann type boundary condition in a bounded or exterior domain in $\mathbb R^n$. The result has been obtained by Grubb and Solonnikov [7, 8, 9, 10] by the systematic use of theory of pseudo-differential operators. In this paper, we give an essentially different proof from [7, 8, 9, 10]. The point of our proof is to use the space $W^{-1}_p$ in order to handle with the equation $\nabla\cdot u = g$ in the half-space model problem as well as in the whole space one. Our result is an extension of the paper by Farwig and Sohr [5] about the Dirichlet zero condition case to the Neumann boundary condition case.

We discuss the existence of the blow-up solution for some nonlinear parabolic system called attractive drift-diffusion equation in two space dimensions. We show that if the initial data satisfies a threshold condition, the corresponding solution blows up in a finite time. This is a system case for the blow-up result of the chemotactic equation proved by Nagai [28] and Nagai-Senba-Suzuki [30] and gravitational interaction of particles by Biler-Nadzieja [7], [8].

A system of nonlinear damped wave equations with symmetric linear part is investigated. A positive steady-state bifurcates from the trivial solution as a parameter changes. The spectrum of the linearized operator is studied. Then the stability of the positive steady-state is considered as a solution of the nonlinear hyperbolic system. Asymptotic stability results are found for solutions in $R^{N},$ $ N \ge 1.$ Bifurcation methods are used to find the steady-states, and semigroup methods are used to study stability. Stability results are obtained although the semigroup is not analytic.

The evolution of one-dimensional fluidized beds may be modeled in form of a system of partial differential equations of the compressible Navier-Stokes type where the viscosity depends on the density, which may vanish, the source term is nonlinear, and the constitutive law for the pressure blows up for finite values of the density. A finite-differences scheme is used to solve an approximated problem in Lagrangian coordinates, which we show to be equivalent to the corresponding problem in Eulerian coordinates. We then prove compactness and convergence properties of the sequence of solutions of the approximated problems and partially identify the limit as a solution of the original problem.

In this paper we deal with semilinear boundary value problems for systems of equations whose nonlinear part involves linear combination of periodic functions and such that the linear part has a multidimensional solution space. This kind of problems is very important in applications, specially in mechanics and electric circuits theory. By using the Liapunov-Schmidt reduction and topological degree techniques, together with a careful analysis of the oscillatory behavior of some integrals associated to the bifurcation equation, we give a qualitative and quantitative description of the range of the corresponding nonlinear operator. Also, we provide some multiplicity results.