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Search Results: 1 - 10 of 2038 matches for " Matthieu Alfaro "
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Generation, motion and thickness of transition layers for a nonlocal Allen-Cahn equation
Matthieu Alfaro
Mathematics , 2009,
Abstract: We investigate the behavior, as a small parameter tends to zero, of a nonlocal Allen-Cahn equation. Given a rather general initial data, we perform a rigorous analysis of both the generation and the motion of interface, and obtain a new estimate for its thickness. The limit problem involves motion by mean curvature together with a nonlocal effect (which allows the possibility of nontrivial stationary state).
The Singular Limit of a Chemotaxis-Growth System with General Initial Data
Matthieu Alfaro
Mathematics , 2009,
Abstract: We study the singular limit of a system of partial differential equations which is a model for an aggregation of amoebae subjected to three effects: diffusion, growth and chemotaxis. The limit problem involves motion by mean curvature together with a nonlocal drift term. We consider rather general initial data. We prove a generation of interface property and study the motion of interfaces. We also obtain an optimal estimate of the thickness and the location of the transition layer that develops.
Slowing Allee effect vs. accelerating heavy tails in monostable reaction diffusion equations
Matthieu Alfaro
Mathematics , 2015,
Abstract: We focus on the spreading properties of solutions of monostable reaction-diffusion equations. Initial data are assumed to have heavy tails, which tends to accelerate the invasion phenomenon. On the other hand, the nonlinearity involves a weak Allee effect, which tends to slow down the process. We study the balance between the two effects. For algebraic tails, we prove the exact separation between "no acceleration and acceleration". This implies in particular that, for tails exponentially unbounded but lighter than algebraic , acceleration never occurs in presence of an Allee effect. This is in sharp contrast with the KPP situation [19]. When algebraic tails lead to acceleration despite the Allee effect, we also give an accurate estimate of the position of the level sets.
Sharp interface limit of the Fisher-KPP equation when initial data have slow exponential decay
Matthieu Alfaro,Arnaud Ducrot
Mathematics , 2010,
Abstract: We investigate the singular limit, as $\ep \to 0$, of the Fisher equation $\partial_t u=\ep \Delta u + \ep ^{-1}u(1-u)$ in the whole space. We consider initial data with compact support plus perturbations with {\it slow exponential decay}. We prove that the sharp interface limit moves by a constant speed, which dramatically depends on the tails of the initial data. By performing a fine analysis of both the generation and motion of interface, we provide a new estimate of the thickness of the transition layers.
Interface dynamics of the porous medium equation with a bistable reaction term
Matthieu Alfaro,Danielle Hilhorst
Mathematics , 2011,
Abstract: We consider a degenerate partial differential equation arising in population dynamics, namely the porous medium equation with a bistable reaction term. We study its asymptotic behavior as a small parameter, related to the thickness of a diffuse interface, tends to zero. We prove the rapid formation of transition layers which then propagate. We prove the convergence to a sharp interface limit whose normal velocity, at each point, is that of the underlying degenerate travelling wave.
Propagating interface in a monostable reaction-diffusion equation with time delay
Matthieu Alfaro,Arnaud Ducrot
Mathematics , 2012,
Abstract: We consider a monostable time-delayed reaction-diffusion equation arising from population dynamics models. We let a small parameter tend to zero and investigate the behavior of the solutions. We construct accurate lower barriers --- by using a non standard bistable approximation of the monostable problem--- and upper barriers. As a consequence, we prove the convergence to a propagating interface.
Convergence of a mass conserving Allen-Cahn equation whose Lagrange multiplier is nonlocal and local
Matthieu Alfaro,Pierre Alifrangis
Mathematics , 2013,
Abstract: We consider the mass conserving Allen-Cahn equation proposed in \cite{Bra-Bre}: the Lagrange multiplier which ensures the conservation of the mass contains not only nonlocal but also local effects (in contrast with \cite{Che-Hil-Log}). As a parameter related to the thickness of a diffuse internal layer tends to zero, we perform formal asymptotic expansions of the solutions. Then, equipped with these approximate solutions, we rigorously prove the convergence to the volume preserving mean curvature flow, under the assumption that classical solutions of the latter exist. This requires a precise analysis of the error between the actual and the approximate Lagrange multipliers.
Generation of interface for an Allen-Cahn equation with nonlinear diffusion
Matthieu Alfaro,Danielle Hilhorst
Mathematics , 2009,
Abstract: In this note, we consider a nonlinear diffusion equation with a bistable reaction term arising in population dynamics. Given a rather general initial data, we investigate its behavior for small times as the reaction coefficient tends to infinity: we prove a generation of interface property.
Sharp interface limit of the Fisher-KPP equation
Matthieu Alfaro,Arnaud Ducrot
Mathematics , 2009,
Abstract: We investigate the singular limit, as $\ep \to 0$, of the Fisher equation $\partial_t u=\ep \Delta u + \ep ^{-1}u(1-u)$ in the whole space. We consider initial data with compact support plus, possibly, perturbations very small as $\Vert x \Vert \to \infty$. By proving both generation and motion of interface properties, we show that the sharp interface limit moves by a constant speed, which is the minimal speed of some related one-dimensional travelling waves. We obtain an estimate of the thickness of the transition layers. We also exhibit initial data "not so small" at infinity which do not allow the interface phenomena.
On the validity of formal asymptotic expansions in Allen-Cahn equation and FitzHugh-Nagumo system with generic initial data
Matthieu Alfaro,Hiroshi Matano
Mathematics , 2011,
Abstract: Formal asymptotic expansions have long been used to study the singularly perturbed Allen-Cahn type equations and reaction-diffusion systems, including in particular the FitzHugh-Nagumo system. Despite their successful role, it has been largely unclear whether or not such expansions really represent the actual profile of solutions with rather general initial data. By combining our earlier result and known properties of eternal solutions of the Allen-Cahn equation, we prove validity of the principal term of the formal expansions for a large class of solutions.
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