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We present a novel microscopic tridomain model describing the electrical activity in cardiac tissue with dynamical gap junctions. The microscopic tridomain system consists of three PDEs modeling the tissue electrical conduction in the intra- and extra-cellular domains, supplemented by a nonlinear ODE system for the dynamics of the ion channels and the gap junctions. We establish the global existence and uniqueness of the weak solutions to our microscopic tridomain model. The global existence of solution, which constitutes the main result of this paper, is proved by means of an approximate non-degenerate system, the Faedo–Galerkin method, and an appropriate compactness argument. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to introduce an optimal control for a nonlocal calcium dynamic model in a cardiac cell acting on ryanodine receptors. The optimal control problem is considered as a coupled nonlocal reaction‐diffusion system with a transmission boundary condition covering the sarcoplasmic reticulum and cytosolic domain. We establish the well‐posedness result of the adjoint problem using Faedo‐Galerkin approximation, a priori estimates, and compactness arguments. The numerical discretization of direct and adjoint problems is realized by using the implicit Euler method in time and the finite element for spatial discretization. Moreover, we obtain the stability result in the L2‐norm for the direct and the adjoint discrete problems. Finally, in order to illustrate the control of our calcium dynamic model, we present some numerical experiments devoted to constant and nonlocal diffusions using the proposed numerical scheme. [ABSTRACT FROM AUTHOR]

This work is concerned with the study of the convergence analysis for an optimal control of bidomain-bath model. The bidomain-bath model equations describe the cardiac bioelectric activity at the tissue and bath volumes where the control acts at the boundary of the tissue domain. We establish the existence and convergence of the unique weak solution of the direct bidomain-bath model by a discrete Galerkin approach. The convergence proof is based on deriving a series of a priori estimates and using a general L 2 -compactness criterion. Moreover, the well-posedness of the adjoint problem and the first order necessary optimality conditions are shown. Comparing to the direct problem, the convergence proof of the adjoint problem is based on using a general L 1 -compactness criterion. The numerical tests are demonstrated which achieve the successful cardiac defibrillation by utilizing less total current. Finally, the robustness of the Newton optimization algorithm is presented for different finer mesh geometries. [ABSTRACT FROM AUTHOR]

This work is dedicated to study the cardiac defibrillation problem by using an optimal thoracic electroshock treatment. The problem is formulated as an optimal control problem in a 3D domain surrounded by the bath and including the heart. The control corresponds to the thoracic electroshock and the model describing the electrical activity in the heart is the bidomain model. The bidomain model is coupled with the quasi-static Maxwell's equation to consider the effect of an external bathing medium. The existence and uniqueness of a weak solution for the direct problem is assessed as well as the existence of a weak solution for the adjoint problem. The numerical discretization is realized using a finite element method for the spatial discretization and linearly implicit Runge–Kutta methods for the temporal discretization of the partial differential equations. The numerical results are demonstrated for the termination of re-entry waves. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with a model of the indirect transmission of an epidemic disease between two spatially distributed host populations having non-coincident spatial domains with nonlocal and cross-diffusion, the epidemic disease transmission occurring through a contaminated environment. The mobility of each class is assumed to be influenced by the gradient of the other classes. We address the questions of existence of weak solutions and existence and uniqueness of classical solution by using, respectively, a regularization method and an interpolation result between Banach spaces. Moreover, we propose a finite volume scheme and proved the well-posedness, nonnegativity and convergence of the discrete solution. The convergence proof is based on deriving a series of a priori estimates and by using a general L p compactness criterion. Finally, the numerical scheme is illustrated by some examples. [ABSTRACT FROM AUTHOR]

Our motivation is a mathematical model describing the spatial propagation of an epidemic disease through a population. In this model, the pathogen diversity is structured into two clusters and then the population is divided into eight classes which permits to distinguish between the infected/uninfected population with respect to clusters. In this paper, we prove the weak and the global existence results of the solutions for the considered reaction-diffusion system with Neumann boundary. Next, mathematical Turing formulation and numerical simulations are introduced to show the pattern formation for such systems. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we are concerned with a system of nonlinear partial differential equations modeling a predator–prey system with cross-diffusion in heterogeneous habitats. Predators are assumed to feed on preys with a Holling type II functional response to prey density and preys are assumed to follow a logistic growth in the absence of predation. The mobility of each classes is assumed to be influenced by the gradient of other classes. The existence result is proved by means of an approximation system, the Faedo–Galerkin method, and the compactness method. The global existence of classical solutions is proved under certain restrictions on the coefficients. [ABSTRACT FROM AUTHOR]

We consider a reaction–diffusion system of 2 × 2 equations modeling the spread of early tumor cells. The existence of weak solutions is ensured by a classical argument of Faedo–Galerkin method. Then, we present a numerical scheme for this model based on a finite volume method. We establish the existence of discrete solutions to this scheme, and we show that it converges to a weak solution. Finally, some numerical simulations are reported with pattern formation examples. [ABSTRACT FROM AUTHOR]

Abstract: We analyze a finite volume scheme for the bidomain model, which is a degenerate reaction–diffusion system modeling electrophysiological waves in cardiac tissue. We establish existence and uniqueness of solutions to the finite volume scheme, and show that it converges to a weak solution of the bidomain model. The convergence proof is based on deriving series of a priori estimates and using a general compactness criterion. [Copyright &y& Elsevier]

We prove existence results for distributional solutions of anisotropic nonlinear elliptic systems with a measure valued right-hand side. The functional setting involves anisotropic Sobolev spaces as well as weak Lebesgue (Marcinkiewicz) spaces. In a special case we also prove maximal regularity and uniqueness results. Some of the obtained results are applied, along with an anisotropic variant of the div-curl lemma in the Hardy one space, to prove that the space of anisotropic harmonic maps into spheres is compact in the weak topology of the relevant anisotropic Sobolev space. [ABSTRACT FROM AUTHOR]