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In this paper, we developed an optimal control of a reaction–diffusion mathematical model, describing the spatial spread of dengue infection. Compartments for human and vector populations are considered in the model, including a compartment for the aquatic phase of mosquitoes. This enabled us to discuss the vertical transmission effects on the spread of the disease in a two-dimensional domain, using demographic data for different scenarios. The model was analyzed, establishing the existence and convergence of the weak solution for the model. The convergence of the numerical scheme to the weak solution was proved. For numerical approximation, we adopted the finite element scheme to solve direct and adjoint state systems. We also used the nonlinear gradient descent method to solve the optimal control problem, where the optimal management of government investment was proposed and leads to more effective dengue fever infection control. These results may help us understand the complex dynamics driven by dengue and assess the public health policies in the control of the disease. [ABSTRACT FROM AUTHOR]

We study the homogenization of a novel microscopic tridomain system, allowing for a more detailed analysis of the properties of cardiac conduction than the classical bidomain and monodomain models. In (Acta Appl.Math.179 (2022) 1–35), we detail this model in which gap junctions are considered as the connections between adjacent cells in cardiac muscle and could serve as alternative or supporting pathways for cell-to-cell electrical signal propagation. Departing from this microscopic cellular model, we apply the periodic unfolding method to derive the macroscopic tridomain model. Several difficulties prevent the application of unfolding homogenization results, including the degenerate temporal structure of the tridomain equations and a nonlinear dynamic boundary condition on the cellular membrane. To prove the convergence of the nonlinear terms, especially those defined on the microscopic interface, we use the boundary unfolding operator and a Kolmogorov–Riesz compactness's result. [ABSTRACT FROM AUTHOR]

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]

In this paper, we are dealing with a rigorous homogenization result at two different levels for the bidomain model of cardiac electro-physiology. The first level associated with the mesoscopic structure such that the cardiac tissue consists of extracellular and intracellular domains separated by an interface (the sarcolemma). The second one related to the microscopic structure in such a way that the intracellular medium can only be viewed as a periodical layout of unit cells (mitochondria). At the interface between intra- and extracellular media, the fluxes are given by nonlinear functions of ionic and applied currents. A rigorous homogenization process based on unfolding operators is applied to derive the macroscopic (homogenized) model from our meso-microscopic bidomain model. We apply a three-scale unfolding method in the intracellular problem to obtain its homogenized equation at two levels. The first level upscaling of the intracellular structure yields the mesoscopic equation. The second step of the homogenization leads to obtain the intracellular homogenized equation. To prove the convergence of the nonlinear terms, especially those defined on the microscopic interface, we use the boundary unfolding method and a Kolmogorov-Riesz compactness’s result. Next, we use the standard unfolding method to homogenize the extracellular problem. Finally, we obtain, at the limit, a reaction-diffusion system on a single domain (the superposition of the intracellular and extracellular media) which contains the homogenized equations depending on three scales. Such a model is widely used for describing the macroscopic behavior of the cardiac tissue, which is recognized to be an important messengers between the cytoplasm (intracellular) and the other extracellular inside the biological cells. [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 paper studies the feasibility of optimal sparse boundary control of the bidomain-bath model in cardiac electrophysiology. We incorporated the Luo–Rudy phase I (LR-I) ventricular action potential model, which accounts for more biophysical details of cellular dynamics. The boundary optimal control framework involves a standard quadratic objective functional that is to be minimized and the L 1 -norm of the control accounting for the sparsity. The existence of the bidomain-bath model using the LR-I is demonstrated, and the derivation of the first-order optimality system is shown. Then we provide the well-posedness and the sparsity of the optimal control problem. The numerical realization of such a boundary control framework for a successful defibrillation study is demonstrated. [ABSTRACT FROM AUTHOR]

The monodomain model is a widely used model in electrocardiology to simulate the propagation of electrical potential in the myocardium. In this paper, we investigate a positive nonlinear control volume finite element scheme, based on Godunov's flux approximation of the diffusion term, for the monodomain model coupled to a physiological ionic model (the Beeler–Reuter model) and using an anisotropic diffusion tensor. In this scheme, degrees of freedom are assigned to vertices of a primal triangular mesh, as in conforming finite element methods. The diffusion term which involves an anisotropic tensor is discretized on a dual mesh using the diffusion fluxes provided by the conforming finite element reconstruction on the primal mesh and the other terms are discretized by means of an upwind finite volume method on the dual mesh. The scheme ensures the validity of the discrete maximum principle without any restriction on the transmissibility coefficients. By using a compactness argument, we obtain the convergence of the discrete solution and as a consequence, we get the existence of a weak solution of the original model. Finally, we illustrate by numerical simulations that the proposed scheme successfully removes nonphysical oscillations in the propagation of the wavefront and maintains conduction velocity close to physiological values. [ABSTRACT FROM AUTHOR]

• Derivation of a new nonlocal kinetic-fluid model by use the micro-macro decomposition method. • Proof of the existence of weak solutions for the proposed system. • Performance of the proposed schemes during the transition from the kinetic regime to the macroscopic regime. • Demonstration of the phenomenon of pattern formation. In this article we propose a nonlocal cross-diffusion–fluid system describing the dynamics of multiple interacting populations living in a Newtonian fluid. First, we derive our nonlocal cross-diffusion–fluid system from a nonlocal kinetic-fluid model by the micro-macro decomposition method. Second, we prove the existence of weak solutions for the proposed system by applying the nonlinear Galerkin method with a priori estimates and compactness arguments. On the basis of micro-macro decomposition, we propose and develop an asymptotic-preserving numerical scheme. Finally, we discuss the computational results for the proposed system. [ABSTRACT FROM AUTHOR]

We present a virtual element method (VEM) for a nonlocal reaction–diffusion system of the cardiac electric field. For this system, we analyze an |$H^1$| -conforming discretization by means of VEM that can make use of general polygonal meshes. Under standard assumptions on the computational domain, we establish the convergence of the discrete solution by considering a series of a priori estimates and by using a general |$L^p$| compactness criterion. Moreover, we obtain optimal order space-time error estimates in the |$L^2$| norm. Finally, we report some numerical tests supporting the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we apply a rigorous homogenization method based on unfolding operators to a microscopic bidomain model representing the electrical activity of the heart at a cellular level. The heart is represented by an arbitrary open bounded connected domain with smooth boundary and the cardiac cells' (myocytes) domain is viewed as a periodic region. We start by proving the well posedness of the microscopic problem by using Faedo–Galerkin method and L 2 -compactness argument on the membrane surface without any restrictive assumptions on the conductivity matrices. Using the unfolding method in homogenization, we show that the sequence of solutions constructed in the microscopic model converges to the solution of the macroscopic bidomain model. Because of the nonlinear ionic function, the proof is based on the surface unfolding method and Kolmogorov compactness argument. [ABSTRACT FROM AUTHOR]

We are interested in modeling the interaction of calcium dynamics in a medium including sarcolemma and sarcoplasmic reticulum. The governing equations consist of a nonlinear reaction–diffusion system representing the various calcium fluxes and theirs buffers in the two media. We address the question of existence of weak solutions by using a fixed-point approach. We propose a finite element method for this system, we establish the existence of the discrete solution, and we show that the discrete solution generated by the given scheme converges to the corresponding weak solution for the model studied. Finally, we give some 2D and 3D numerical examples to our model. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new nonlinear model describing the dynamical interaction of two species within a viscous flow. The proposed model is a cross‐diffusion system coupled with the Brinkman problem written in terms of velocity fluid, vorticity, and pressure and describing the flow patterns driven by an external source depending on the distribution of species. In the first part, we derive macroscopic models from the kinetic‐fluid equations by using the micro‐macro decomposition method. On the basis of the Schauder fixed‐point theory, we prove the existence of weak solutions for the derived model in the second part. The last part is devoted to developing a one‐dimensional finite volume approximation for the kinetic‐fluid model, which is uniformly stable along the transition from kinetic to macroscopic regimes. Our computation method is validated with various numerical tests. [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]

In this work, we have designed conforming and nonconforming virtual element methods (VEM) to approximate non-stationary nonlocal biharmonic equation on general shaped domain. By employing Faedo–Galerkin technique, we have proved the existence and uniqueness of the continuous weak formulation. Upon applying Brouwer's fixed point theorem, the well-posedness of the fully discrete scheme is derived. Further, following [J. Huang and Y. Yu, A medius error analysis for nonconforming virtual element methods for Poisson and biharmonic equations, J. Comput. Appl. Math. 386 (2021) 113229], we have introduced Enrichment operator and derived a priori error estimates for fully discrete schemes on polygonal domains, not necessarily convex. The proposed error estimates are justified with some benchmark examples. [ABSTRACT FROM AUTHOR]

• Proposing a new coupled stochastic kinetic-fluid model. • Derivation of the proposed model by the micro-macro and Odd-Even methods. • Performance of a stable numerical scheme in the hydrodynamic limit in 2D. • Chemotaxis organism and convection of cells caused by the Reynolds number. • Gravitational effect on the dynamic of cells and of fluid motion. A numerical approach to solve a new coupled stochastic kinetic–fluid model is proposed in this paper. The approach yields a stable numerical scheme in the hydrodynamic limit as the Knudsen number goes to zero along the transition from kinetic regime to macroscopic regime. The method is based on the Odd-Even decomposition of the kinetic stochastic chemotaxis–fluid model which leads to an implicit-explicit scheme based on a two steps splitting strategy. The method is numerically validated by various tests in two dimensional space. [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 analyze the effects of introducing the fractional-in-space operator into a Lotka-Volterra competitive model describing population super-diffusion. First, we study how cross super-diffusion influences the formation of spatial patterns: a linear stability analysis is carried out, showing that cross super-diffusion triggers Turing instabilities, whereas classical (self) super-diffusion does not. In addition we perform a weakly nonlinear analysis yielding a system of amplitude equations, whose study shows the stability of Turing steady states. A second goal of this contribution is to propose a fully adaptive multiresolution finite volume method that employs shifted Grünwald gradient approximations, and which is tailored for a larger class of systems involving fractional diffusion operators. The scheme is aimed at efficient dynamic mesh adaptation and substantial savings in computational burden. A numerical simulation of the model was performed near the instability boundaries, confirming the behavior predicted by our analysis. [ABSTRACT FROM AUTHOR]

This paper is devoted to the analysis of the uniform null controllability for a family of nonlinear reaction-diffusion systems approximating a parabolic-elliptic system which models the electrical activity of the heart. The uniform--with respect to the degenerating parameter--null controllability of the approximating system by means of a single control is shown. The proof is based on the combination of Carleman estimates and weighted energy inequalities. [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]

This paper is concerned with the mathematical analysis of a coupled elliptic-parabolic system modeling the interaction between the propagation of electric potential and subsequent deformation of the cardiac tissue. The problem consists in a reaction-diffusion system governing the dynamics of ionic quantities, intra- and extra-cellular potentials, and the linearized elasticity equations are adopted to describe the motion of an incompressible material. The coupling between muscle contraction, biochemical reactions and electric activity is introduced with a so-called active strain decomposition framework, where the material gradient of deformation is split into an active (electrophysiology-dependent) part and an elastic (passive) one. Under the assumption of linearized elastic behavior and a truncation of the updated nonlinear diffusivities, we prove existence of weak solutions to the underlying coupled reaction-diffusion system and uniqueness of regular solutions. The proof of existence is based on a combination of parabolic regularization, the Faedo-Galerkin method, and the monotonicity-compactness method of Lions. A finite element formulation is also introduced, for which we establish existence of discrete solutions and show convergence to a weak solution of the original problem. We close with a numerical example illustrating the convergence of the method and some features of the model. [ABSTRACT FROM AUTHOR]

In this paper, we consider a reaction-diffusion system describing three interacting species in the food chain structure with nonlocal and cross diffusion. We propose a semi-implicit finite volume scheme for this system, we establish existence and uniqueness of the discrete solution, and it is also showed that the discrete solution generated by the given scheme converges to the corresponding weak solution for the model studied. The convergence proof is based on the use of the discrete Sobolev embedding inequalities with general boundary conditions and a space-time L1 compactness argument that mimics the compactness lemma due to Kruzhkov. Finally we give some numerical examples. [ABSTRACT FROM AUTHOR]

This paper deals with construction and convergence analysis of a finite volume scheme for compressible/incompressible (gas-water) flows in porous media. The convergence properties of finite volume schemes or finite element scheme are only known for incompressible fluids. We present a new result of convergence in a two or three dimensional porous medium and under the only consideration that the density of gas depends on global pressure. In comparison with incompressible fluid, compressible fluids requires more powerful techniques; especially the discrete energy estimates are not standard. [ABSTRACT FROM AUTHOR]

This work is intended to provide a convenient tool for the mathematical analysis of a particular kind of fi-nite volume approximation which can be used, for instance, in the context of nonlinear and/or anisotropic diffusion operators in three dimensions (3D). Following the so-called DDFV (discrete duality finite vol-ume) approach developed by Hermeline (1998, Une méthode de volumes finis pour les équations ellip-tiques du second ordre. C. R. Math. Acad. Sci. Paris, 326, 1433-1436 (in French); 2000, A finite volume method for the approximation of diffusion operators on distorted meshes. J. Comput. Phys., 160, 481-499) and by Domelevo & Omnès (2005, A finite volume method for the Laplace equation on almost arbitrary two-dimensional grids. M2AN Math. Model. Numer. Anal., 39, 1203-1249) in 3D, we consider a 'double' covering ξ of a 3D domain by a rather general primal mesh and by a well-chosen 'dual' mesh. The associated discrete divergence operator divξ is obtained by the standard finite volume approach. A simple and consistent discrete gradient operator ∇ξ is defined by a local affine interpolation that takes into account the geometry of the double mesh. Under mild geometrical constraints on the choice of the dual volumes, we show that --divξ and ∇ξ are linked by the 'discrete duality property', which is an ana-logue of the integration-by-parts formula. The primal mesh need not be conformai, and its interfaces can be general polygons. We give several numerical examples for anisotropic linear diffusion problems; good convergence properties are observed. The sequel, Andreianov et al. (201 la, On 3D DDFV discretization of gradient and divergence operators. II. Discrete functional analysis tools and applications to degenerate parabolic problems. HAL preprint available at http://hal.archives-ouvertes.fr/hal-00567342) to this paper will summarize some key discrete functional analysis tools for DDFV schemes and give applications to proving convergence of DDFV schemes for several nonlinear degenerate parabolic partial differential equations. [ABSTRACT FROM AUTHOR]

Abstract: This Note is devoted to the analysis of the null controllability of a nonlinear reaction–diffusion system, approximating a parabolic–elliptic system, modeling electrical activity in the heart. The uniform, with respect to the degenerating parameter, null controllability of the approximating system by a single control force acting on a subdomain is shown. The proof needs a precise estimate with respect to the degenerating parameter and it is done combining Carleman estimates and energy inequalities. [Copyright &y& Elsevier]

Abstract: In the present paper, an optimal control problem constrained by the tridomain equations in electrocardiology is investigated. The state equations consisting in a coupled reaction–diffusion system modeling the propagation of the intracellular and extracellular electrical potentials, and ionic currents, are extended to further consider the effect of an external bathing medium. The existence and uniqueness of solution for the tridomain problem and the related control problem is assessed, and the primal and dual problems are discretized using a finite volume method which is proved to converge to the corresponding weak solution. In order to illustrate the control of the electrophysiological dynamics, we present some preliminary numerical experiments using an efficient implementation of the proposed scheme. [Copyright &y& Elsevier]

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]

The main goal of this paper is to propose a convergent finite volume method for a reaction-diffusion system with cross-diffusion. First, we sketch an existence proof for a class of cross-diffusion systems. Then the standard two-point finite volume fluxes are used in combination with a nonlinear positivity-preserving approximation of the cross-diffusion coefficients. Existence and uniqueness of the approximate solution are addressed, and it is also shown that the scheme converges to the corresponding weak solution for the studied model. Furthermore, we provide a stability analysis to study pattern-formation phenomena, and we perform two-dimensional numerical examples which exhibit formation of nonuniform spatial patterns. From the simulations it is also found that experimental rates of convergence are slightly below second order. The convergence proof uses two ingredients of interest for various applications, namely the discrete Sobolev embedding inequalities with general boundary conditions and a spacetime L1 compactness argument that mimics the compactness lemma due to Kruzhkov. The proofs of these results are given in the Appendix. [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: This paper is aimed at studying the formation of patches in a cross-diffusion system without reaction terms when the diffusion matrix can be negative but with positive self-diffusion. We prove existence results for small data and global a priori bounds in space–time Lebesgue spaces for a large class of ‘diffusion’ matrices. This result indicates that blow-up should occur on the gradient. One can tackle this issue using a relaxation system with global solutions and prove uniform a priori estimates. Our proofs are based on a duality argument à la M. Pierre which we extend to treat degeneracy and growth of the diffusion matrix. We also analyze the linearized instability of the relaxation system and a Turing type mechanism can occur. This gives the range of parameters and data for which instability may occur. Numerical simulations show that patterns arise indeed in this range and the solutions tend to exhibit patches with stiff gradients on bounded solutions, in accordance with the theory. [Copyright &y& Elsevier]

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]

Abstract: Spatially two-dimensional, possibly degenerate reaction–diffusion systems, with a focus on models of combustion, pattern formation and chemotaxis, are solved by a fully adaptive multiresolution scheme. Solutions of these equations exhibit steep gradients, and in the degenerate case, sharp fronts and discontinuities. This calls for a concentration of computational effort on zones of strong variation. The multiresolution scheme is based on finite volume discretizations with explicit time stepping. The multiresolution representation of the solution is stored in a graded tree (“quadtree”), whose leaves are the non-uniform finite volumes on whose borders the numerical divergence is evaluated. By a thresholding procedure, namely the elimination of leaves with solution values that are smaller than a threshold value, substantial data compression and CPU time reduction is attained. The threshold value is chosen such that the total error of the adaptive scheme is of the same order as that of the reference finite volume scheme. Since chemical reactions involve a large range of temporal scales, but are spatially well localized (especially in the combustion model), a locally varying adaptive time stepping strategy is applied. For scalar equations, this strategy has the advantage that consistence with a CFL condition is always enforced. Numerical experiments with five different scenarios, in part with local time stepping, illustrate the effectiveness of the adaptive multiresolution method. It turns out that local time stepping accelerates the adaptive multiresolution method by a factor of two, while the error remains controlled. [Copyright &y& Elsevier]

Abstract: We prove existence and uniqueness of a renormalized solution to nonlinear elliptic equations with variable exponents and data. The functional setting involves Lebesgue–Sobolev space with variable exponents . [Copyright &y& Elsevier]

Abstract: We are concerned with a system of nonlinear partial differential equations modeling the Lotka–Volterra interactions of predators and preys in the presence of prey-taxis and spatial diffusion. The spatial and temporal variations of the predator''s velocity are determined by the prey gradient. We prove the existence of weak solutions by using Schauder fixed-point theorem and uniqueness via duality technique. The linearized stability around equilibrium is also studied. A finite volume scheme is build and numerical simulation show interesting phenomena of pattern formation. [Copyright &y& Elsevier]

We consider a fully parabolic model for chemotaxis with volume-filling effect and a nonlinear diffusion that degenerates in a two-sided fashion. We address the questions of existence of weak solutions and of their regularity by using, respectively, a regularization method and the technique of intrinsic scaling. [ABSTRACT FROM AUTHOR]

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]

We prove the well-posedness (existence and uniqueness)of renormalized entropy solutions to the Cauchy problem for quasi-linear anisotropic degenerate parabolic equations with L1 data. This paper complements the work by Chen and Perthame [Ann. Inst. H. Poincaré Anal. Non Linéaire, 20 (2003), pp. 645-668], who developed a pure L1 theory based on the notion of kinetic solutions. [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence of weak solutions for a reaction-diffusion system with general anisotropic diffusivities and transport effects, supplemented with either mixed boundary conditions or no-flux boundary conditions. Initial conditions and external forcing terms are in L1; this does not allow us to use classical variational formulations. Our motivation is a mathematical model describing the spatial propagation in heterogeneous environments of Feline Immunodeficiency Virus, a feline retrovirus. [Copyright &y& Elsevier]

Abstract: We define renormalized solutions for the problems of the kind in , . Here β is a maximal monotone graph in , and , , is the fractional Laplace operator which is a particular case of Lévy diffusions. We prove well-posedness in the framework of renormalized solutions. Then the Cauchy problem for the associated evolution equations can be solved using the Crandall–Liggett semigroup technique. [ABSTRACT FROM AUTHOR]