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 withfractionaldiffusion. 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 fastin 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.
It now remains to obtain a similar bound for a given 4B0 1 < λ < 1. Such estimate can be obtained by redoing the argument in Section 3.
Acknowledgement. J.Coville carried out this work in the framework of Archim`ede Labex (ANR-11-LABX-0033) and of the A*MIDEX project (ANR-11-IDEX-0001-02), funded by the “Investissements d’Avenir” French Government program managed by the French National Research Agency (ANR). He has also received funding from the ANR DEFI project NONLOCAL (ANR-14-CE25-0013). C. Gui is supported by NSF grants DMS-1601885 and DMS-1901914 and Simons Foundation Award 617072. M. Zhao is supported by the National Natural Science Foundation of China No.11801404.
Such a reaction-diffusion equation is often used in ecology and population dynamics to model spatial propagation phenomena [ 1 ]. In this context, the unknown function u denotes the density of some species, and the parameter α stands for its per capita growth rate. The function χ stands for the motility of individuals, and therefore the heterogeneity means that the motility depends on local environmental conditions. In the past decade, there has been several series of works dedicated to the study of a variation of ( 1.1 ) with constant motility (say χ ≡ 1 to fix the ideas) but with shifting growth rates, that typically reads
should underline the fact that in the fractional case, the speed of propagation does not depend anymore on the direction. They proved that the speed of propagation is exponential in time with a precise exponent depending on a periodic principal eigenvalue.
The objective of this work is to provide an alternative proof of this property using an asymptotic approach known as "approximation of geometric optics". We will be interested in the longtime behavior of the solution n. We demonstrate that in the set n (x, t) | |x| < e d+2α |λ1|t
the unique positive zero of f . In the moving frame with speed c to the right, a traveling front u(t, x) = φ(x − ct) is invariant in time. Such fronts are known to exist if and only if c ≥ 2pD 0 f 0 (0) and they are stable with respect to some natural classes of perturbations,
see e.g. [8, 28, 29, 42, 46]. For heterogeneous equations such as (1.1) in periodic media, standard traveling fronts do not exist in general and the notion of fronts is replaced by the more general one of pulsating fronts . For (1.1), in the case of the existence (and uniqueness) of a periodic positive steady state p(x) of (2.2), a pulsating front connecting 0 and p is a solution of the type u(t, x) = φ(x · e − ct, x) with c 6= 0, e is a unit vector (e ∈ S N −1 ),
Bibliographical remarks Existence of travelling waves for reaction-diffusionequations have been studied since the well known works of Kolmogorov, Petrovskii and Piskunov . It has been generalised in the late seventies by Aronson and Weinberger  and Fife and McLeod  for the specific problem of a one dimensional bistable wave. Since these pioneering works, an important effort has been done to study front propagation and transition waves in inhomogeneous reaction-diffusionequations and it is impossible to cite them all. For our purpose let us mention the paper of Berestycki and Nirenberg  where they showed existence of travelling waves in cylinders with an inhomogeneous advection field. The notion of transition fronts, or invasion fronts, has been introduced in  and precisely defined and studied in . The specific problem of transition waves when heterogeneities come from the boundary have received a much more recent attention. In , the authors showed that for a bistable nonlinearity, if the domain Ω is a succession of two half rectangular cylinders, one can find conditions on them width for the propagation to be blocked. The problem of transition fronts for exterior domains has been treated in  where the authors devised geometrical conditions for the invasion to be complete. Finally, as already said, cylinder with varying cross sections has just been studied in .
Generally speaking, for diffusion and reaction coefficients a(x) and f (x, u) satisfying (1.2), the questions of the existence of front-like solutions connecting 0 and 1 and of the existence of propagation speeds are still open, even under the additional assumption (1.11). Based on the work in this present paper and the companion paper , one can conclude that the spatial period L plays an important role in answering these questions. Namely, under various assumptions, there are transition fronts propagating with nonzero speeds (pulsating fronts with nonzero speeds) when L is small, while there are both stationary and non-stationary transition fronts propagating with zero speed when L is large. However, so far there is no explicit condition to guarantee the existence of transition fronts in general, and no example is known to show their non-existence.
1 Introduction and main results
1.1 Motivation: models on climate change
Reactiondiffusion problems are often used to model the evolution of biological species. In 1937, Kolmogorov, Petrovskii and Piskunov in , Fisher in  used reaction diffu- sion to investigate the propagation of a favourable gene in a population. One of the main notions introduced in [15, 10] is the notion of travelling waves, i.e solution of the form u(t, x) = U (x − ct) for x ∈ R, t > 0 and some constant c ∈ R. Since then a lot of papers have been dedicated to reactiondiffusionequations and travelling waves in settings mod- elling all sorts of phenomena in biology.
Existence results for transition fronts
Several articles study the existence of transition fronts in non homogeneous settings. Shen [106, 107], Nadin and Rossi  and Berestycki and Hamel  prove the existence of transition fronts (almost planar) when f is heterogeneous with respect to time in the bistable, monostable and KPP case. Vakulenko and Volpert  also proved the existence of another type of generalised transition fronts for perturbed bistable reactiondiffusion systems. Mellet, Roquejoffre and Sire , Nolen and Ryzhik , Mellet et al  and Zlatoˇs  study the existence and the stability of transition fronts for combustion type heterogeneities. In all these papers it has been proved that for combustion like equationsin dimension 1, transition fronts always exist, are unique and stable. Nolen et al  and Zlatos  build a multitude of transition fronts for non homogeneous KPP equations using the linearisation around 0. In a recent paper Tuo, Zhu and Zlatos  use this same approach to built transition fronts of monostable type equations. Thus there exist several heterogeneous frameworks where the existence of transitions fronts is proved, nevertheless this existence is questionned by Nolen et al , for some class of monostable type reactiondiffusionequationsin dimension 1. Indeed they show that when the nonlinearity f has a strongly localised heterogeneity then there exist some situations where transition fronts do not exist. This last result show that the existence of transition fronts in heterogeneous settings is not guarantee for monostable equation in dimension 1. Moreover Lewis and Keener  highlight the existence of blocking phenomena for reactiondiffusionequationsin dimension 1 in media with heterogeneous excitability. The same kind of phenomena is observed by Grindrod and Lewis  in dimension 2 in domains with varying diameters, studying an eikonal approximation as the limit of a propagation model. Lou, Matano and Nakamura [79, 74] study the existence of front-like solutions through the notion of recurrent waves, for curvature dependent motions of plane curves and prove that the speed of propagation is slowed down in the presence of undulations. All these papers thus indicates that in a heterogeneous framework, a multitude of phenomena can be observed (non existence of transition fronts, slowing down or failure of the propagation...). Chapters 1 and 3 of this thesis come up in this context. We study the existence of transition fronts for heterogeneous bistable reactiondiffusionequations where the problem is set in a domain Ω Ĺ R n .
The setup we consider is in striking contrast with previous available results in homogenization of integro-differential problems. In [2, 3], Arisawa analyzed periodic homogenization for equa- tions with purely Lévy operators, and rather light interaction between the slow and fast variable. Homogenization results for nonlocal equationswith variational structure have been recently stud- ied in [26, 30]. This paper is closely related to , where periodic homogenization for uniformly elliptic Bellman-Isaacs equations was obtained by Schwab. Later on these results were extended to stochastic homogenization in . The arguments in both papers are completely different than ours, and are based on the obstacle problem method, previously introduced in [18, 19] in order to establish stochastic homogenization and rates of convergence for fully nonlinear, uniformly ellip- tic partial differential equations. Periodic homogenization for nonlocal Hamilton-Jacobi equationswith coercive gradient terms has been addressed in , where techniques similar to ours appear, except that here we cannot rely on the gradient coercivity.
s (x, 0) = µ 0 (φ(x)), with µ 0 a periodic function and φ a smooth increasing function such
that φ ′ (x) → 0 as x → +∞. If φ increases sufficiently fast, then these authors proved that
w ∗ = w ∗∗ and it is possible to compute this speed. We refer to  for a precise definition
of “sufficiently fast”. An example of such a φ is φ(x) = ln(1 + |x|) α , α > 1. This result is proved by constructing appropriate test-functions in the definition of the generalized principal eigenvalues (10). As in the random stationary ergodic setting, it is necessary to consider test- functions which are not necessarily bounded but satisfy 1 x ln φ(x) → 0 as |x| → +∞. When φ increases slowly for example, when φ(x) = ln(1 + |x|) α , α ∈ (0, 1) , then it was proved in  that w ∗ = 2 √ min R µ 0 and w ∗ = 2√max R µ 0 , which provides an example of coefficients
In the cases where the critical travelling waves are unique up to translation in time, such as ignition-type or monostable equations, these waves satisfy a property which is close to the translation property introduced in  (Proposition 4.1). We derive from this result that if the coefficients are homogeneous/periodic, then the critical transition waves are planar/pulsating travelling waves in the ignition-type and monostable frame- works (Propositions 4.2 and 4.3) as well as in the bistable setting for homogeneous co- efficients. If the heterogeneity of the coefficients is compactly supported, as in , then critical travelling waves are spatial transition waves with minimal speed if such waves exist, and bump-like solutions otherwise (Proposition 4.5). If the equation is monostable and the coefficients are random stationary ergodic in space, then the wave and its inter- face also satisfy such a dependence in space and admit a propagation speed, in a sense (Proposition 4.7). In the monostable setting, if the coefficients are “recurrent at infinty”, then critical travelling waves attract the solution of the Cauchy problem with a Heaviside initial datum (Theorem 5.2) along a subsequence. Lastly, if the equation is bistable, then there might exist non-trivial steady states which block the propagation and in this case the identification of critical transition waves depends on the normalization of these waves (Proposition 6.1).
Mots-clés : réaction-diffusion, ondes progressives, fronts, propagation, vitesse.
Abstract : the aim of the thesis is the study of enhancement of propagationinreaction- diffusionequations, through a new mechanism involving a line withfastdiffusion. We answer the question of the influence of such a coupling with strong diffusion on propagation by gen- eralizing a result of Berestycki, Roquejoffre and Rossi (2013). The model under study was proposed to give a mathematical understanding of the influence of transportation networks on biological invasions. The first chapter shows existence and uniqueness of travelling waves solu- tions with a continuation method. The transition occurs through a singular perturbation – new in this context – connecting the system with a Wentzell boundary value problem. The second chapter is concerned with the speed of the waves : we show that it grows as the square root of the diffusivity on the line, generalizing and showing the robustness of the result by Berestycki, Roquejoffre and Rossi. Moreover, the growth ratio is characterized as the unique admissible velocity for the waves of an hypoelliptic a priori degenerate system. The last part is about the dynamics : we show that the waves attract a large class of initial data. In particular, we shed light on a new mechanism of attraction which enables the waves to attract initial data with size independent of the diffusivity on the line : this is a new result, in the sense than usually, enhancement of propagation has to be paid by strengthening the assumptions on the initial data for invasion to happen.
the condition was algebraic and not asymptotic), there exists indeed such a pulsating front. While the forthcoming main ideas might be generalizable to systems with periodic diffusion and interspecific competition rates, an existence result is lacking. Therefore we naturally stick with the aforementioned system. Let us recall moreover that the fully heterogeneous problem (non-periodic non-constant coefficients) is, as far as we know, still completely open at this time. Let us recall as well that several important results about scalar reaction–diffusionequationsin periodic media have been established recently (about “KPP”-type, see [16, 17, 117, 118, 121]; about “ignition”-type and monostable non-linearities, see ; about bistable non-linearities, see [57, 56, 144]). The first author used extensively the results about “KPP”-type equationsin [Gir17]. In the forthcoming work, we will use the whole collection of results. Especially, we will use several times, in slightly different contexts, the sliding method of Berestycki–Hamel .
The proof of Theorem 1.1 consists in showing that the solution u of (1.1) and (1.5) with this function f satisfies the desired conclusions listed in Theorem 1.1. For the proof of (1.7), the strategy is to estimate u from above and below at some larger and larger times and at some further and further points. More precisely, on the one hand, the fact that µ takes its maximal value µ + on the large intervals [x n + 1, y n − 1] lead to the existence of time-growing bumps on these spatial intervals, whereas, on the other hand, the fact that µ takes its minimal value µ − on the large intervals [y n , x n+1 ] slows down the propagationin some time intervals when the growing bumps are still negligible (see the following lemmas for further details and Figure 1 below).
L , + ∞) have been given
in [25, 28, 29].
Uniqueness of the speed and further qualitative properties of pulsating fronts
In this paper, we first discuss the question of the uniqueness of the speed of pulsating fronts for the bistable equation (1.1), under assumptions (1.2) and (1.3), as well as the monotonicity and the uniqueness of pulsating fronts with nonzero speed. In , qualitative properties of tran- sition fronts, which are more general than pulsating fronts, were investigated in unstructured heterogeneous media. Some results of  can be applied to the pulsating fronts of the periodic equation (1.1), provided that the propagation speeds are not zero. More precisely, [9, Theo- rems 1.11 and 1.14] (see also  for x-independent reactions f = f (u)) lead to the uniqueness of the speed and of the fronts (up to shifts in time) in the class of pulsating fronts with nonzero speed, as well as the negativity of ∂ ξ φ for a pulsating front φ(x −ct, x/L) with c 6= 0 and ξ = x−ct.
∂ t u − ∆u = f (t, u), x ∈ R N , t ∈ R,
where f = f (t, u) is a KPP monostable nonlinearity which depends in a general way on t ∈ R. A typical f which satisfies our hypotheses is f (t, u) = µ(t)u(1 − u), with µ ∈ L ∞ (R) such that ess inf t∈R µ(t) > 0. We first prove the existence of generalized transition waves (recently defined in [4, 22]) for a given class of speeds. As an appli- cation of this result, we obtain the existence of random transition waves when f is a random stationary ergodic function with respect to t ∈ R. Lastly, we prove some spreading properties for the solution of the Cauchy problem.
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-diffusionequations 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-diffusionequationswith nonlinear fractionaldiffusion.
(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-diffusion problems.
These models are also amenable to modeling invasion phenomena, because (when posed on the whole space) they typically have solutions called travelling waves in the form y(x − ct) for certain speeds c, linking the states 0 and 1, see the pioneering work [ 10 ].
For such problems, it is thus a requirement for the solution to satisfy y > 0, a condition which is fulfilled with non-negative Dirichlet boundary conditions. We might consider using controls that are above 1, taking into account the possibility for releases at 0 or L to be above the capacity of the system.