数理解析研究所講究録
1191 巻 2001 年 29-42

29Some existence, monotonicity and uniquenessresults for pulsating travelling fronts inperiodic media and periodic domains
F. Hamel
CNRS-Universit\’e Paris VI, Laboratoire d’Analyse Num\’erique,$\mathrm{B}.\mathrm{C}$. $187$4, place Jussieu, F-75252 Paris Cedex 05, Fkance1

Introduction

This study is devoted to the analysis of some front propagation phenomena for a class of
advection-diffusion-reaction equations in a general class of periodic domains with underlying
periodic diffusion and velocity fields. In the case where the coefficients of the eq.uation are
invariant in some given direction and the domain itself is invariant in that direction, then one
can speak about travelling fronts which move in that direction with constant speed and whose
profiles do not change as time runs. In periodic domains or media, the notion of travelling
fronts has to be replaced by the notion of pulsating (or periodic) travelling fronts: a pulsating
travelling front propagates in some direction with some unknown effective speed but its profilechanges periodically as time runs. Pulsating travelling fronts appear in various physical models
and can propagate in several classes of periodic domains such as straight or oscillating infinite
cylinders, the whole space, or domains with periodic holes, etc. Various existence, uniquenessand monotonicity results are given for two types of reaction terms. For a combustion-type
nonlinearity, the pulsating travelling fronts exist, their speed is unique and the fronts are
increasing in the time variable and unique up to translation in time. For another class ofnonlinearity arising either in combustion or biological models, the set of possible speeds is asemi-infinite interval, closed and bounded from below, and for each speed, a time-increasing
pulsating travelling front exists. The results can all be stated in a same general class of periodic
media and domains (see section 6), and, as well as more general ones, they are proved in two
papers [7] and [8] written with H. Berestycki and with H. Berestycki and N. Nadirashvili.

2

Travelling fronts and pulsating travelling fronts instraight infinite cylinders

Let us first deal with the case of a straight infinite cylinder$\Omega=\{(x, y), x\in R, y\in\omega\}$30

where is a smooth bounded and connected subset of $R^{N-1}$ and let us consider the classicalsolutions $u(t, x, y)$ of the following advection-diffusion-reaction equation$\omega$$\frac{\partial u}{\partial t}-\triangle u+q(x, y)\cdot\nabla_{x,y}u=f(u)$together with Neumann boundary conditions on$\partial_{\nu}u=0$,,$t\in R,$

(2.1)$(x, y)\in\overline{\Omega}$$\partial\Omega$(2.2)$(t, x, y)\in R\mathrm{x}\partial\Omega$where $\nu=\iota/(x, y)=\nu(y)$ is the outward unit normal toand. These Neumannboundary condition mean that there is no flux of across the wall of the cylinder.
$q_{N}(x, y))$ is given in
The underlying velocity field $q(x, y)=(q_{1}(x, y),$, bounded inand one assumes that$\partial\Omega$$\partial_{\nu}u=\frac{\partial u}{\partial\nu}$$u$$\overline{\Omega}$$\cdots,$$C^{1}(\overline{\Omega})$

in

$=$

$0$

$q(x+L, y)$$=$$q(x, y)$$\int_{(0,L)\cross\omega}q_{1}(x, y)dxdy$

$=$

$0$

$q\cdot\nu$

$=$

$0$

$\mathrm{d}\mathrm{i}\mathrm{v}q$$\forall(x, y)\in\overline{\Omega}$,$\overline{\Omega}$

(2.3)

$\{$

on

$\partial\Omega$where the period of with respect to the variable is some given positive number. This field
is divergence-free (which corresponds to the incompressibility assumption for the underlying
medium) and may represent some turbulent fluctuations with respect to a mean field.
Such semilinear parabolic equations can arise in the modelling of thermodiffusive premixed
flame propagation with a unit Lewis number and a simple chemistry, and then represents
an adimensionalized temperature (see $e.g$ . [9], [52], [58]. These equations can also come from
biolog.ical models of population dynamics where stands for the relative concentration ofsome substance [1], [20]. One of our goals is to analyze the influence of periodic advection,and of other periodic phenomena, on the propagation of fronts (flames in combustion theory).
Related questions in combustion theory have been treated in [2], [16], [57]. In dimension $N\geq 2$ ,equation (2.1) can then arise in turbulent combustion models to describe the propagation of apremixed flame in an array of vortical cells. Generally speaking, equation (2.1) is a transport
equation for a passive quantity in a periodic excitable medium.
Two main types of nonlinearities are considered here. Namely, the given function isassumed to be Lipschitz-continuous in $[0,1]$ and to be of one of the following types : either
$L$

$x$

$q$

$q$

$u$

$u$

$u$

$f$

$\exists\theta\in(0,1),$
$\{$

$f$

$f(s)=0$ for all $s\in[0, \theta],$ $f(s)>0$ for all $s\in(\theta, 1),$ $f(1)=0$$\exists\mu\in(0,1-\theta))$

or

$f$

is nonincreasing on $[1-\mu, 1]$ ,$\{$$f>0$ on $(0,1)$ , $f(0)=f(1)=0$,
is nonincreasing on $[1-\mu, 1]$, $f\in C^{1,\delta}([0,1])$ .

$\exists\mu>0$

$f$

(2.4)

(2.5)

$\exists\delta>0$

Case (2.4) is usually referred to as the combustion nonlinearity with positive ignition temperature. Case (2.5) can be viewed as a combustion nonlinearity with ignition temperatureequal to $0[36]$ , or can also be thought of as the production rate of a population in biologicalmodels [1], [22], [38], in which case the quantity represents the density of a population.
$\theta[36]$

$u$

31

One is interested in some particular solutions of (2.1-2.2), namely the pulsating travelling
fronts, which propagate in a given direction, say to the left, with an unknown effective speed$c\neq 0$ , in the sense that$u(t+ \frac{L}{c},$$x,$$y)=u(t, x+L, y)$ for all $t\in B\S,$$(x, y)\in\overline{\Omega}$

(2.6)

.

Such fronts are assumed to have prescribed limiting conditions as $xarrow\pm\infty$ :$\forall t\in R$,$u(t, -\infty, y)=0$ ,$u(t, +\infty, y)=1$

uniformly with respect to .

(2.7)

$y$

Such pulsating fronts (which correspond to flames with pulsating shapes in combustion
theory) are of particular interest since, in periodic media, they can describe the behavior atlarge time of the solutions of the related Cauchy problem with front-like initial conditions.
However, the question of the stability of the pulsating solutions is not addressed here.
The first analyses of the propagation phenomena for advection-diffusion-reaction equations
like (2.1) have dealt with the case of planar travelling fronts, for one-dimensional equations$u_{t}=u_{xx}+f(u)$ with a zero velocity field $q=0$ . Such travelling fronts $u(t, x)$ move withconstant speed and their shape does not change as time runs : they satisfy (2.6) for any$L\in R$ and can be written as $u(t, x)=\phi(x+ct)$ . Travelling fronts are then particular types of
pulsating traveling fronts. Since the pioneering paper of Kolmogorov, Petrovsky and Piskunov[38] in 1937 for nonlinearities of the type (2.5), there have been many papers on the questionsof existence, uniqueness or stability properties of planar travelling fronts for various kindsof reaction terms $f(u)$ , more general than (2.4) or (2.5), arising in combustion or biologicalmodels (see $e.g$ . Aronson and Weinberger [1], Fife and, Kanel’ [36]). In thecase (2.4), there exists a unique speed (which is positive) and a unique-up to translation-front . In the case (2.5), travelling fronts with speed exist if and only if
for some(positive) minimal speed and, for any given, the fronts with speed are unique up totranslation. Many papers have also been devoted to the study of planar travelling fronts for
systems of one-dimensional diffusion-reaction equations [5], [12], [14], [19], [42], [51].
For the one-dimensional equation $u_{t}=u_{xx}+f(x, u)$ with no advection and with a function
similar to (2.5), Hudson and Zinner [35] have got the existence of a semi-infinite lineof possible speeds of pulsating travelling fronts, as well as the formula (7.2) below for .
These existence and uniqueness results have almost entirely been generalized in the multidimensional case of straight infinite cylinders,
with shear flows $q=$
by Berestycki, Larrouturou, Lions [10] and Berestycki, Nirenberg [13]. In the case of shear
flows, the velocity field is -periodic in for all and the equation (2.1) is invariant bytranslation in the variable . In this framework, travelling fronts are solutions of the type$u(t, x, y)=\phi(x+ct, y)$ (the problem for travelling fronts is then reduced to a semilinear elliptic equation for the function ). The known results for these travelling fronts are the following:
if is of type (2.4), there exists a unique speed and a unique travelling front $\phi(x+ct, y)(\phi$is increasing in $s=x+ct$ and unique up to translation in ) whereas if is of type (2.5), there
exists a speed such that travelling fronts $\phi(x+ct, y)$ exist if and only if
and, for each, the front is increasing and unique up to translation in if $f’(0)>0$ . The cases
givenof monotone shear flows along the main direction of the cylinder, and of almost parallel flows$\mathrm{M}\mathrm{c}\mathrm{L}\mathrm{e}\mathrm{o}\mathrm{d}[21]$

$c$

$u$

$c\geq c^{*}$$c$$c\geq c^{*}$$c^{*}$$c$$f$$[c^{*}, +\infty)$$c^{*}$$\Omega=R\cross\omega$

$L$

$q$

$(\alpha(y), 0, \cdots , 0)$

$L$

$x$

$x$

$\phi$

$f$

$c$

$f$

$s$

$c\geq c^{*}$$c^{*}$$c\geq c^{*}$$\phi$$s$32

with more general reaction terms have been considered in [27], [28]. Lastly, similar existenceor uniqueness results with Dirichlet conditions on
have been obtained in [24] and [50].
Many works have been devoted to the behavior at large time, and especially to the convergence to travelling fronts, of solutions of Cauchy problems for equations like (2.1) under alarge class of initial conditions. These works have been initiated by Kolmogorov, Petrovsky
and Piskunov [38] in the one-dimensional case with no advection (see also [1], [15], [21], [49])and followed by the study of the stability of travelling waves in infinite cylinders with shear
flows (see [11], [40], [46], [47]). So far, few works have dealt with the question of the stabilityof pulsating travelling fronts in periodic media like the real line or the whole space [39], [53].
The above results for shear flows can be for the most part generalized for pulsating travelling
fronts in straight infinite cylinders with periodic advection :$\partial\Omega$$q$

Theorem 2.1 [7] Let be a velocity field satisfying (2.3).
1) If satisfies (2.4), there exists a unique solution $(c, u)$ of
andbeing increasing in and unique up to translation in . Moreover, $0<u<1$ and $c>0$ .
2) If satisfies (2.5), there exists a positive real number such that: if $c<c^{*}$ , there is no; ifsolution $(c, u)$ of
and, there exists a solution $(c, u)$ , such that, then any solution of$0<u<1$ and is increasing in ; if $f’(0)>0$ andandis increasing in .
$q$

$f$

$(\mathit{2}.\mathit{6})-(\mathit{2}.7),$$(\mathit{2}.\mathit{1})-(\mathit{2}.\mathit{2})$

$t$

$u$

$t$

$f$

$c^{*}$$(\mathit{2}.\mathit{6})-(\mathit{2}.7)$$(\mathit{2}.\mathit{1})-(\mathit{2}.\mathit{2})$$c\geq c^{*}$

$t$

$u$

$c\geq c^{*}$$u$$(\mathit{2}.\mathit{1})-(\mathit{2}.\mathit{2})$$t$$(\mathit{2}.\mathit{6})-(\mathit{2}.7)$

Remark 2.2 If satisfies (2.5) and the additional assumption $f’(0)>0$ , one conjectures
that, for each speed, the solutions are unique up to translation in .
$f$$c\geq c^{*}$

$t$

$u$

Remark 2.3 The function may not be increasing in the variable . This indeed can beobserved in some remarquable experiments carried out by P. Ronney and collaborators [45] on-Couette cells in the framework of autocatalytic chemical waves.
$u$

$x$

$\mathrm{s}\mathrm{o}\mathrm{m}\mathrm{e}|\mathrm{T}\mathrm{a}\mathrm{y}\mathrm{l}\mathrm{o}\mathrm{r}$3Cylinder type domains with periodic boundaries

The periodicity of the velocity field can actually derive directly from the periodicity of thedomain. That is the case when, instead of a straight infinite cylinder, one considers an infinite
cylinder with a smooth and oscillating boundary :$\Omega$(3.1)$\Omega=\{(x, y)\in R^{N}, x\in R, y\in\omega(x)\}$where the function
is periodic with period $L>0$ . Straight infinite cylinders
correspond to the case where $\omega=constant$ . Let now be avelocity field satisfying$x\vdash\Rightarrow\omega(x)$$q$$C^{1}(\overline{\Omega})-$

in

$=$

$0$

$q(x+L, y)$$=$$q(x, y)$$\int_{\{x\in(0,L),y\in\omega(x)\}}q_{1}(x, y)dxdy$

$=$

$0$

$q\cdot\nu$

$=$

$0$

$\mathrm{d}\mathrm{i}\mathrm{v}q$,$\forall(x, y)\in\overline{\Omega}$$\Omega$

(3.2)

$\{$

on

$\partial\Omega$.

In the case where is of the “bistable” type and where $q=0$ , some conditions for the
existence or non-existence of pulsating travelling fronts have been given by Matano [41].
In the cases where is of the types (2.4) or (2.5), the same result as Theorem 2.1 holds :
$f$

$f$

Theorem 3.1 [7] Under the assumptions$(\mathit{3}.\mathit{1})-(\mathit{3}.\mathit{2})$, parts 1) and 2) of Theorem 2.1 hold.

33

4

Fronts in the whole space with periodic flows

A natural question about pulsating travelling fronts concerns the case where the domain. Let us consider the advection-diffusion-reaction equation
the whole space$\Omega$is$R^{N}$$\frac{\partial u}{\partial t}-\triangle u+q(x)\cdot\nabla_{x}u=f(u)$,$t\in R,$ $x\in R^{N}$

(4.1)

.

If the velocity field in (4.1) is equal to a constant vector , then planar travelling frontsof the type $u(t, x)=\phi(x\cdot e+ct)$ , propagating in a given direction $-e\in S^{N-1}$ , exist in both
set of planar speeds for thecases (2.4) or (2.5), and the set of possible speeds is equal to.
, translated with the shiftequation withand is periodic with respect toSimilarly, if is a shear flow $q=\alpha(x)e$ where$u(t,x)=\phi(x\cdot
e+ct, x\cdot e_{2}, \cdots, x\cdot e_{N})$ ,
type
of
the
frontsthe variables orthogonal to , travelling, also exist. In that
where has been completed into an orthonormal basis $(e, e_{2}, \cdots, e_{N})$ of$u(x,t)=\phi_{0}(x\cdote’+c_{0}t)$exist for any direction
last case, planar travelling fronts of the type$e’\in S^{N-1}$ such that $e’\perp e$ , where the couple
does not depend on and is the uniquesolution of $\phi_{0}’’-c_{0}\phi_{0}’+f(\phi_{0})=0$ with $\phi_{0}(-\infty)=0,$ $\phi_{0}(+\infty)=1$ . Furthermore, it caneasily be checked in that case that, provided that $q=\alpha(x)e$ is not constant, there exists noor the directions perpendicular to . This
other thantravelling front in a direction
example shows that, even for shear flows, the notion of travelling fronts is not sufficient todescribe the propagation of fronts in most of the directions of $S^{N-1}$ .
Let now be a divergence-free velocity field , of class $C^{1}(R^{N}),$ -periodic with respect tosuch thatthe space variables, in the sense that there exists an $N$-uple$q_{0}$$q$$\mathrm{t}\mathrm{h}\mathrm{e}$$q\equiv 0$$q_{0}\cdot e$$e\cdot\nabla\alpha=0$$q$$\alpha$$e$$R^{N}$$e$$(c_{0}, \phi_{0})$

$e’$

$q$

$\pm e$

$e$

$L$

$q$

$q$

$(L_{i})\in(R_{+}^{*})^{N}$$\forall k\in\prod_{i=1}^{N}L_{i}\mathbb{Z}$

$\{$

,

$0$

$q(x+k)$$=$$q(x)$$\int_{\Pi_{i=1}^{N}(0,L_{i})}q(x)dx$

$=$

$0$

$\forall x\in R^{N}$

.

in

$=$

$\mathrm{d}\mathrm{i}\mathrm{v}q$,$R^{N}$(4.2).

Under the above assumptions, pulsating travelling fronts for (4.1) are the solutions $u(t, x)$which propagate in a given direction, say $-e\in S^{N-1}$ , with an effective speed $c\neq 0$ :$\forall k\in\prod_{i=1}^{N}L_{i}\mathbb{Z}$,$\forall x\in R^{N}$,$u(t+ \frac{k\cdot e}{c},$

$x)$

$=$

$u(t, x+k)$

(4.3)

$\{$

$\forall t\in R$,$u(t, x)$$arrow$$0$,$u(t, x)$$arrow$1,$x\cdot earrow+\infty$$x\cdot earrow-\infty$

where the above limits hold locally in and uniformly in the variables orthogonal to .
The questions of the existence and uniqueness of pulsating travelling fronts have been solved
by Xin [54], [56] in the case of a combustion nonlinearity satisfying (2.4), under the additional
assumption $f’(1)<0$ : for each given $e\in S^{N-1}$ , there exists a unique solution $u(t, x)$ of (4.1)
and (4.3), and is increasing and unique up to translation in . This result, which actuallyholds for more general equations involving space-dependent diffusion terms (see also section6) has been proved through a continuation method based on some invertibility properties oflinearized operators. This method does not seem to easily extend to the case of a nonlinearity
$e$

$t$

$f$

$u$

$t$

34

satisfying (2.5), whereas the method used in [7] allows for the following Theorem 4.1, similar
to Theorems 2.1 and 3.1. Before stating this result, let us mention that the homogenization
limit with velocity fields or diffusion matrices involving very small scales has been carried out
by Freidlin [23], Heinze [31] and Xin [56]. Lastly, the question of front propagation in random
media has been considered in [23] and [56].
Let us now turn to the statement of the following existence, monotonicity and uniquenessresult of pulsating travelling fronts for the equation (4.1) :$f$

Theorem 4.1 [7] Let be avelocity field satisfying
and let $e\in S^{N-1}$ be a unit vector.
,
and
If is of the type (2.4), there exists a unique solution $(c, u)=(c(e), u(e))$ of
the function being increasing and unique up to translation in . If is of the type (2.5),
there exists $c^{*}=c^{*}(e)>0$ such that no solution $(c, u)$ exists if $c<c^{*}$ , and, for eachtime-increasing solution exists and all solutions are increasing in if $f’(0)>0$ .
$C^{1}$$q$$(\mathit{4}\cdot \mathit{2})$$f$$(\mathit{4}\cdot \mathit{1})$$u$$(\mathit{4}\cdot \mathit{3})$

$f$

$t$

$c\geq c^{*},$

$u$

5

$a$

$t$

$u$

Periodic media with holes

Another class of periodic domains and media is the case where the domains have periodic
holes. For instance, consider first the case of the whole space with periodic holes; namely, letbe a domain with a smooth boundary and such that$\Omega$$\exists(L_{i})_{1\leq i\leq N}\in(R_{+}^{*})^{N}$,,

Let $\nu=\nu(x)$ be the outward unit normal to$\Omega$

. Let

$q$

be a

,

$C^{1}(\overline{\Omega})$

in

$0$

$q(x+k)$$=$$q(x)$$\int_{\Pi_{i=1}^{N}(0,L_{i})}\mathrm{n}\Omega q(x)dx$

$=$

$0$

$q\cdot\nu$

$=$

$0$

$\forall x\in\overline{\Omega}$

,

(5.1)

.

velocity field such that$=$$\mathrm{d}\mathrm{i}\mathrm{v}q$$\forall k\in\prod_{i=1}^{N}L_{i}\mathbb{Z}$$\Omega+k=\Omega$$\forall k\in\prod_{i=1}^{N}L_{i}\mathbb{Z}$$\Omega$

(5.2)

$\{$

A pulsating travelling front in a$\mathrm{d}\mathrm{i}\mathrm{r}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}-e\in S^{N-1}$

on$\partial\Omega$

is a solution $(c, u)$ (with $c\neq 0$ ) of$=$$f(u)$ ,$\partial_{\nu}u$

$=$

$0$

$x)$

$=$

$u(t, x+k)$$\frac{\partial u}{\partial t}-\triangle u+q(x)\cdot\nabla_{x}u$.

,$t\in R,$$t\in R,$$x\in\overline{\Omega}$$x\in\partial\Omega$

(5.3)

$\{$
$\forall k\in\prod_{i=1}^{N}L_{i}\mathbb{Z}$,$\forall t\in R$$\forall x\in\overline{\Omega}$,,$u(t+ \frac{k\cdot e}{c},$$u(t, x)x\cdot earrow-\inftyarrow 0$,$u(t, x)$$x\cdot earrow+\inftyarrow$1,

where the above limits hold locally in and uniformly in the variables orthogonal to .
For a nonlinearity satisfying (2.4), the existence of pulsating travelling fronts has been
proved by Heinze [32] in the limit of asymptotically small holes, by using a perturbation
technique around the homogenized equation.
With the method used in [7], the same result as for the whole space holds :
$t$

$e$

$f$

Theorem 5.1 [7] If (5.1) and (5.2) are satisfied, Theorem 4.1 holds for the solutions of (5.3).

35

6

General periodic domains

The results presented above can all be written in a more general framework which we describe
now. Let be a connected unbounded open set, with a smooth boundary, and such that$\Omega$$\exists 1\leq d\leq N,$$\exists L_{1},$$\cdots,$$L_{d}>0,$$\forall k=(k_{i})_{1\leq i\leq d}\in\prod_{i=1}^{d}L_{i}\mathbb{Z}$

$\{$

and

$\Omega$

is bounded with respect to the variables$x_{d+1},$,$\Omega+\sum_{i=1}^{d}k_{i}e_{i}=\Omega$$\cdots,$ $x_{N}$

(6.1)

,

where
is the canonical basis of. Let us denote by $x=(x_{1}, \cdots , x_{d})$ the first$y=(x_{d+1},\cdots,x_{N})$
coordinates and by
the last $N-d$ ones. Let $\nu=\nu(x, y)$ be the outward
unit normal to . Let be the periodicity cell defined by$R^{N}$$(e_{i})_{1\leq i\leq N}$

$d$

$C$

$\Omega$$C=\{(x, y)\in\Omega, x\in(0, L_{1})\cross\cdots\cross(0, L_{d})\}$.

We say that a field $v(x, y)$ defined in is -periodic with respect to the variable if $v(x+k, y)=$$v(x, y)$ for all
and for all.
Note that that class of domains includes all domains described above : the infinite cylinders with straight or oscillating boundaries, the whole space with or without periodic holes.
Domains of the class (6.1) also include infinite cylinders or slabs with periodic holes.
Let $q=(q_{1}, \cdots, q_{N})$ denote a globally
vector field defined in and such that
$L$$\Omega$$x$$k\in L_{1}\mathbb{Z}\cross\cdots\cross L_{d}\mathbb{Z}$$(x, y)\in\overline{\Omega}$$C^{1}$$\overline{\Omega}$

$=$

$0$

$\int_{C}q_{i}dxdy$

$=$

$0$

$q\cdot\nu$

$=$

$0$

$\mathrm{d}\mathrm{i}\mathrm{v}q$

is -periodic w.r.t.
$L$

$q$

in

$\overline{\Omega}$

$x$

(6.2)

$\{$

$\forall 1\leq i\leq d,$

Furthermore, let $A(x, y)=(A_{ij}(x, y))_{1\leq i,j\leq N}$ be a globally$\exists 0<c_{1}\leq c_{2}$,$\forall\xi\in R^{N}$

,

,

on

$\partial\Omega$$C^{1}(\overline{\Omega})$.

matrix field such that$\forall(x, y)\in\overline{\Omega}$$c_{1}| \xi|^{2}\leq\sum_{1\leq i,j\leq N}A_{ij}(x, y)\xi_{i}\xi_{j}\leq c_{2}|\xi|^{2}$

$\{$

$A$

is symmetric and -periodic w.r.t.
$L$

$x$

.

(6.3)

In the sequel, if andare two vectors inand $B$ is an $N\cross N$-matrix, then $zBz’$
denotes the number $zBz’:= \sum_{1\leq i,j\leq N}z_{i}B_{ij}z_{j}’$ .
Let be any given unit vector inand let be of the type (2.4) or (2.5). Let us now
study the questions of the existence and of the qualitative properties of pulsating travelling
fronts $u(t, x, y)$ , moving in direction $-e$ with an effective speed $c\neq 0$ , and solving$z$$R^{N}$$z’$$R^{d}$$e$$f$$\frac{\partial u}{\partial t}-\mathrm{d}\mathrm{i}\mathrm{v}(A\nabla u)+q\cdot\nabla u$$=$$f(u)$ ,$\nu A\nabla u$

$=$

$0$

$y)$

$=$

$u(t, x+k, y)$ for all $(t, x, y)\in R\cross\overline{\Omega}$,$t\in R,$$t\in R,$$(x, y)\in\overline{\Omega}$$(x, y)\in\partial\Omega$$\{$$\forall k\in\prod_{i=1}^{d}L_{i}\mathbb{Z}$,$u(t+ \frac{k\cdot e}{c},$ $x,$$u(t, x, y)arrow 0$ ,$x\cdot earrow-\infty$$u(t, x, y)$$arrow$$x\cdot earrow+\infty$

1 for each $(t, y)$ ,

(6.4)

36

orthogwhere the above limits hold locally in and uniformly in and in the directions ofonal to .
That framework for the propagation of pulsating travelling fronts contains all situations
described in the previous sections. Note that the Laplace operator has been replaced with ageneral heterogeneous diffusion operator. Such operators have also been consideredin the onedimensional case or in the case of the whole space (see [44], [53], [54], [55], [56]).
In that general framework, the foilowing Theorem, generalizing Theorems 2.1, 3.1, 4.1 and
5.1, holds :$t$$lR^{d}$$y$$e$$\mathrm{d}\mathrm{i}\mathrm{v}(A\nabla u)$. Let and
Theorem 6.1 [7] Let be a domain satisfying (6.1). Let be a unit vector in
be two globallyand (6.3).
vector and matrix fields satisfying
1) Let be a nonlinearity of the ignition temperature type (2.4). There exists a unique solution $(c, u)=(c(e), u(e))$ of (6.4), the function being increasing and unique up to translationin . Moreover, $0<u<1$ and $c(e)>0$ .
2) Let be a nonlinearity of the type (2.5). There exists $c^{*}(e)>0$ such that problem (6.4)
has no solution $(c, u)$ if $c<c^{*}(e)$ while, for each $c\geq c^{*}(e)$ , it has a solution $(c, u)$ such thatis increasing in . Moreover, if $f’(\mathrm{O})>0$ , then any solution of (6.4) is increasing in .
$\Omega$$A$$lR^{d}$$e$$C^{1}(\overline{\Omega})$$q$$(\theta.\mathit{2})$

$f$

$u$

$t$

$f$

$u$

$t$

$t$

$u$

Remark 6.2 Theorems 2.1, 3.1, 4.1 and 5.1 hold in the general case where the Laplace operator is replaced with a divergence type operator
together with Neumann typeboundary conditions $\nu A\nabla u=0$ on.
$\mathrm{d}\mathrm{i}\mathrm{v}(A\nabla u)$$\partial\Omega$

Remark 6.3 All above theorems work in the case where the nonlinearity $f(u)$ is replacedwith $h(x, y)f(u)$ if is a continuous, positive function which is -periodic w.r.t. (see [7]).
$L$

$h$

$x$

At this stage, the question of the uniqueness of the pulsating travelling fronts for each speed, in the case where satisfies (2.5), remains open, even under the assumption $f’(0)>0$ .
Another related open problem concerns the case where the function is of the bistable
type, namely, there exists $\theta\in(0,1)$ such that $f(0)=f(\theta)=f(1),$ $f<0$ on$f>0$ onand is nonincreasing in a right neighborhood of and in a left neighborhood. Someconditions for the existence or nonexistence of pulsating travelling fronts in infinite cylinders
with periodic boundary have been given by Matano [41]. Other existence, nonexistence or
stability results have been obtained by Xin [53], [55] and Papanicolaou and Xin [44] in the caseof the whole space with almost uniform diffusion and advection coefficients, and by Nakamura[43] for the one-dimensional case with periodic diffusion coefficient.
Lastly, let us mention here that the methods used in [7] to prove the uniqueness and monotonicity properties of the pulsating travelling fronts in the case of a nonlinearity with positive
ignition temperature (2.4) actually work and lead to the same uniqueness and monotonicity
results in the case of a bistable nonlinearity .
$c\geq c^{*}$

$f$

$f$

$(0, \theta),$$(\theta, 1)$

$f$

$0$

$\mathrm{o}\mathrm{f}1$

$f$

$f$

7

Further results : formulas for the speeds

One of the most important questions related to the front propagation phenomena is the determination of the speed of propagation of the travelling fronts, or of the pulsating travelling37

fronts in the periodic framework. In the theory of combustion for instance, the determinationof the burning velocity of a deflagration flame is a fundamental question.
Many works have been devoted to finding some formulas for the speeds of propagation of
travelling waves for advection-diffusion-reaction equations more general than those arising in
combustion models. The first formula comes back to the paper of Kolmogorov, Petrovsky andPiskunov [38] and concerns the minimal speed $c^{*}=2\sqrt{f’(0)}$ of planar travelling fronts for theequation $u_{t}=u_{xx}+f(u)$ with nonlinearities of the “Fisher-KPP” ([22], [38]) type$f(\mathrm{O})=f(1)=0,$

and$\exists\mu>0,$$f$$f>0,$ $f(s)\leq f’(0)s$ on ] $0,1$

$[$

(7.1)

is nonincreasing on $[1-\mu, 1]$ .

Other formulas of the variational type have been derived for such one-dimensional equations.
Let us for instance mention the formula,$c^{*}= \min_{0\rho:[,1]arrow R,\rho(0)=0,\rho(0)>0,\rho>0}$

in $(0,1]$$\sup_{u\in(0,1]}(\rho’(u)+\frac{f(u)}{\rho(u)})$of Hadeler and Rothe [26] for nonlinearities of the type (2.5). The latter implies $2\sqrt{f’(0)}\leq$and gives $c^{*}=2\sqrt{f’(0)}$ in the case (7.1). Integral formulations havebeen given by Benguria and Depassier [4]. Other variational formulas have been obtained forsystems of one-dimensional equations [42], [48], [51], or for equations with discrete diffusion[30]. Some formulas have been generalized by Hamel [29], Heinze, Papanicolaou and Stevens[33] in the multidimensional case with shear flows, and by Hudson and Zinner [34] in thediscrete case. For instance, in the case (2.4), the unique speed of travelling fronts $\phi(x+ct, y)$solving (2.1) in a cylinder
with a shear flow $q=(\alpha(y), 0, \cdots, 0)$ , is given by$c^{*}\leq 2\sqrt{\sup_{(0,1]}f(u)/u}$$c$$\Omega=R\cross\omega$$c= \min_{w\in \mathcal{E}}\sup_{x_{1}(,y)\in\overline{\Omega}}(\frac{\Delta w+f(w)}{\partial_{x}w}-\alpha(y))=\max_{w\in \mathcal{E}}\inf_{(x_{1},y)\in\overline{\Omega}}(\frac{\Delta w+f(w)}{\partial_{x}w}-\alpha(y))$

where$0<w<1,$in,on$w(-\infty, \cdot)=0,$ $w(+\infty, \cdot)=1\}$ and $p>N$ (see [29]). In the case (2.5) with $f’(0)>0$ , theminimal speed for travelling fronts is equal to$\mathcal{E}=\{w\in W_{loc}^{2,p}(\Omega),$$\triangle w\in C(\overline{\Omega}),$$\partial_{x}w>0$$\overline{\Omega},$$\partial_{\nu}w=0$$\partial\Omega$$c^{*}$$c^{*}= \min_{w\in \mathcal{E}}\sup_{x_{1}(,y)\in\overline{\Omega}}(\frac{\triangle w+f(w)}{\partial_{x}w}-\alpha(y))$.

Explicit formulas for the speeds of propagation of travelling waves have been obtainedin some asymptotic cases, like in the limit of high activation energies (see [12] in the onedimensional case, and [6] in the multi-dimensional case). Formal asymptotics in the case ofshear flows with large amplitude have been derived by Audoly, Berestycki and Pomeau in [3].
We also refer to [17], [18] and [37] for some a priori bounds of the speeds of propagationof the solutions of the Cauchy problem associated to (2.1) with front-like initial conditions.
Namely, Constantin, Kiselev, Oberman and Ryzhik have defined the notion of bulk burning
rate $V(t)=| \omega|^{-1}\int_{R\cross\omega}u_{t}(t, x, y)dxdy$ (is the Lebesgue-measure of ), and, from a subtle$|\omega|$$\omega$38

decomposition of the velocity field into positive and negative parts, they have obtained somelower bounds for $V(t)$ (or for the time-average of $V(t)$ ) if is a solution of the corresponding
Cauchy problem with front-like initial conditions [17], [37]. These bounds have been obtained
both for shear-like percolating or cellular flows and especially lead to some lower bounds forthe effective speed of any pulsating travelling front solving (2.1-2.2) and (2.6-2.7), since, forsuch a solution , one has $T^{-1} \int_{t_{0}}^{t_{0}+T}V(t)dt=c$ with $T=L/c$ , for any $t_{0}\in R$ .
For pulsating travelling fronts in periodic media, the only formula, derived by Hudson
and Zinner [35], concerns the minimal speed of propagation in the one-dimensional case $u_{t}=$$f(x, 0)=f(x, \overline{u}(x))=0$$u_{xx}+f(x, u)$ , whereis 1-periodic in $f(x, u)>0$ for $u\in$ ]and $\mu(x)=f_{u}’(x, 0)=\sup_{u\in]0,\overline{u}(x)[}f(x, u)/x$ . Namely, Hudson and Zinner have obtained thefollowing formula for the minimal speed :
$q$

$u$

$c$

$u$

$f$

$c^{*}= \min_{r>0}$$0,$ $\overline{u}(x)[,$$x,$$\max_{x\in[0,1]}\frac{\psi’’+2r\psi’+(r^{2}+\mu(x))\psi}{r\psi}$$\{\psi=\psi(x)\in C^{2}(R), \min_{\psi>0,\psi}1-\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{i}\mathrm{o}\mathrm{d}\mathrm{i}\mathrm{c}\}$

.

(7.2)

In the paper [8], the question of the determination of the minimal speed of pulsating
travelling waves solving (6.4) in a domain of the class (6.1) is considered under the additional
assumption that the function satisfies (7.1).
From Theorem 6.1, under the assumptions $(6.1)-(6.3)$ and for each given unit direction, there exists a minimal speedofof the pulsating travelling fronts. Our goal in [8] has.
been to find an explicit formula for the minimal speed:
equivalentfollowing
variational formulas for
We have obtained the
$f$

$e$

$lR^{d}$$c^{*}(e)$$c^{*}(e)$$c^{*}(e)$(7.3)$c^{*}(e)= \min\{c, \exists\lambda>0, \mu_{c}(\lambda)=f’(0)\}$is the principal eigenvalue of the elliptic operator
on the set $E$ of -periodic withrespect to functions $\psi(x, y)$ such that
. Here, denotes the vectoron. Thus, under the KPP assumption (7.1), the minimal speed
canbe explicitely given in terms of , the domain , the coefficients and and of $f’(0)$ . In thegeneral case where satisfies (2.5) and $f’(\mathrm{O})>0$ , the minimal speed
is always greaterthan or equal to the right hand side of (7.3). Note also that the formula (7.3) is similar to thatof Berestycki and Nirenberg [13] for travelling waves in infinite cylinders with shear flows.
As observed in [56], the above formula (7.3) is equivalent to the following one :where$-L_{c,\lambda}\psi=-\mathrm{d}\mathrm{i}\mathrm{v}(A\nabla\psi)$$\mu_{c}(\lambda)$$L$$\lambda(\mathrm{d}\mathrm{i}\mathrm{v}(A\tilde{e}\psi)+\tilde{e}A\nabla\psi)+q\cdot\nabla\psi+(\lambda q\cdot\tilde{e}+\lambda c-\lambda^{2}\tilde{e}A\tilde{e})\psi$$\nu A(\tilde{e}\lambda\psi+\nabla\psi)=0$$x$-$\partial\Omega$$\tilde{e}$$\tilde{e}=(e_{1}, \cdots, e_{d}, 0, \cdots, 0)$$c^{*}(e)$$e$$\Omega$

$q$

$f$

$A$

$c^{*}(e)$(7.4)$c^{*}(e)= \min_{\lambda>0}\frac{-k(\lambda)}{\lambda}$

is the principal eigenvalue of the operator
on the same set $E$ of functions
as above.
Note that the formula (7.4) is similar to that of G\"artner and Freidlin [25] for the asymptotic
speed of propagation of solutions of Cauchy problem inwith compactly supported initialconditions and periodic diffusion coefficients (see [8] for a further study of the asymptotic
speeds of propagation). Note also that when $\Omega=lR^{N},$ $A=I$ and $q=0$ , this formula (7.4)
gives the KPP formula $c^{*}(e)=2\sqrt{f’(0)}$ for the minimal speed of planar fronts.
where$k(\lambda)$$-L_{\lambda}\psi=-\mathrm{d}\mathrm{i}\mathrm{v}(A\nabla\psi)-\lambda(\mathrm{d}\mathrm{i}\mathrm{v}(A\overline{e}\psi)+$$\tilde{e}A\nabla\psi)+q\cdot\nabla\psi+(\lambda q\cdot\tilde{e}-\lambda^{2}\tilde{e}A\tilde{e}-f’(\mathrm{O}))\psi$$\psi$$R^{N}$

39

Lastly, the following formula also holds(7.5)$c^{*}(e)= \min_{\lambda>0}\min_{\psi\in F}(x,y)\in\overline{\Omega}\max\frac{L_{\lambda}\psi}{\lambda\psi}$$\psi>0$ in
where $F=$ { $\psi\in E,$}. This formula is obtained from (7.4) and fromsome characterizations of principal eigenvalues of elliptic operators. This formula (7.5) for theminimal speed of multidimensional pulsating fronts generalizes the formula (7.2) of Hudson andZinner [35] for the minimal speed of pulsating travelling fronts in the case of one-dimensional
equations of the type $u_{t}=u_{xx}+f(x, u)$ .
$\overline{\Omega}$$\psi\in C^{2}(\overline{\Omega}),$8

Short sketch of the proofs

The monotonicity and uniqueness results stated in part 1) of Theorem 6.1, in the case where thefunction satisfies (2.4), are based on a sliding method in another set of variables $(s, x, y)=$$(x\cdot\tilde{e}+ct, x, y)$ , for which the equation is elliptic degenerate, and on the parabolic maximum
principle in the original variables $(t, x, y)$ (remember that for travelling fronts with constant
speed , the equation of the profile of the front is elliptic in some variables, say $(x+ct, y)$ in thecase of an infinite straight cylinder). The existence of a solution $(c, u)$ in part 1) of Theorem 6.1
is obtained as a limit of solutions of regularized elliptic equations in approximated boundeddomains. The main difficulty is to deal with the degeneracy of the equations and to provethat the solution obtained at the limit is not trivial. One especially proves some Bernsteintype gradient estimates and one uses some exponentially decaying upper solutions in somesemi-infinite domains.
In the case where the function satisfies (2.5), the existence of a solution for the minimalof the type (2.4) and apspeed
is obtained as a limit of solutions for nonlinearities
proximating (with small ignition temperatures ). The existence of solutions for any speed$c\geq c^{*}(e)$ is obtained through a method using sub- and super-solutions, and the non-existenceof solutions with speeds $c<c^{*}(e)$ follows from a sliding method and from a comparison withsuitable sub-solutions.
$f$

$c$

$f$

$c^{*}(e)$$f_{\theta}$$f$$\theta$

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