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Matthieu Alfaro

Publications and source records attributed to Matthieu Alfaro.

54 records · Page 3Linked to original sources
arXiv2015

Asymptotic analysis of a monostable equation in periodic media

We consider a multidimensional monostable reaction-diffusion equation whose nonlinearity involves periodic heterogeneity. This serves as a model of invasion for a population facing spatial heterogeneities. As a rescaling parameter tends to zero, we prove the convergence to a limit interface, whose motion is governed by the minimal speed (in each direction) of the underlying pulsating fronts. This dependance of the speed on the (moving) normal direction is in contrast with the homogeneous case and makes the analysis quite involved. Key ingredients are the recent improvement \cite{A-Gil} of the well-known spreading properties \cite{Wein02}, \cite{Ber-Ham-02}, and the solution of a Hamilton-Jacobi equation.

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arXiv2015

Varying the direction of propagation in reaction-diffusion equations in periodic media

We consider a multidimensional reaction-diffusion equation of either ignition or monostable type, involving periodic heterogeneity, and analyze the dependence of the propagation phenomena on the direction. We prove that the (minimal) speed of the underlying pulsating fronts depends continuously on the direction of propagation, and so does its associated profile provided it is unique up to time shifts. We also prove that the spreading properties \cite{Wein02} are actually uniform with respect to the direction.

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arXiv2014

Explicit solutions for replicator-mutator equations: extinction vs. acceleration

We consider a class of nonlocal reaction-diffusion problems, referred to as replicator-mutator equations in evolutionary genetics. By using explicit changes of unknown function, we show that they are equivalent to the heat equation and, therefore, compute their solution explicitly. Based on this, we then prove that, in the case of beneficial mutations in asexual populations, solutions dramatically depend on the tails of the initial data: they can be global, become extinct in finite time or, even, be defined for no positive time. In the former case, we prove that solutions are accelerating, and in many cases converge for large time to some universal Gaussian profile. This sheds light on the biological relevance of such models.

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arXiv2013

Convergence of a mass conserving Allen-Cahn equation whose Lagrange multiplier is nonlocal and local

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.

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arXiv2013

Bistable travelling waves for nonlocal reaction diffusion equations

We are concerned with travelling wave solutions arising in a reaction diffusion equation with bistable and nonlocal nonlinearity, for which the comparison principle does not hold. Stability of the equilibrium $u\equiv 1$ is not assumed. We construct a travelling wave solution connecting 0 to an unknown steady state, which is "above and away", from the intermediate equilibrium. For focusing kernels we prove that, as expected, the wave connects 0 to 1. Our results also apply readily to the nonlocal ignition case.

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arXiv2012

Travelling waves in a nonlocal reaction-diffusion equation as a model for a population structured by a space variable and a phenotypical trait

We consider a nonlocal reaction-diffusion equation as a model for a population structured by a space variable and a phenotypical trait. To sustain the possibility of invasion in the case where an underlying principal eigenvalue is negative, we investigate the existence of travelling wave solutions. We identify a minimal speed $c^*>0$, and prove the existence of waves when $c\geq c^*$ and the non existence when $0\leq c

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arXiv2012

Propagating interface in a monostable reaction-diffusion equation with time delay

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.

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arXiv2012

Rapid travelling waves in the nonlocal Fisher equation connect two unstable states

In this note, we give a positive answer to a question addressed in \cite{Nad-Per-Tan}. Precisely we prove that, for any kernel and any slope at the origin, there do exist travelling wave solutions (actually those which are "rapid") of the nonlocal Fisher equation that connect the two homogeneous steady states 0 (dynamically unstable) and 1. In particular this allows situations where 1 is unstable in the sense of Turing. Our proof does not involve any maximum principle argument and applies to kernels with {\it fat tails}.

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arXiv2011

On the validity of formal asymptotic expansions in Allen-Cahn equation and FitzHugh-Nagumo system with generic initial data

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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arXiv2011

Interface dynamics of the porous medium equation with a bistable reaction term

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.

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arXiv2011

Convergence to a propagating front in a degenerate Fisher-KPP equation with advection

We consider a Fisher-KPP equation with density-dependent diffusion and advection, arising from a chemotaxis-growth model. We study its behavior as a small parameter, related to the thickness of a diffuse interface, tends to zero. We analyze, for small times, the emergence of transition layers induced by a balance between reaction and drift effects. Then we investigate the propagation of the layers. Convergence to a free-boundary limit problem is proved and a sharp estimate of the thickness of the layers is provided.

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arXiv2010

Sharp interface limit of the Fisher-KPP equation when initial data have slow exponential decay

We investigate the singular limit, as $\ep \to 0$, of the Fisher equation $\partial_t u=\ep Δu + \ep ^{-1}u(1-u)$ 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.

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arXiv2009

Sharp interface limit of the Fisher-KPP equation

We investigate the singular limit, as $\ep \to 0$, of the Fisher equation $\partial_t u=\ep Δu + \ep ^{-1}u(1-u)$ 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.

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arXiv2009

The singular limit of the Allen-Cahn equation and the FitzHugh-Nagumo system

We consider an Allen-Cahn type equation with a bistable nonlinearity associated to a double-well potential whose well-depths can be slightly unbalanced, and where the coefficient of the nonlinear reaction term is very small. Given rather general initial data, we perform a rigorous analysis of both the generation and the motion of interface. More precisely we show that the solution develops a steep transition layer within a small time, and we present an optimal estimate for its width. We then consider a class of reaction-diffusion systems which includes the FitzHugh-Nagumo system as a special case. Given rather general initial data, we show that the first component of the solution vector develops a steep transition layer and that all the results mentioned above remain true for this component.

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arXiv2009

The Singular Limit of a Chemotaxis-Growth System with General Initial Data

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.

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arXiv2009

Generation, motion and thickness of transition layers for a nonlocal Allen-Cahn equation

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).

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arXiv2009

Motion by anisotropic mean curvature as sharp interface limit of an inhomogeneous and anisotropic Allen-Cahn equation

We study the singular limit of a spatially inhomogeneous and anisotropic reaction-diffusion equation. We use a Finsler metric related to the anisotropic diffusion term and work in relative geometry. We prove a weak comparison principle and perform an analysis of both the generation and the motion of interfaces. The limit problem involves motion by anisotropic mean curvature. We also prove an optimal estimate of the thickness of the transition layer.

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