Monotone systems involving variable-order nonlocal operators

In this paper, we study the existence and uniqueness of bounded viscosity solutions for parabolic Hamilton-Jacobi monotone systems in which the diffusion term is driven by variable-order nonlocal operators whose kernels depend on the space-time variable. We prove the existence of solutions via Perron's method, and considering Hamiltonians with linear and superlinear nonlinearities related to their gradient growth we state a comparison principle for bounded sub and supersolutions. Moreover, we present steady-state large time behavior with an exponential rate of convergence

Fractional reaction-diffusion problems

Cette thèse porte sur deux problèmes différents : dans le premier, nous étudions le comportement en temps long des solutions des équations de réaction diffusion 1d-fractionnaire de type Fisher-KPP lorsque la condition initiale est asymptotiquement de type front et décroît à l'infini plus lentement que, où et est l'indice du laplacien fractionnaire (Chapitre 2). Dans le second problème, nous étudions la propagation asymptotique en temps des solutions de systèmes coopératifs de réaction-diffusion (Chapitre 3). Dans le premier problème, nous démontrons que les ensembles de niveau des solutions se déplacent exponentiellement vite en temps quand t tend vers l'infini. De plus, une estimation quantitative du mouvement de ces ensembles est obtenue en fonction de la décroissance à l'infini de la condition initiale. Dans le second problème, nous montrons que la vitesse de propagation est exponentielle en temps et nous trouvons un exposant précis qui dépend du plus petit ordre des laplaciens fractionnaires considérés et de la non-linéarité. Nous notons aussi que cet indice ne dépend pas de la direction spatiale de propagation.This thesis deals with two different problems: in the first one, we study the large-time behavior of solutions of one-dimensional fractional Fisher-KPP reaction diffusion equations, when the initial condition is asymptotically front-like and it decays at infinity more slowly than a power , where and is the order of the fractional Laplacian (Chapter 2); in the second problem, we study the time asymptotic propagation of solutions to the fractional reaction diffusion cooperative systems (Chapter 3). For the first problem, we prove that the level sets of the solutions move exponentially fast as time goes to infinity. Moreover, a quantitative estimate of motion of the level sets is obtained in terms of the decay of the initial condition. In the second problem, we prove that the propagation speed is exponential in time, and we find a precise exponent depending on the smallest index of the fractional laplacians and of the nonlinearity, also we note that it does not depend on the space direction

SOME RESULTS FOR THE LARGE TIME BEHAVIOR OF HAMILTON-JACOBI EQUATIONS WITH CAPUTO TIME DERIVATIVE

We obtain some Hölder regularity estimates for an Hamilton-Jacobi with fractional time derivative of order α ∈ (0, 1) cast by a Caputo derivative. The Hölder seminorms are independent of time, which allows to investigate the large time behavior of the solutions. We focus on the Namah-Roquejoffre setting whose typical example is the Eikonal equation. Contrary to the classical time derivative case α = 1, the convergence of the solution on the so-called projected Aubry set, which is an important step to catch the large time behavior, is not straightforward. Indeed, a function with nonpositive Caputo derivative for all time does not necessarily converge; we provide such a counterexample. However, we establish partial results of convergence under some geometrical assumptions

Optimal Monetary Provisions and Risk Aversion in Plural Form Franchise Networks A Model of Incentives with Heterogeneous Agents

Existing literature on franchising has extensively studied the presence of plural form distribution networks, where two types of vertical relationships-integration versus franchising-co-exist. However, despite the importance of monetary provisions in franchise contracts, their definition in the case of plural form networks had not been addressed. In this paper, we focus more precisely on the " share parameters " in integrated (company-owned retail outlet) and decentralized (franchised outlet) vertical contracts, respectively the commission rate and the royalty rate. We develop an agency model of payment mechanism in a two-sided moral hazard context, with one principal and two heterogenous agents distinguished by different levels of risk aversion. We define the optimal monetary provisions, and demonstrate that even in the case of segmented markets, with no correlation between demand shocks, the two rates (commission rate, royalty rate) are negatively interrelated

Some existence and regularity results for a non-local transport-diffusion equation with fractional derivatives in time and space

We study the existence of global weak solutions of a nonlinear transport-diffusion equation with a fractional derivative in the time variable and under some extra hypotheses, we also study some regularity properties for this type of solutions. In the system considered here, the diffusion operator is given by a fractional Laplacian and the nonlinear drift is assumed to be divergence free and it is assumed to satisfy some general stability and boundedness properties in Lebesgue spaces

Some existence and regularity results for a non-local transport-diffusion equation with fractional derivatives in time and space

We study the existence of global weak solutions of a nonlinear transport-diffusion equation with a fractional derivative in the time variable and under some extra hypotheses, we also study some regularity properties for this type of solutions. In the system considered here, the diffusion operator is given by a fractional Laplacian and the nonlinear drift is assumed to be divergence free and it is assumed to satisfy some general stability and boundedness properties in Lebesgue spaces

Some existence and regularity results for a non-local transport-diffusion equation with fractional derivatives in time and space

We study the existence of global weak solutions of a nonlinear transport-diffusion equation with a fractional derivative in the time variable and under some extra hypotheses, we also study some regularity properties for this type of solutions. In the system considered here, the diffusion operator is given by a fractional Laplacian and the nonlinear drift is assumed to be divergence free and it is assumed to satisfy some general stability and boundedness properties in Lebesgue spaces

SOME RESULTS FOR THE LARGE TIME BEHAVIOR OF HAMILTON-JACOBI EQUATIONS WITH CAPUTO TIME DERIVATIVE

International audienceWe obtain some Hölder regularity estimates for an Hamilton-Jacobi with fractional time derivative of order α ∈ (0, 1) cast by a Caputo derivative. The Hölder seminorms are independent of time, which allows to investigate the large time behavior of the solutions. We focus on the Namah-Roquejoffre setting whose typical example is the Eikonal equation. Contrary to the classical time derivative case α = 1, the convergence of the solution on the so-called projected Aubry set, which is an important step to catch the large time behavior, is not straightforward. Indeed, a function with nonpositive Caputo derivative for all time does not necessarily converge; we provide such a counterexample. However, we establish partial results of convergence under some geometrical assumptions