System Of Degenerate Parabolic Equations Research Articles

Existence of weak solutions to time-dependent mean-field games

Here, we establish the existence of weak solutions to a wide class of time-dependent monotone mean-field games (MFGs). These MFGs are given as a system of degenerate parabolic equations with initial and terminal conditions. To construct these solutions, we consider a high-order elliptic regularization in space–time. Then, applying Schaefer’s fixed-point theorem, we obtain the existence and uniqueness for this regularized problem. Using Minty’s method, we prove the existence of a weak solution to the original MFG. Finally, the paper ends with a discussion on congestion problems and density constrained MFGs.

Controllability of a system of degenerate parabolic equations with non-diagonalizable diffusion matrix

In this paper we study the null controllability of some non diagonalizable degenerate parabolic systems of PDEs, we assume that the diffusion, coupling and controls matrices are constant and we characterize the null controllability by an algebraic condition so called Kalman's rank condition.

Stability analysis of the POD reduced order method for solving the bidomain model in cardiac electrophysiology

In this paper we show the numerical stability of the Proper Orthogonal Decomposition (POD) reduced order method used in cardiac electrophysiology applications. The difficulty of proving the stability comes from the fact that we are interested in the bidomain model, which is a system of degenerate parabolic equations coupled to a system of ODEs representing the cell membrane electrical activity. The proof of the stability of this method is based on a priori estimates controlling the gap between the reduced order solution and the Galerkin finite element one. We present some numerical simulations confirming the theoretical results. We also combine the POD method with a time splitting scheme allowing a faster solving of the bidomain problem and show numerical results. Finally, we conduct numerical simulation in 2D illustrating the stability of the POD method in its sensitivity to the ionic model parameters. We also perform 3D simulation using a massively parallel code. We show the computational gain using the POD reduced order model. We also show that this method has a better scalability than the full finite element method.

Some Estimates of a System of Degenerate Parabolic Equations

The Dirichlet problem on a coupled system for degenerate parabolic equations is investigated in this paper. The aim is to show the existence, uniqueness of maximum solution, the continuity on the coupled functions, and time-dependent estimates.

A Two‐Species Cooperative Lotka‐Volterra System of Degenerate Parabolic Equations

We consider a cooperating two‐species Lotka‐Volterra model of degenerate parabolic equations. We are interested in the coexistence of the species in a bounded domain. We establish the existence of global generalized solutions of the initial boundary value problem by means of parabolic regularization and also consider the existence of the nontrivial time‐periodic solution for this system.

Positive periodic solutions and optimal control for a distributed biological model of two interacting species

The paper deals with the existence of positive periodic solutions to a system of degenerate parabolic equations with delayed nonlocal terms and Dirichlet boundary conditions. Taking in each equation a meaningful function as a control parameter, we show that for a suitable choice of a class of such controls we have, for each of them, a time-periodic response of the system under different assumptions on the kernels of the nonlocal terms. Finally, we consider the problem of the minimization of a cost functional on the set of pairs: control-periodic response. The considered system may be regarded as a possible model for the coexistence problem of two biological populations, which dislike crowding and live in a common territory, under different kind of intra- and inter-specific interferences.

Finite element approximation of a two-layered liquid film in the presence of insoluble surfactants

We consider a system of degenerate parabolic equations modelling a thin film, consisting of two layers of immiscible Newtonian liquids, on a solid horizontal substrate. In addition, the model includes the presence of insoluble surfactants on both the free liquid-liquid and liquid-air interfaces, and the presence of both attractive and repulsive van der Waals forces in terms of the heights of the two layers. We show that this system formally satisfies a Lyapunov structure, and a second energy inequality controlling the Laplacian of the liquid heights. We introduce a fully practical finite element approximation of this nonlinear degenerate parabolic system, that satisfies discrete analogues of these energy inequalities. Finally, we prove convergence of this approximation, and hence existence of a solution to this nonlinear degenerate parabolic system.

Blow-up and global existence for a coupled system of degenerate parabolic equations in a bounded domain

This article deals with the conditions that ensure the blow-up phenomenon or its absence for solutions of the system u t = Δ u l + u p 1 v q 1 and v t = Δ v m + u p 2 v q 2 with homogeneous Dirichlet boundary conditions. The results depend crucially on the sign of the difference p 2 q 1 - ( l - p 1 ) ( m - q 2 ) , the initial data, and the domain ω

A System of Degenerate Parabolic Equations from Plasma Physics: The Large Time Behavior

The authors study the large time behavior of solutions of a strongly coupled system of degenerate parabolic equations describing the heat and mass transfer in a one-dimensional plasma. The model was introduced by Hyman and Rosenau, and the existence of solutions was studied in our earlier paper. The results of this paper describe how the large time behavior depends on some of the parameters in the equations.

On a certain system of degenerate parabolic equations which arises in hydrodynamics

On a certain system of degenerate parabolic equations which arises in hydrodynamics

A System of Degenerate Parabolic Equations

The system of two nonlinear equations which arises in plasma physics is considered. The equations are of degenerate parabolic type. The global existence theorem for the Cauchy problem is proved. The proof is based on the Lagrangian transformation, thus using a particular structure of the system.

Numerical solution of multicomponent alloy solidification by multigrid techniques

The solidification of an N-component alloy is described by an initial boundary value problem for a system of degenerate parabolic equations modeling heat conduction and mass diffusion. Discretizing implicitly in time and by piecewise linear finite elements in the space variables, at each time step the solution of a system of quasivariational inequalities is required. For the numerical solution of that system, a multigrid algorithm is developed by making use of game theoretic concepts and duality arguments from convex analysis. Finally, the efficiency of the algorithm is demonstrated by displaying numerical results for a ternary alloy.