Coexistence and stability of solutions for a class of reaction-diffusion systems.
Collision patterns on mollusc shells.
Comparison of time stepping schemes on the cable equation.
Comparison results and steady states for the Fujita equation with fractional laplacian
Comparison results between minimal barriers and viscosity solutions for geometric evolutions
Competition of Species with Intra-Specific Competition
Intra-specific competition in population dynamics can be described by integro-differential equations where the integral term corresponds to nonlocal consumption of resources by individuals of the same population. Already the single integro-differential equation can show the emergence of nonhomogeneous in space stationary structures and can be used to model the process of speciation, in particular, the emergence of biological species during evolution [S. Genieys et al., Math. Model. Nat. Phenom....
Compétition Réaction-Diffusion et comportement asymptotique d’un problème d’obstacle doublement non linéaire
Le but de cet article est l’étude de la compétition Réaction-Diffusion pour un problème de type où est un opérateur de Lerray-Lions, est une fonction continue croissante et la réaction est une fonction croissante qui dépend de l’espace . On suppose que les coefficients de diffusion et de Réaction dépendent du paramètre avec et/ou tends vers lorsque . Dans le cas où, le coefficient de réaction est très rapide, nous étudions le comportement asymptotique lorsque de la solution...
Computational studies of conserved mean-curvature flow
The paper presents the results of numerical solution of the evolution law for the constrained mean-curvature flow. This law originates in the theory of phase transitions for crystalline materials and describes the evolution of closed embedded curves with constant enclosed area. It is reformulated by means of the direct method into the system of degenerate parabolic partial differential equations for the curve parametrization. This system is solved numerically and several computational studies are...
Computational studies of non-local anisotropic Allen-Cahn equation
The paper presents the results of numerical solution of the Allen-Cahn equation with a non-local term. This equation originally mentioned by Rubinstein and Sternberg in 1992 is related to the mean-curvature flow with the constraint of constant volume enclosed by the evolving curve. We study this motion approximately by the mentioned PDE, generalize the problem by including anisotropy and discuss the computational results obtained.
Concentration in the Nonlocal Fisher Equation: the Hamilton-Jacobi Limit
The nonlocal Fisher equation has been proposed as a simple model exhibiting Turing instability and the interpretation refers to adaptive evolution. By analogy with other formalisms used in adaptive dynamics, it is expected that concentration phenomena (like convergence to a sum of Dirac masses) will happen in the limit of small mutations. In the present work we study this asymptotics by using a change of variables that leads to a constrained Hamilton-Jacobi equation. We prove the convergence analytically...
Connections in scalar reaction diffusion equations with Neumann boundary conditions
Constructing universal pattern formation processes governed by reaction-diffusion systems.
Construction of a natural norm for the convection-diffusion-reaction operator
In this work, we construct, by means of the function space interpolation theory, a natural norm for a generic linear coercive and non-symmetric operator. We look for a norm which is the counterpart of the energy norm for symmetric operators. The natural norm allows for continuity and inf-sup conditions independent of the operator. Particularly we consider the convection-diffusion-reaction operator, for which we obtain continuity and inf-sup conditions that are uniform with respect to the operator...
Continuation of the connection matrix for singularly perturbed hyperbolic equations
Continuity with respect to data and parameters of weak solutions to a Stefan-like problem.
Continuous host-macroparasite models with application to aquaculture.
Control problems for convection-diffusion equations with control localized on manifolds
We consider optimal control problems for convection-diffusion equations with a pointwise control or a control localized on a smooth manifold. We prove optimality conditions for the control variable and for the position of the control. We do not suppose that the coefficient of the convection term is regular or bounded, we only suppose that it has the regularity of strong solutions of the Navier–Stokes equations. We consider functionals with an observation on the gradient of the state. To obtain optimality...
Control problems for convection-diffusion equations with control localized on manifolds
We consider optimal control problems for convection-diffusion equations with a pointwise control or a control localized on a smooth manifold. We prove optimality conditions for the control variable and for the position of the control. We do not suppose that the coefficient of the convection term is regular or bounded, we only suppose that it has the regularity of strong solutions of the Navier–Stokes equations. We consider functionals with an observation on the gradient of the state. To obtain...
Convective Instability of Reaction Fronts in Porous Media
We study the influence of natural convection on stability of reaction fronts in porous media. The model consists of the heat equation, of the equation for the depth of conversion and of the equations of motion under the Darcy law. Linear stability analysis of the problem is fulfilled, the stability boundary is found. Direct numerical simulations are performed and compared with the linear stability analysis.
Convergence of approximate solutions to scalar conservation laws by degenerate diffusion