18 results on '"Matthieu Alfaro"'
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2. Quantifying the Threshold Phenomenon for Propagation in Nonlocal Diffusion Equations
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Matthieu Alfaro, Arnaud Ducrot, and Hao Kang
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Computational Mathematics ,Applied Mathematics ,Analysis - Published
- 2023
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3. Lotka–Volterra competition-diffusion system: the critical competition case
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Matthieu Alfaro and Dongyuan Xiao
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Applied Mathematics ,Analysis - Published
- 2023
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4. Populations facing a nonlinear environmental gradient: Steady states and pulsating fronts
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Matthieu Alfaro and Gwenaël Peltier
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010101 applied mathematics ,Applied Mathematics ,Modeling and Simulation ,010102 general mathematics ,0101 mathematics ,01 natural sciences - Abstract
We consider a population structured by a space variable and a phenotypical trait, submitted to dispersion, mutations, growth and nonlocal competition. This population is facing an environmental gradient: to survive at location [Formula: see text], an individual must have a trait close to some optimal trait [Formula: see text]. Our main focus is to understand the effect of a nonlinear environmental gradient. We thus consider a nonlocal parabolic equation for the distribution of the population, with [Formula: see text], [Formula: see text]. We construct steady states solutions and, when [Formula: see text] is periodic, pulsating fronts. This requires the combination of rigorous perturbation techniques based on a careful application of the implicit function theorem in rather intricate function spaces. To deal with the phenotypic trait variable [Formula: see text] we take advantage of a Hilbert basis of [Formula: see text] made of eigenfunctions of an underlying Schrödinger operator, whereas to deal with the space variable [Formula: see text] we use the Fourier series expansions. Our mathematical analysis reveals, in particular, how both the steady states solutions and the fronts (speed and profile) are distorted by the nonlinear environmental gradient, which are important biological insights.
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- 2021
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5. Evolution and spread of multi-adapted pathogens in a spatially heterogeneous environment
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Quentin Griette, Matthieu Alfaro, Gaël Raoul, and Sylvain Gandon
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The emergence and the spread of multi-adapted pathogens is often viewed as a slow process resulting from the incremental accumulation of single adaptations. In bacteria, for instance, multidrug resistance to antibiotics may result from the sequential acquisition of single drug resistance to different antibiotics. In phytopathogens, the ability to infect different resistant varieties of crops may also result from the accumulation of distinct virulence genes. Here we use a general epidemiological model to analyse the evolution of pathogen adaptations throughout an epidemic spreading in a heterogeneous host population where selection varies periodically in space. This spatially heterogeneous selection may result from the use of different drugs, different vaccines or different crop varieties in agriculture. We study both the transient evolution of pathogen adaptation at the front of the epidemic and the long-term evolution far behind the epidemic front. We identify five different types of epidemic profiles that may arise from different combinations of spatial heterogeneity and the cost of multi-adaptation. In particular, we show that multi-adaptation can drive epidemic spread, while the evolution of single-adaptation may only occur in a second phase, when the pathogen specializes on local selective pressures. Indeed, a generalist pathogen with multiple adaptations can outpace the spread of a coalition of specialist pathogens when selection varies frequently in space. This result is amplified in finite host populations because demographic stochasticty can lead to the extinction of maladapted pathogens specialised to a local selective pressure. Our work has important implications for the management of multiple drugs and vaccines against pathogens but also for the optimal deployment of resistant varieties in agriculture.
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- 2022
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6. Interplay of nonlinear diffusion, initial tails and Allee effect on the speed of invasions
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Thomas Giletti, Matthieu Alfaro, Institut Montpelliérain Alexander Grothendieck (IMAG), Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS), Institut Élie Cartan de Lorraine (IECL), and Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS)
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AMS Subject Classifications: 35K65 ,Acceleration (differential geometry) ,92D25 ,01 natural sciences ,Theoretical Computer Science ,Allee effect ,symbols.namesake ,Mathematics - Analysis of PDEs ,Mathematics (miscellaneous) ,Position (vector) ,Reaction–diffusion system ,35K67 ,FOS: Mathematics ,[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP] ,Nonlinear diffusion ,Statistical physics ,0101 mathematics ,Diffusion (business) ,spreading properties ,Mathematics ,porous medium diffusion ,35B40 ,010102 general mathematics ,acceleration ,heavy tails ,010101 applied mathematics ,Nonlinear system ,reaction-diffusion equations ,fast diffusion ,symbols ,Porous medium ,Analysis of PDEs (math.AP) - Abstract
We focus on the spreading properties of solutions of monostable equations with non-linear diffusion. We consider both the porous medium diffusion and the fast diffusion regimes. Initial data may have heavy tails, which tends to accelerate the invasion phenomenon. On the other hand, the nonlinearity may involve a weak Allee effect, which tends to slow down the process. We study the balance between these three effects (nonlin-ear diffusion, initial tail, KPP nonlinearity/Allee effect), revealing the separation between "no acceleration" and "acceleration". In most of the cases where acceleration occurs, we also give an accurate estimate of the position of the level sets., arXiv admin note: text overlap with arXiv:1505.04626
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- 2020
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7. On the modelling of spatially heterogeneous nonlocal diffusion: deciding factors and preferential position of individuals
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Matthieu Alfaro, Thomas Giletti, Yong-Jung Kim, Gwenaël Peltier, Hyowon Seo, Laboratoire de Mathématiques Raphaël Salem (LMRS), Université de Rouen Normandie (UNIROUEN), Normandie Université (NU)-Normandie Université (NU)-Centre National de la Recherche Scientifique (CNRS), Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), Department of Mathematical Sciences, KAIST, KAIST, Institut Montpelliérain Alexander Grothendieck (IMAG), Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS), Kyung Hee University (KHU), and ANR-20-CE40-0011,DEEV,Modèles intégro-différentiels venant de la biologie évolutive(2020)
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Applied Mathematics ,nonlocal diffusion ,deciding factors ,focusing kernels ,Agricultural and Biological Sciences (miscellaneous) ,shape of steady states ,Diffusion ,Mathematics - Analysis of PDEs ,Modeling and Simulation ,FOS: Mathematics ,Humans ,AMS Subject Classifications : 92B05 (General biology and biomathemat-ics), 45K05 (Integro partial differential equations), 35B36 (Pattern for-mations in context of PDEs) ,[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP] ,heterogeneity ,Analysis of PDEs (math.AP) - Abstract
We develop general heterogeneous nonlocal diffusion models and investigate their connection to local diffusion models by taking a singular limit of focusing kernels. We reveal the link between the two groups of diffusion equations which include both spatial heterogeneity and anisotropy. In particular, we introduce the notion of deciding factors which single out a nonlocal diffusion model and typically consist of the total jump rate and the average jump length. In this framework, we also discuss the dependence of the profile of the steady state solutions on these deciding factors, thus shedding light on the preferential position of individuals.
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- 2022
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8. Density dependent replicator-mutator models in directed evolution
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Matthieu Alfaro and Mario Veruete
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Applied Mathematics ,Context (language use) ,Term (logic) ,Expression (computer science) ,Directed evolution ,Mathematics - Analysis of PDEs ,Mutation (genetic algorithm) ,FOS: Mathematics ,Discrete Mathematics and Combinatorics ,Initial value problem ,Statistical physics ,Diffusion (business) ,Selection (genetic algorithm) ,Analysis of PDEs (math.AP) ,Mathematics - Abstract
We analyze a replicator-mutator model arising in the context of directed evolution [23], where the selection term is modulated over time by the mean-fitness. We combine a Cumulant Generating Function approach [13] and a spatio-temporal rescaling related to the Avron-Herbst formula [1] to give of a complete picture of the Cauchy problem. Besides its well-posedness, we provide an implicit/explicit expression of the solution, and analyze its large time behaviour. As a by product, we also solve a replicator-mutator model where the mutation coefficient is socially determined, in the sense that it is modulated by the mean-fitness. The latter model reveals concentration or anti diffusion/diffusion phenomena., 19 pages, 7 figures
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- 2020
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9. Erratum: Quantitative Estimates of the Threshold Phenomena for Propagation in Reaction-Diffusion Equations
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Matthieu Alfaro, Arnaud Ducrot, and Grégory Faye
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Modeling and Simulation ,Analysis - Published
- 2021
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10. Travelling waves for a non-monotone bistable equation with delay: existence and oscillations
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Thomas Giletti, Arnaud Ducrot, and Matthieu Alfaro
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Physics ,education.field_of_study ,Diffusion equation ,Degree (graph theory) ,Bistability ,General Mathematics ,010102 general mathematics ,Population ,01 natural sciences ,010101 applied mathematics ,Nonlinear system ,Traveling wave ,0101 mathematics ,Non monotone ,education ,Constant (mathematics) ,Mathematical physics - Abstract
We consider a bistable ($0\textless{}\theta\textless{}1$ being the three constant steady states) delayed reaction diffusion equation, which serves as a model in population dynamics. The problem does not admit any comparison principle. This prevents the use of classical technics and, as a consequence, it is far from obvious to understand the behaviour of a possible travelling wave in $+\infty$. Combining refined {\it a priori} estimates and a Leray Schauder topological degree argument, we construct a travelling wave connecting 0 in $-\infty$ to \lq\lq something" which is strictly above the unstable equilibrium $\theta$ in $+\infty$. Furthemore, we present situations (additional bound on the nonlinearity or small delay) where the wave converges to 1 in $+\infty$, whereas the wave is shown to oscillate around 1 in $+\infty$ when, typically, the delay is large.
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- 2017
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11. Propagation phenomena in monostable integro-differential equations: Acceleration or not?
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Jérôme Coville, Matthieu Alfaro, Institut Montpelliérain Alexander Grothendieck (IMAG), Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS), Biostatistique et Processus Spatiaux (BioSP), Institut National de la Recherche Agronomique (INRA), 'ANR JCJC' project MODEVOL ANR-13-JS01-0009, ANR 'DEFI' project NONLOCAL ANR-14-CE25-0013, ANR-13-JS01-0009,MODEVOL,Modèles mathématiques pour la biologie évolutive(2013), ANR-14-CE25-0013,NONLOCAL,Phénomènes de propagation et équations non locales(2014), and Biostatistique et Processus Spatiaux (BIOSP)
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Kernel (set theory) ,Differential equation ,Applied Mathematics ,010102 general mathematics ,Mathematical analysis ,Acceleration (differential geometry) ,01 natural sciences ,010101 applied mathematics ,Nonlinear system ,Mathematics - Analysis of PDEs ,Bounded function ,[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP] ,Initial value problem ,0101 mathematics ,Dispersion (water waves) ,Degeneracy (mathematics) ,Analysis ,Mathematics - Abstract
International audience; We consider the homogeneous integro-differential equation$\partial _t u=J*u-u+f(u)$ with a monostable nonlinearity $f$. Our interest is twofold: we investigate the existence/non existence of travelling waves, and the propagation properties of the Cauchy problem.When the dispersion kernel $J$ is exponentially bounded, travelling waves are known to exist and solutions of the Cauchy problem typically propagate at a constant speed \cite{Schumacher1980}, \cite{Weinberger1982}, \cite{Carr2004}, \cite{Coville2007a}, \cite{Coville2008a}, \cite{Yagisita2009}. %When the dispersion kernel $J$ is exponentially bounded, travelling waves are known to exist when $f$ belongs to one of the three main class of non-linearities (bistable, ignition or monostable), and solutions of the Cauchy problem typically propagate at a constant speed \cite{Schumacher1980}, \cite{Wei-82},\cite{Bates1997},\cite{Chen1997}, \cite{Carr2004}, \cite{Coville2007a}, \cite{Coville2008a}, \cite{Yagisita2009,Yagisita2009a}. On the other hand, when the dispersion kernel $J$ has heavy tails and the non-linearity $f$ is non degenerate, i.e $f'(0)>0$, travelling waves do not exist and solutions of the Cauchy problem propagate by accelerating \cite{Medlock2003}, \cite{Yagisita2009}, \cite{Garnier2011}. For a general monostable non-linearity, a dichotomy between these two types of propagation behaviour is still not known. The originality of our work is to provide such dichotomy by studying the interplay between the tails of the dispersion kernel and the Allee effect induced by the degeneracy of $f$, i.e. $f'(0)=0$. First, for algebraic decaying kernels, we prove the exact separation between existence and non existence of travelling waves. This in turn provides the exact separation between non acceleration and acceleration in the Cauchy problem. In the latter case, we provide a first estimate of the position of the level sets of the solution.
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- 2017
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12. Superexponential growth or decay in the heat equation with a logarithmic nonlinearity
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Matthieu Alfaro, Rémi Carles, Institut Montpelliérain Alexander Grothendieck (IMAG), and Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS)
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Logarithm ,Applied Mathematics ,Gaussian ,010102 general mathematics ,Mathematical analysis ,Double exponential function ,Ode ,01 natural sciences ,010101 applied mathematics ,Nonlinear system ,symbols.namesake ,Mathematics - Analysis of PDEs ,Line (geometry) ,symbols ,[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP] ,Heat equation ,Uniqueness ,0101 mathematics ,Analysis ,Mathematics - Abstract
We consider the heat equation with a logarithmic nonlinearity, on thereal line. For a suitable sign in front of the nonlinearity, weestablish the existence and uniqueness of solutions of the Cauchyproblem, for a well-adapted class of initial data. Explicitcomputations in the case of Gaussian data lead to various scenariiwhich are richer than the mere comparison with the ODE mechanism,involving (like in the ODE case) double exponential growth or decayfor large time. Finally, we prove that such phenomena remain, in the case of compactlysupported initial data., Comment: 14 pages
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- 2017
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13. Explicit solutions in evolutionary genetics and applications
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Rémi Carles and Matthieu Alfaro
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Human evolutionary genetics ,Calculus ,Applied mathematics ,Heat equation ,General Medicine ,Finite time ,Mathematics - Abstract
We show that the solution to a nonlocal reaction–diffusion equation, present in evolutionary genetics, can be related explicitly to the solution of the heat equation with the same initial data. As a consequence, we show different possible scenario for the solution: it can be either well-defined for all time, or become extinct in finite time, or even be defined for no positive time. In the former case, we give the leading-order asymptotic behavior of the solution for large time, which is universal.
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- 2015
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14. On the validity of formal asymptotic expansions in Allen-Cahn equation and FitzHugh-Nagumo system with generic initial data
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Hiroshi Matano, Matthieu Alfaro, Institut de Mathématiques et de Modélisation de Montpellier (I3M), Centre National de la Recherche Scientifique (CNRS)-Université Montpellier 2 - Sciences et Techniques (UM2)-Université de Montpellier (UM), Graduate School of Mathematical Sciences (GSMS), and The University of Tokyo (UTokyo)
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Large class ,Singular perturbation ,Applied Mathematics ,010102 general mathematics ,Principal (computer security) ,Type (model theory) ,Fitzhugh nagumo ,01 natural sciences ,Term (time) ,010101 applied mathematics ,Mathematics - Analysis of PDEs ,FOS: Mathematics ,[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP] ,Discrete Mathematics and Combinatorics ,Applied mathematics ,0101 mathematics ,Asymptotic expansion ,Allen–Cahn equation ,Analysis of PDEs (math.AP) ,Mathematics - Abstract
International audience; 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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- 2012
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15. Sharp interface limit of the Fisher-KPP equation
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Arnaud Ducrot, Matthieu Alfaro, Institut de Mathématiques et de Modélisation de Montpellier (I3M), Centre National de la Recherche Scientifique (CNRS)-Université Montpellier 2 - Sciences et Techniques (UM2)-Université de Montpellier (UM), Institut de Mathématiques de Bordeaux (IMB), and Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)
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Physics ,Singular perturbation ,Applied Mathematics ,media_common.quotation_subject ,010102 general mathematics ,Mathematical analysis ,Motion (geometry) ,Fisher equation ,General Medicine ,Infinity ,Space (mathematics) ,01 natural sciences ,010101 applied mathematics ,Mathematics - Analysis of PDEs ,FOS: Mathematics ,Traveling wave ,Sharp interface ,[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP] ,Limit (mathematics) ,0101 mathematics ,Analysis ,Analysis of PDEs (math.AP) ,media_common - Abstract
We investigate the singular limit, as $\varepsilon\to 0$, of the Fisher equation $\partial_t u=\varepsilon\Delta u + \varepsilon^{-1}u(1-u)$ in the whole space. We consider initial data with compact support plus, possibly, perturbations very small as $||x|| \to \infty$. 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. Moreover, we obtain a new 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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- 2012
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16. Convergence rate of the Allen-Cahn equation to generalized motion by mean curvature
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Jérôme Droniou, Hiroshi Matano, Matthieu Alfaro, Institut de Mathématiques et de Modélisation de Montpellier (I3M), Centre National de la Recherche Scientifique (CNRS)-Université Montpellier 2 - Sciences et Techniques (UM2)-Université de Montpellier (UM), Graduate School of Mathematical Sciences (GSMS), and The University of Tokyo (UTokyo)
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Off phenomenon ,Mean curvature ,010102 general mathematics ,Mathematical analysis ,Motion (geometry) ,01 natural sciences ,010101 applied mathematics ,Mathematics (miscellaneous) ,Rate of convergence ,[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP] ,0101 mathematics ,Computer Science::Data Structures and Algorithms ,Allen–Cahn equation ,Mathematical physics ,Mathematics - Abstract
We investigate the singular limit, as \({\varepsilon \to 0}\), of the Allen-Cahn equation \({u^\varepsilon_t=\Delta u^\varepsilon+\varepsilon^{-2}f(u^\varepsilon)}\), with f a balanced bistable nonlinearity. We consider rather general initial data u0 that is independent of \({{\varepsilon}}\). It is known that this equation converges to the generalized motion by mean curvature — in the sense of viscosity solutions—defined by Evans, Spruck and Chen, Giga, Goto. However, the convergence rate has not been known. We prove that the transition layers of the solutions \({u^{\varepsilon}}\) are sandwiched between two sharp “interfaces” moving by mean curvature, provided that these “interfaces” sandwich at t = 0 an \({\mathcal O({\varepsilon}|\,{\rm ln}\,{\varepsilon}|)}\) neighborhood of the initial layer. In some special cases, which allow both extinction and pinches off phenomenon, this enables to obtain an \({\mathcal O({\varepsilon}|\,{\rm ln}\,{\varepsilon}|)}\) estimate of the location and the thickness measured in space-time of the transition layers. A result on the regularity of the generalized motion by mean curvature is also provided in the Appendix.
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- 2011
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17. Sharp interface limit of the Fisher-KPP equation when initial data have slow exponential decay
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Matthieu Alfaro, Arnaud Ducrot, Institut de Mathématiques et de Modélisation de Montpellier (I3M), Centre National de la Recherche Scientifique (CNRS)-Université Montpellier 2 - Sciences et Techniques (UM2)-Université de Montpellier (UM), Tools of automatic control for scientific computing, Models and Methods in Biomathematics (ANUBIS), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS), Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS), Université Montpellier 2 - Sciences et Techniques (UM2)-Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Inria Bordeaux - Sud-Ouest, and Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)
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Applied Mathematics ,Mathematical analysis ,010102 general mathematics ,Constant speed ,Fisher equation ,Motion (geometry) ,Space (mathematics) ,01 natural sciences ,010101 applied mathematics ,Mathematics - Analysis of PDEs ,FOS: Mathematics ,Sharp interface ,Traveling wave ,[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP] ,Discrete Mathematics and Combinatorics ,Limit (mathematics) ,Exponential decay ,0101 mathematics ,Analysis of PDEs (math.AP) ,Mathematics - Abstract
International audience; 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.
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- 2011
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18. Varying the direction of propagation in reaction-diffusion equations in periodic media
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Thomas Giletti, Matthieu Alfaro, Institut de Mathématiques et de Modélisation de Montpellier (I3M), Centre National de la Recherche Scientifique (CNRS)-Université Montpellier 2 - Sciences et Techniques (UM2)-Université de Montpellier (UM), Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), Institut de Mathématiques et de Modélisation de Montpellier ( I3M ), Université Montpellier 2 - Sciences et Techniques ( UM2 ) -Université de Montpellier ( UM ) -Centre National de la Recherche Scientifique ( CNRS ), Equations aux dérivées partielles ( EDP ), Institut Élie Cartan de Lorraine ( IECL ), and Université de Lorraine ( UL ) -Centre National de la Recherche Scientifique ( CNRS ) -Université de Lorraine ( UL ) -Centre National de la Recherche Scientifique ( CNRS )
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Statistics and Probability ,[ MATH.MATH-GM ] Mathematics [math]/General Mathematics [math.GM] ,Type (model theory) ,01 natural sciences ,law.invention ,Mathematics - Analysis of PDEs ,[ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP] ,pulsating traveling front ,[MATH.MATH-GM]Mathematics [math]/General Mathematics [math.GM] ,law ,0103 physical sciences ,Reaction–diffusion system ,FOS: Mathematics ,[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP] ,0101 mathematics ,spreading properties ,35K57, 35B10 ,monostable nonlinearity ,periodic media ,Physics ,Applied Mathematics ,010102 general mathematics ,Mathematical analysis ,General Engineering ,ignition nonlin earity ,Computer Science Applications ,Ignition system ,Multivibrator ,010307 mathematical physics ,Analysis of PDEs (math.AP) - Abstract
International audience; We consider a multidimensional reaction-diffusion equation of either ignition or monostable type, involving periodic heterogeneity, and analyze the dependence of thepropagation phenomena on the direction. We prove that the(minimal) speed of the underlying pulsating fronts dependscontinuously on the direction of propagation, and so does itsassociated profile provided it is unique up to time shifts. Wealso prove that the spreading properties \cite{Wein02} areactually uniform with respect to thedirection.
- Published
- 2015
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