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In this paper, we are concerned with the study of the mathematical analysis for an optimal control of a nonlocal degenerate aggregation model. This model describes the aggregation of organisms such as pedestrian movements, chemotaxis, animal swarming. We establish the wellposedness (existence and uniqueness) for the weak solution of the direct problem by means of an auxiliary nondegenerate aggregation equation, the Faedo-Galerkin method (for the existence result) and duality method (for the uniqueness). Moreover, for the adjoint problem, we prove the existence result of minimizers and first-order necessary conditions. The main novelty of this work concerns the presence of a control to our nonlocal degenerate aggregation model. Our results are complemented with some numerical simulations. [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]

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: A finite volume method for solving the degenerate chemotaxis model is presented, along with numerical examples. This model consists of a degenerate parabolic convection–diffusion PDE for the density of the cell-population coupled to a parabolic PDE for the chemoattractant concentration. It is shown that discrete solutions exist, and the scheme converges. [Copyright &y& Elsevier]

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]