c ∗∗ = f ′ (0)/2α. All these results will be formally established in Chapter 1. Additionally we
recall the earlier work in the case α ∈ (0, 1) by Berestycki, Roquejoffre and Rossi , where it is proved that there is invasion of the unstable state by the stable one, also in , the authors derive a class of integro-differential reaction-diffusion equations from simple principles. They then prove an approximation result for the first eigenvalue of linear integro-differential opera- tors of the fractionaldiffusion type, they also prove the convergence of solutions of fractional evolution problem to the steady state solution when the time tends to infinity. For a large class of nonlinearities, Engler  has proved that the invasion has unbounded speed. For another type of integro-differential equations Garnier  also establishes that the position of the level sets moves exponentially in time for algebraically decaying dispersal kernels. And in a recent paper Stan and V´azquez  study the propagation properties of nonnegative and bounded solutions of the class of reaction-diffusion equations with nonlinear fractionaldiffusion.
We propose and study a posteriori error estimates for convection–diffusion–reaction prob- lems with inhomogeneous and anisotropic diffusion approximated by weighted interior- penalty discontinuous Galerkin methods. Our twofold objective is to derive estimates without undetermined constants and to analyze carefully the robustness of the estimates in singularly perturbed regimes due to dominant convection or reaction. We first derive locally computable estimates for the error measured in the energy (semi)norm. These estimates are evaluated using H(div, Ω)-conforming diffusive and convective flux recon- structions, thereby extending previous work on pure diffusionproblems. The resulting estimates are semi-robust in the sense that local lower error bounds can be derived us- ing suitable cutoff functions of the local P´ eclet and Damk¨ ohler numbers. Fully robust estimates are obtained for the error measured in an augmented norm consisting of the energy (semi)norm, a dual norm of the skew-symmetric part of the differential operator, and a suitable contribution of the interelement jumps of the discrete solution. Numerical experiments are presented to illustrate the theoretical results.
Part I of the thesis has been devoted to a rigorous analysis of the asymptotic location of the level sets of the solution to two different problems.
In Chapter 1, we have applied our method on a Fisher-KPP model in periodic media with fractionaldiffusion. We have been able to construct precise explicit subsolutions and supersolutions. Thus, we have proved that the transition between the unstable state and the stable one occurs exponentially fast in time, and we have obtained the precise exponent that appears in this exponential speed of propagation. This has led to the proof of the convergence of the solution to its stationary state on a set that expands with an exponential in time speed. Numerical simulations have been carried out to understand the dependence of the speed of propagation on the initial condition at lower order in time. Although the different numerical results, done for the homogeneous model in dimension two, have given a precise idea of what is happening, a mathematical proof should be undertaken. Indeed, it seems that there is a symmetrisation of the solution, in the sense of Jones in [ 77 ]. Proving this observation requires an estimate of the gradient of the solution, which is not done in this thesis. This geometric result of symmetrisation could also be studied in periodic media. Moreover, as suggested by numerical investigations, it seems that the diffusive term of the reaction-diffusion equation only plays a role for small times. It would be interesting to show it rigorously. Finally, one could think of further perspectives. A first one consists in getting similar results for integro-differential equations, and thus obtaining more precise asymptotics as the ones proved in [ 65 ]. More general heterogeneous media might also be analysed, media for which the notion of generalised eigenvalues is needed.
The GLS formulation was developed as a generalization of the SUPG method, but has recently seen extensive developments in the field of time-harmonic acoustics problems. The singular diffu- sion problem is a particular case corresponding to evanescent waves. Harari and Hughes  and Harari et al.  applied GLS to the particular case of a constant source term and Dirichlet boundary conditions. For such conditions and assuming very low diffusivity, GLS and GGLS lead to identical solutions and both improve significantly over the Galerkin method. Harari and Hughes  and Valentin and Franca  combined GLS and GGLS into GLS/gradient least-squares (GLSGLS) in order to provide stability for both convective-dominated problems and problems in which the singular behavior comes from a large undifferentiated term compared with the second-order term. Applications of GLS and GGLS show superiority of one or the other depending on model problem, choice of stability parameters and discretization used. Harari  concludes that GLS is superior to GGLS for non-uniform meshes. However, Harari comes to this conclusion solely on a stability analysis for internal nodes and without considering the effect of the boundary conditions. No applications are presented in the paper to support the stability analysis and the ensuing superiority of GLS over GGLS. Harari and Haham  compared Galerkin, GLS and GGLS formulations for the solution to elastic wave problems. The paper concludes that the GLS method is not sufficient to correctly handle the directionality of the solution. Alternatively, they mention that the GGLS method yields high accuracy in both magnitude and phase for all directions of propagation. Ilinca and H´etu  discussed the issue of global conservation for GLS and GGLS in the context of the singular diffusion problem. They have shown that although GGLS is globally conservative, GLS is not. This causes GLS to produce wrong results when Robin boundary conditions are imposed. In a recent work, Hauke et al.  propose to solve advection–diffusion–reaction equations by a subgrid scale/gradient subgrid scale method, which combines two types of stabilization [8, 9]. The resulting formulation provides nodally exact solutions in the one-dimensional case and it recovers the SUPG in the advective–diffusive limit and for linear elements.
More than 50 years have passed since Montroll & Weiss  introduced the continuous-time random walk (CTRW) formalism to account for a broad variety of anomalous diffusion mechanisms. Despite countless achievements in the past decades, non-Gaussian diffusion is still topical in many different fields such as statistical physics, condensed matter physics or biology – as testified by the present special issue of EPJB. In the present paper, we propose an original application of fractionaldiffusion to describe the dynamics of supply and demand in financial markets. In the past few years, the concave nature of the impact of traded volume on asset prices – coined the “square-root impact law” – has made its way among the most firmly established stylized facts of modern finance [2,3,4,5,6,7]. Several attempts have been made to build theoretical mod- els that account for non-linear market impact, see e.g. [8,9]. Following the ideas of T´oth et al. , the notion of a locally linear “latent” order-book model (LLOB) was introduced in [10,11]. The latter model builds upon cou- pled continuous reaction-diffusion equations for the dy- namics of the bid and the ask sides of the latent order book  and allows one to compute the price trajectory conditioned to any execution profiles. In the slow execu- tion limit, the LLOB model was shown to match the lin- ear propagator model that relates past order flow to price changes through a power law decaying kernel.
In the context of biological complex systems multi- agent simulation, we present an interaction-agent model for reaction-diffusionproblems that enables interaction with the simulation during the execution, and we establish a mathematical validation for our model. We use two types of interaction-agents: on one hand, in a chemical reactor with no spatial dimension -e.g. a cell-, a reaction-agent rep- resents an autonomous chemical reaction between several reactants, and modifies the concentration of reaction prod- ucts. On the other hand, we use interface-agents in order to take into account the spatial dimension that appears with diffusion : interface-agents achieve the matching transfer of reactants between cells. This approach, where the simula- tion engine makes agents intervene in a chaotic and asyn- chronous way, is an alternative to the classical model - which is not relevant when the limits conditions are fre- quently modified- based on partial derivative equations. We enounciate convergence results for our interaction-agent methods, and illustrate our model with an example about coagulation inside a blood vessel.
handles the reaction part of the evolution. As already mentioned, the WFR metric will allow to suitable control both steps in a uniﬁed metric framework. We will ﬁrst state a general convergence result for scalar reaction-diﬀusion equations, and then illustrate on a few particular examples how the general idea can be adapted to treat e.g. prey-predator systems or very degenerate Hele-Shaw diﬀusion problems. In this work we do not focus on optimal results and do not seek full generality, but rather wish to illustrate the eﬃciency of the general approach.
speed c(e) in the direction e is constant. Other proofs of this result, using PDE tools, can be found in  and . In the case of the fractional Laplacian and a constant environment, Cabré and Roquejoffre in  proved the front position is exponential in time (see also for instance  for some heuristic and numerical works predicting such behavior and  for an alternative proof). Then in , Cabré, Coulon and Roquejoffre investigate the speed of propagation in a periodic environment modeled by equation (1) but considering the fractional Laplacian instead of the operator L α . One
 S. Heinze, Large convection limits for KPP fronts, preprint.
 J. Huang, W. Shen, Speeds of spread and propagation for KPP models in time almost and space periodic media, SIAM J. Appl. Dyn. Syst. 8 (2009), 790-821.
 W. Hudson, B. Zinner, Existence of travelling waves for reaction-diffusion equations of Fisher type in periodic media, In: Boundary Value Problems for Functional-Differential Equations, J. Henderson (ed.), World Scientific, 1995, 187-199.
(Dated: September 4, 2016)
For reaction-diffusion processes with at most bimolecular reactants, we derive well-behaved, numerically-tractable, exact Langevin equations that govern a stochastic variable related to the response field in field theory. Using duality relations, we show how the particle number and other quantities of interest can be computed. Our work clarifies long-standing conceptual issues encoun- tered in field theoretical approaches and paves the way to systematic numerical and theoretical analyses of reaction-diffusionproblems.
Remark 3 The result obtained in Theorem 3.2 can be of course generalizes in the case when the generator L is just an operator satisfying a spectral gap inequality on the domain considered Ω. For example one can consider the case of a fractional Laplacian, a p-Laplacian,..., and others. (A problem may remain to prove the existence of a non- negative solution in some of these cases.)
pests are active (here, the adult stage).
Additionally, the reaction-diffusion model can be replaced by other types of models such as individual based models (Gross et al., 1992; Kareiva and Shigesada, 1983; Marsh and Jones, 1988) or integral models (Hamel et al., 2010; Kawasaki and Shigesada, 2007; Kot et al., 1996). The method proposed in Section 3.2.1 for modeling the nest density as a function of adult density and environmental factors is independent of the underlying model for adult dispersal. However, as al- ready emphasized, the choice of the reaction-diffusion approach was encouraged by computational speed advantages and by an observation of the spatial genetic structure of the PPM. It is also note- worthy that reaction-diffusion models are in good agreement with the dispersal properties of some species, at least qualitatively (Murray, 2002; Okubo and Levin, 2002; Shigesada and Kawasaki, 1997; Turchin, 1998); these models can describe constant speed as well as accelerating range ex- pansions (Hamel and Roques, 2010; Roques et al., 2010). Furthermore, recent developments in the field of inverse problems show that most parameters of reaction-diffusion models can be de- termined using only spatially-incomplete measurements of the population density (Cristofol and Roques, 2008; Roques and Cristofol, 2010). Although the measurements in these papers are not of the binary type, nor are they blurred by an observation uncertainty, such a one-to-one and onto relationship between partial measurements of the population density and parameters of the model suggests that reaction-diffusion is a good framework for estimating the effects of environmental covariates.
5.1.1 Setting and main results
Motivations Reaction-diffusion equations have drawn a lot of attention from the mathematical community over the last decades, but most usually in spatially homogeneous setting, while the lit- erature devoted to spatially heterogeneous domains only started developing recently. This growing interest led to many interesting questions regarding the possible effects of spatial heterogeneity on, for instance, the dynamics of the equation, or on optimization and control problems: how do these heterogeneities impact the dynamics or the criteria under consideration? Can the results obtained in the homogeneous case be obtained in the heterogeneous one, which is more relevant for applications? In this article, we study some of these questions and the influence of spatial heterogeneity from the angle of control theory. Some of our proofs and results are, however, of independent interest for reaction-diffusion equations.
Our goal in this paper is to state similar results for coupled parabolic systems of the form ( 1.1 ) with internal control. This framework raises several difficulties. Indeed, for boundary control or equations satisfying a maximum principle, there is an equivalence between nonnegativity of the state and nonnegativity of the control, which is useful as control-constrained problems are better understood in general. This equivalence does not hold anymore for System ( 1.1 ): the state might remain positive even if the control is negative. Moreover, due to the coupling terms, the asymptotic behaviour of the trajectories is difficult to know precisely. This means that we cannot rely on dissipativity or stabilization to a steady state to obtain controllability, with the notable exception of the case where the diffusion matrix D is equal to the identity matrix.
∂ t q + v∇ x q + ∂ y (f (y, S)q) = Λ(y, S)(hqi − q) . (0.1)
Here q(t, x, v, y) denotes the probability density function of bacteria at time t, position x ∈ R d , velocity
v ∈ V with V the sphere (or the ball) with radius V 0 , and the intra-cellular molecular content y ∈ R. The
function f (y, S) takes into account the slowest reaction in the chemotactic signal transduction pathways for a given external effective signal S. The right hand side terms in ( 0.1 ) describes the velocity jump process where Λ(y, S) is the tumbling frequency. The specific forms of f (y, S) and Λ(y, S) depend on different types of bacteria, where a linear cartoon description for f (y, S) is used in [ 11 ] and more sophisticated forms for E.coli chemotaxis have been studied in [ 16 , 20 ]. The frequency Λ(y, S) is determined by the regulation of the flagellar motors by biochemical pathways [ 16 ] and it usually has steep transition with respect to y.
The above described link between X(t) and Eq. (1) in Ω = R d persists in bounded domain provided we consider
boundary conditions equivalent to restrictions imposed to the sample paths of X(t) [21, 39]. However, most boundary problems associated with space-fractional p.d.es remain still open. It is only in specific cases that we actually know slight modifications that transform the sample paths of X(t) into the ones of a closely related random walk whose p.d.f satisfies Eq. (1) and prescribed boundary conditions. This occurs in the simple case of homogeneous Dirichlet conditions [40–43] which we use for checks. Associating Eq. (1) with non-homogeneous Dirichlet conditions re- turns problems whose well-posedness is not assessed: solutions do exist but uniqueness apparently depends on how we interpret these boundary conditions in the case of space-fractional equations . We disregard here Neumann conditions because it is only in too miscellaneous cases that they have been proved to correspond to sample paths transformations compatible with density satisfying Eq. (1). Just note that Neumann conditions for space-fractional
4.2 The rescaled equation
In order to prove the convergence for w ∈ (0, w) in Theorem 2.1, we will first determine the limit of v ε (t, x) := u(t/ε, x/ε) as ε → 0 by using homogenization techniques. To do
so, we follow the ideas developed by Majda and Souganidis in , which are based on the half-limits method. There is indeed a deep link between homogenization problems and spreading properties for reaction-diffusion equations. This link will be discussed in details in our forthcoming work . We also refer to  for a detailed discussion on homogenization problems and on the existence of approximate correctors, a notion which is close to that of generalized principal eigenvalues.
monostable type of nonlinearity, see for example [2, 25, 31, 33]. On the other hand, thanks to the work of Fife and Mc Leod , and Alfaro  we can see that accelerated transitions will never occur when the non-linearity considered is bistable or of ignition type.
In this spirit, in this paper we are interested in propagation acceleration phenomena that are caused by anomalous diffusions such as super diffusions, which plays important roles in various physical, chemical, biological and geological processes. (See, e.g.,  for a brief summary and references therein.) A typical feature of such anomalous diffusions is related to L´evy stochastic processes which may possesses discontinuous ”jumps” in their paths and have long range dispersal, while the standard diffusion is related to the Brownian motion. Analytically, certain L´evy processes (α stable) may be modeled by their infinitesimal generators which are fractional Laplace operators (−∆) s u with 0 < s < 1, whose
general Freidlin-G¨ artner type formula for the spreading speeds of the solutions in any direction. This formula holds under general assumptions on the reaction and for so- lutions emanating from initial conditions with general unbounded support, whereas most of earlier results were concerned with more specific reactions and compactly supported or almost-planar initial conditions. We also prove some results of inde- pendent interest on some conditions guaranteeing the spreading of solutions with large initial support and the link between these conditions and the existence of trav- eling fronts with positive speed. Furthermore, we show some flattening properties of the level sets of the solutions if initially supported on subgraphs. We also investigate the special case of asymptotically conical-shaped initial conditions. For Fisher-KPP equations, we prove some asymptotic one-dimensional symmetry properties for the elements of the Ω-limit set of the solutions, in the spirit of a conjecture of De Giorgi for stationary solutions of Allen-Cahn equations. Lastly, we show some logarithmic- in-time estimates of the lag of the position of the solutions with respect to that of a planar front with minimal speed, for initial conditions which are supported on subgraphs with logarithmic growth at infinity. The proofs use a mix of ODE and PDE methods, as well as some geometric arguments. The paper also contains some related conjectures and open problems.
5. K. Diethelm and A. D. Freed, On the solution of nonlinear fractional order differen- tial equations used in the modeling of viscoplasticity. In: Scientific Computing in Chem- ical Engineering II-Computational Fluid Dynamics, Reaction Engineering and Molecular Properties (F. Keil, W. Mackens, H. Voss, and J. Werther, Eds.), pp. 217–224, Springer- Verlag, Heidelberg, 1999.