We introduce a class of abstract doubly nonlinear evolution equations associated with subdifferential operators depending on the unknown. This is modeled on quasi-variational inequalities for systems of elliptic-parabolic partial differential equations that arise from the flows of multi-component fluids in partially saturated porous media. We prove the existence of a solution by employing the theory of time-dependent subdifferential evolution equations and its generalization. Application examples are given for models of fluid flows and population diffusion with a temperature-dependent constraint.

We consider the initial value problem for the incompressible magnetohydrodynamics system with the Coriolis force in the whole space R³. We prove the global in time existence and the uniqueness of solutions for large initial data in the scaling critical Sobolev space when the speed of rotation is sufficiently high. In order to control the large magnetic fields, we introduce a modified linear solution for the velocity, and show its smallness in a suitable space-time norm by means of the dispersive effect of the Coriolis force.

The transcendency of the characteristic equation of a linear delay differential equation (DDE) with a delay parameter is a main feature that creates a difference from ordinary differential equations. Here we discuss a method to find the stability condition of the characteristic equation of a planar system of linear DDEs by using the critical delay, which is the threshold delay value dividing the stability and instability regions of the characteristic equation. This method gives a clear understanding of the nature of stability of characteristic equations of some class of linear DDEs, and the obtained results are considered to be an extension of the previous result obtained by Hara and Sugie [Funkcial. Ekvac., 39 (1996), 69-86].

In this paper, a nonclassical algebraic solution of a 3-variable irregular Garnier system is constructed. Diarra-Loray have studied classification of algebraic solutions of irregular Garnier systems. There are two type of the algebraic solutions: classical type and pull-back type. They have shown that there are exactly three nonclassical algebraic solutions for N-variables irregular Garnier systems with N > 1. Explicit forms of two of the three solutions are already given. The solution constructed in the present paper is the remaining algebraic solution.