This paper is concerned with a weakly coupled system of quasilinear parabolic equations where the coefficients are allowed to be discontinuous and the reaction functions may depend on continuous delays. By the method of upper and lower solutions and the associated monotone iterations and by difference ratios method and various estimates, we obtained the existence and uniqueness of the global pi...

Abstract. In this paper we report some important results that help in analizing the border collision bifurcations that occur in n-dimensional discontinuous maps. For this purpose, we use the piecewise linear approximation in the neighborhood of the plane of discontinuity. Earlier, Feigin had made a similar analysis for general n-dimensional piecewise smooth continuous maps. Proceeding along sim...

The paper proposes a scheme by combining the Runge-Kutta discontinuous Galerkin method with a δ-mapping algorithm for solving hyperbolic conservation laws with discontinuous fluxes. This hybrid scheme is particularly applied to nonlinear elasticity in heterogeneous media and multi-class traffic flow with inhomogeneous road conditions. Numerical examples indicate the scheme’s efficiency in resol...

We consider the system of equations governing an incompressible immiscible two-phase flow within an heterogeneous porous medium made of two different rock types. Since the capillary pressure function depends on the rock type, the capillary pressure field might be discontinuous at the interface between the rocks. We introduce multivalued phase pressures to give a sense to the transmission condit...

This paper introduces economists to some fixed point theorems for discontinuous mappings with non-convex images on a non-convex domain. These theorems have recently been developed based on a new approach by mathematical economists and mathematicians. The new method of proof is first transformed into a sort of metatheorem, which is then used to obtain a set of necessary and sufficient conditions...

In this letter, without assuming the boundedness of the activation functions, we discuss the dynamics of a class of delayed neural networkswith discontinuous activation functions. A relaxed set of sufficient conditions is derived, guaranteeing the existence, uniqueness, and global stability of the equilibrium point. Convergence behaviors for both state and output are discussed. The constraints ...

The formal extension of the conventional theory of dynamic programming to control problems with slow growth and a state constraint x 2 encounters two major drawbacks: on one hand the formal Hamiltonian may happen to be discontinuous; on the other hand, just as in the case of bounded controls, the imposition of a state constraint possibly gives rise to a discontinuous value function. On the cont...

In this paper, a discontinuous dynamic state feedback that robustly stabilizes affine in control uncertain nonlinear systems is proposed. The formulation is based on Hamilton-Jacobi-Isaacs partial differential equations with two boundary conditions. The resulting dynamic state feedback is then expressed in terms of the solution of the related PDE’s. An interesting feature is that the internal s...

We present preliminary results of a discontinuous Galerkin scheme applied to deterministic computations of the transients for the Boltzmann-Poisson system describing electron transport in semiconductor devices. The collisional term models optical-phonon interactions which become dominant under strong energetic conditions corresponding to nanoscale active regions under applied bias. The proposed...

Abstract. This paper provides a theoretical foundation for interior penalty discontinuous Galerkin methods for second order elliptic equations on very general polygonal or polyhedral meshes. The mesh can be composed of any polygons or polyhedra which satisfies certain shape regularity conditions characterized in a recent paper by two of the authors in [18]. The usual H conforming finite element...