### Pulsating traveling waves in the singular limit of a reaction–diffusion system in solid combustion

^{✩}

### R. Monneau

^{a,}

^{∗}

### , G.S. Weiss

^{b}

a*Ecole Nationale des Ponts et Chaussées, CERMICS, 6 et 8 avenue Blaise Pascal, Cité Descartes, Champs-sur-Marne,*
*77455 Marne-la-Vallée Cedex 2, France*

b*Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo, 153-8914 Japan*
Received 6 November 2007; received in revised form 9 September 2008; accepted 9 September 2008

Available online 27 September 2008

**Abstract**

We consider a coupled system of parabolic/ODE equations describing solid combustion. For a given rescaling of the reaction term (the high activation energy limit), we show that the limit solution solves a free boundary problem which is to our knowledge new.

In the time-increasing case, the limit coincides with the Stefan problem with spatially inhomogeneous coefficients. In general it is a parabolic equation with a memory term.

In the first part of our paper we give a characterization of the limit problem in one space dimension. In the second part of the paper, we construct a family of pulsating traveling waves for the limit one phase Stefan problem with periodic coefficients. This corresponds to the assumption of periodic initial concentration of reactant.

©2008 Elsevier Masson SAS. All rights reserved.

*MSC:*35K55; 35R35

*Keywords:*Solid combustion; Singular limit; Pulsating traveling wave; High activation energy; Stefan problem; Free boundary; Self-propagating
high temperature synthesis

**1. Introduction**

For*ε >*0, we consider the system

*∂*_{t}*u** _{ε}*−

*u*

*=1*

_{ε}*εv*_{ε}*g*_{ε}*(u*_{ε}*),*

*∂*_{t}*v** _{ε}*= −1

*εv*_{ε}*g*_{ε}*(u*_{ε}*),*

(1)

✩ G.S. Weiss has been partially supported by the Grant-in-Aid 15740100/18740086 of the Japanese Ministry of Education and partially supported by a fellowship of the Max Planck Society. Both authors thank the Max Planck Institute for Mathematics in the Sciences for the hospitality during their stay in Leipzig.

* Corresponding author.

*E-mail addresses:*monneau@cermics.enpc.fr (R. Monneau), gw@ms.u-tokyo.ac.jp (G.S. Weiss).

*URLs:*http://cermics.enpc.fr/~monneau/home.html (R. Monneau), http://www.ms.u-tokyo.ac.jp/~gw (G.S. Weiss).

0294-1449/$ – see front matter ©2008 Elsevier Masson SAS. All rights reserved.

doi:10.1016/j.anihpc.2008.09.002

where *u** _{ε}* is the normalized temperature,

*v*

*is the normalized concentration of the reactant and the non-negative nonlinearity*

_{ε}*g*

*describes the reaction kinetics. This model has been extensively used in solid combustion (to analyze SHS, i.e. Self-propagating High temperature Synthesis), see for instance Logak and Loubeau [7] and the references therein. We study the limit of high activation energy as*

_{ε}*ε*→0, where we can take for instance

*g**ε**(z)*:=

exp((1−1/(z+1))/ε), *z >*−1,

0, *z*−1

but can also consider more general functions*g** _{ε}*. In this particular case, the non-negative temperature is actually

*u*

*+1 (see Section 6.1).*

_{ε}The analysis of the high activation energy limit has been introduced in the pioneering work of Zeldovich and
Frank-Kamenetskii [11]. The rigorous asymptotic analysis for a system with finite Lewis model has been done by
Berestycki, Nicolaenko and Scheurer in [3]. In the case of infinite Lewis number (which corresponds to system (1) of
solid combustion), Logak and Loubeau proved in [7] the existence of a (planar) traveling wave for system (1) in one
space dimension and gave a rigorous proof of convergence of this planar traveling wave in the limit*ε*→0.

The present paper consists of two parts. In the first part (Sections 4–6), we study the high activation energy limit
*ε*→0 on a bounded domain, and in the second part (Sections 7, 8) we study pulsating traveling waves for the limit
equation on the whole space.

More precisely we show in Section 5 (cf. Theorem 5.1) that in one space dimension, each limit*u*of*u** _{ε}* solves the
Stefan problem

*∂*_{t}*u*−*v*^{0}*∂*_{t}*χ*=*u* (2)

where*v*^{0} is the initial value of *v* and*χ* is the memory term*χ*=*H (esssup*_{(0,t )}*u(*·*, x))*where*H* is the heavyside
function. In higher space dimensions (in Section 4), we get less information on the memory term*χ, except in the case*
that*∂*_{t}*u** _{ε}*0 in which we show that any limit

*u*still satisfies (2) (see Theorem 4.1). In Section 6, we apply our results to two cases considered in the literature.

In Section 7, we show the existence of pulsating traveling waves solutions of (2) for periodic*v*^{0}. Such pulsating
traveling waves exist for any velocity and any direction of propagation. In Section 8, in the case*v*^{0}=constant*>*0,
we give a result of non-existence of non-trivial pulsating traveling waves.

We conclude with miscellaneous remarks in Section 9 and present in Appendix A a result on formal stability of the planar wave for the limit one-phase Stefan problem in one space dimension.

Our approach to the problem is first to reduce the system to a single equation with a right-hand side which turns out to be the time derivative of a term which is non-local in time.

In the first part of the paper, using some a priori estimates, we show the compactness in*L*^{1}(in space–time) of any
truncation of*u** _{ε}* and its convergence to a solution of the limit problem. One main difficulty consists in proving that

*χ*=0 in the region where esssup

_{(0,t )}*u(*·

*, x)*is negative for the limit problem, with

*u*

_{ε}*(t*

_{ε}*, x*

_{ε}*)*being possibly positive for

*x*

*ε*close to

*x*and

*t*

*ε*

*< t. To do this, we analyze the “burnt zone”u*

*ε*

*> κ*(in space–time) for negative

*κ*. Based on the comparison principle and on some integral estimates, we show in one space dimension that if the measure of the

“burnt zone” is small enough in a parabolic cylinder, then in a cylinder of smaller radius it must be the empty set.

Our construction of pulsating traveling waves for the one-phase Stephan problem in the second part of the paper is
based on an integration in time which reduces the problem to an obstacle problem. We then approximate this obstacle
problem by a reaction–diffusion equation for which existence of pulsating traveling waves has been proved in [2] by
Berestycki and Hamel. We conclude the proof passing to the limit in the approximation. The ingredients used involve
Harnack inequality and blow-up arguments. Finally, in order to obtain non-existence of pulsating waves for constant
*v*^{0}we use a Liouville technique.

**2. Notation**

Throughout this article**R*** ^{n}*will be equipped with the Euclidean inner product

*x*·

*y*and the induced norm|

*x*|

*. B*

_{r}*(x)*will denote the open

*n-dimensional ball of centerx,*radius

*r*and volume

*r*

^{n}*ω*

_{n}*.*When the center is not specified, it is assumed to be 0.

When considering a set *A, χ**A* shall stand for the characteristic function of *A,* while *ν* shall typically denote
the outward normal to a given boundary. We will use the distance pardist with respect to the parabolic metric
*d((t, x), (s, y))*=

|*t*−*s*| + |*x*−*y*|^{2}.

The operator*∂** _{t}* will mean the partial derivative of a function in the time direction,the Laplacian in the space
variables and

*L*

*the*

^{n}*n-dimensional Lebesgue measure.*

Finally**W**^{2,1}*p* denotes the parabolic Sobolev space as defined in [6].

**3. Preliminaries**

In what follows,*Ω*is a bounded*C*^{1}-domain in**R*** ^{n}*and

*u** _{ε}*∈

*T*∈*(0,*+∞*)*

**W**^{2,1}_{2}

*(0, T )*×*Ω*
is a strong solution of the equation

*∂**t**u**ε**(t, x)*−*u**ε**(t, x)*= −*v*^{0}_{ε}*(x)∂**t*exp

−1
*ε*

*t*
0

*g**ε*

*u**ε**(s, x)*
*ds* *,*

*u*_{ε}*(0,*·*)*=*u*^{0}* _{ε}* in

*Ω,*∇

*u*

*·*

_{ε}*ν*=0 on

*(0,*+∞

*)*×

*∂Ω*; (3)

here*g**ε* is a non-negative function on**R**satisfying:

(0) *g**ε*is for each*ε*∈*(0,*1)piecewise continuous with only one possible jump at*z*0,*g**ε**(z*0−*)*=*g**ε**(z*0*)*=0 in case of
a jump, and*g**ε*satisfies for each*ε*∈*(0,*1)and for every*z*∈**R**the bound*g**ε**(z)C**ε**(1*+ |*z*|*).*

(1) *g**ε**/ε*→0 as*ε*→0 on each compact subset of*(*−∞*,*0).

(2) For each compact subset*K*of*(0,*+∞*)*there is*c**K**>*0 such that min(g*ε**, c**K**)*→*c**K* uniformly on*K*as*ε*→0.

The initial data satisfy 0*v*_{ε}^{0}*C <*+∞,*v*_{ε}^{0}converges in*L*^{1}*(Ω)*to*v*^{0}as*ε*→0,*(u*^{0}_{ε}*)*_{ε}_{∈}* _{(0,1)}*is bounded in

*L*

^{2}

*(Ω),*it is uniformly bounded from below by a constant

*u*

_{min}, and it converges in

*L*

^{1}

*(Ω)*to

*u*

^{0}as

*ε*→0.

**Remark 3.1.**Assumption (0) guarantees existence of a global strong solution for each*ε*∈*(0,*1).

**4. The high activation energy limit**

The following theorem has been proved in [9]. Let us repeat the statements and its proof for the sake of complete- ness.

**Theorem 4.1.***The family(u*_{ε}*)*_{ε}_{∈}_{(0,1)}*is for eachT* ∈*(0,*+∞*)precompact inL*^{1}*((0, T )*×*Ω), and each limitu* *of*
*(u*_{ε}*)*_{ε}_{∈}_{(0,1)}*as a sequenceε** _{m}*→0,

*satisfies in the sense of distributions the initial-boundary value problem*

*∂*_{t}*u*−*v*^{0}*∂*_{t}*χ*=*u* *in(0,*+∞*)*×*Ω,* (4)

*u(0,*·*)*=*u*^{0}+*v*^{0}*H*
*u*^{0}

*inΩ,* ∇*u*·*ν*=0 *on(0,*+∞*)*×*∂Ω,*
*where*

*χ (t, x)*

∈ [0,1]*,* esssup_{(0,t )}*u(*·*, x)*0,

=1, esssup_{(0,t )}*u(*·*, x) >*0,
*andH* *is the maximal monotone graph*

*H (z)*

=0, *z <*0,

∈ [0,1]*,* *z*=0,

=1, *z >*0.

*Moreover,χis increasing in time anduis a supercaloric function.*

*If(u*_{ε}*)*_{ε}_{∈}_{(0,1)}*satisfies∂*_{t}*u** _{ε}*0

*in(0, T )*×

*Ω, thenuis a solution of the Stefan problem for supercooled water, i.e.*

*∂*_{t}*u*−*v*^{0}*∂*_{t}*H (u)*=*u* *in(0,*+∞*)*×*Ω.*

**Remark 4.2.**Note that assumption (1) is only needed to prove the second statement “If . . . ”.

**Remark 4.3.**It is interesting to observe that even in the time-increasing case our singular limit*selects certain solutions*
of the two-phases Stefan problem. For example,*u(t )*=*(κ*−1)χ_{{}*t <1*}+*κχ*_{{}_{t >1}_{}}is for each*κ*∈*(0,*1)a perfectly valid
solution of the two-phases Stefan problem, but, as easily verified, it cannot be obtained from the ODE

*∂**t**u**ε**(t )*= −*∂**t*exp

−1
*ε*
*t*
0

exp

1−1/

*u**ε**(s)*+1_{+}
*/ε*

*ds* as*ε*→0.

**Proof.** *Step 0 (Uniform bound from below): Sinceu** _{ε}*is supercaloric, it is bounded from below by the constant

*u*

_{min}.

*Step 1 (L*

^{2}

*((0, T )*×

*Ω)-bound): The time-integrated functionv*

_{ε}*(t, x)*:=

_{t}0*u*_{ε}*(s, x) ds,*satisfies

*∂**t**v**ε**(t, x)*−*v**ε**(t, x)*=*w**ε**(t, x)*+*u*^{0}_{ε}*(x)* (5)

where*w** _{ε}*is a measurable function satisfying 0

*w*

_{ε}*C.*Consequently

*T*

0

*Ω*

*(∂*_{t}*v*_{ε}*)*^{2}+1
2

*Ω*

|∇*v** _{ε}*|

^{2}

*(T )*=

*T*0

*Ω*

*w** _{ε}*+

*u*

^{0}

_{ε}*∂*_{t}*v** _{ε}*1
2

*T*0

*Ω*

*(∂*_{t}*v*_{ε}*)*^{2}+*T*
2

*Ω*

*C*+*u*^{0}_{ε}^{2}*,*

implying
*T*
0

*Ω*

*u*^{2}_{ε}*T*

*Ω*

*C*+u^{0}_{ε}^{2}*.* (6)

*Step 2 (L*^{2}*((0, T )*×*Ω)-bound for*∇min(u*ε**, M)): For*
*G**M**(z)*:=

*z*^{2}*/2,* *z < M,*
*Mz*−*M*^{2}*/2,* *zM,*
and any*M*∈**N,**

*Ω*

*G*_{M}*(u*_{ε}*)*−*G*_{M}*u*^{0}_{ε}

+
*T*

0

*Ω*

∇min(u*ε**, M)*^{2}=
*T*
0

*Ω*

−*v*_{ε}^{0}min(u*ε**, M)∂** _{t}*exp

−1
*ε*

*t*
0

*g*_{ε}

*u*_{ε}*(s, x)*
*ds* *.*

As*∂** _{t}*exp(−

^{1}

_{ε}

_{t}0*g*_{ε}*(u*_{ε}*(s, x)) ds)*0,we know that*∂** _{t}*exp(−

^{1}

_{ε}

_{t}0*g*_{ε}*(u*_{ε}*(s, x)) ds)*is bounded in*L*^{∞}*(Ω*;*L*^{1}*((0, T ))),*
and

*T*
0

*Ω*

−*v*^{0}* _{ε}*min(u

*ε*

*, M)∂*

*exp*

_{t}−1
*ε*

*t*
0

*g*_{ε}

*u*_{ε}*(s, x)*

*ds* *C*

*Ω*

sup

*(0,T )*

max

min(u*ε**, M),*0

*CML*^{n}*(Ω).*

*Step 3 (Compactness): Let* *χ**M*:**R**→**R** be a smooth non-increasing function satisfying *χ*_{(}_{−∞}_{,M}_{−}1) *χ**M*
*χ**(*−∞*,M)*and let*Φ**M* be the primitive such that*Φ**M**(z)*=*z*for*zM*−1 and*Φ**M**M. Moreover, let(φ**δ**)**δ*∈*(0,1)*be a
family of mollifiers, i.e.*φ** _{δ}*∈

*C*

_{0}

^{0,1}

*(R*

*; [0,+∞*

^{n}*))*such that

*φ** _{δ}*=1 and suppφ

*δ*⊂

*B*

_{δ}*(0).*Then, if we extend

*u*

*and*

_{ε}*v*

^{0}

*by the value 0 to the whole of*

_{ε}*(0,*+∞

*)*×

**R**

^{n}*,*we obtain by the homogeneous Neumann data of

*u*

*that*

_{ε}*∂**t*

*Φ**M**(u**ε**)*∗*φ**δ*

*(t, x)*

=

*χ*_{M}*(u*_{ε}*)*

*χ*_{Ω}*u** _{ε}*−

*v*

^{0}

_{ε}*∂*

*exp*

_{t}−1
*ε*
*t*
0

*g*_{ε}

*u*_{ε}*(s, x)*
*ds*

∗*φ*_{δ}*(t, x)*

=

**R**^{n}

*χ*_{M}*(u*_{ε}*)(t, y)*

*χ*_{Ω}*(y)u*_{ε}*(t, y)*−*v*_{ε}^{0}*(y)∂** _{t}*exp

−1
*ε*
*t*
0

*g*_{ε}

*u*_{ε}*(s, y)*
*ds*

*φ*_{δ}*(x*−*y) dy*

=

**R**^{n}

*φ*_{δ}*(x*−*y)*

−*χ*_{M}^{}

*u*_{ε}*(t, y)*

*χ*_{Ω}*(y)*∇*u*_{ε}*(t, y)*^{2}−*χ*_{M}

*u*_{ε}*(t, y)*

*v*_{ε}^{0}*(y)∂** _{t}*exp

−1
*ε*
*t*
0

*g*_{ε}

*u*_{ε}*(s, y)*
*ds*

+*χ*_{M}

*u*_{ε}*(t, y)*

*χ*_{Ω}*(y)*∇*u*_{ε}*(t, y)*· ∇*φ*_{δ}*(x*−*y) dy.*

Consequently
*T*
0

**R**^{n}

*∂**t*

*Φ**M**(u**ε**)*∗*φ**δ**C*1*(Ω, C, M, δ, T )*

and
*T*
0

**R**^{n}

∇

*Φ*_{M}*(u*_{ε}*)*∗*φ*_{δ}*C*_{2}*(Ω, M, δ, T ).*

It follows that*(Φ*_{M}*(u*_{ε}*)*∗*φ*_{δ}*)*_{ε}_{∈}* _{(0,1)}*is for each

*(M, δ, T )*precompact in

*L*

^{1}

*((0, T )*×

**R**

^{n}*).*

On the other hand
*T*

0

**R**^{n}

Φ_{M}*(u*_{ε}*)*∗*φ** _{δ}*−

*Φ*

_{M}*(u*

_{ε}*)C*

_{3}

*δ*^{2}
*T*
0

*Ω*

∇*Φ*_{M}*(u*_{ε}*)*^{2} ^{1}^{2} +2(M−*u*_{min}*)TL*^{n}

*B*_{δ}*(∂Ω)*
*C*4*(C, Ω, u*min*, M, T )δ.*

Combining this estimate with the precompactness of*(Φ*_{M}*(u*_{ε}*)*∗*φ*_{δ}*)*_{ε}_{∈}* _{(0,1)}*we obtain that

*Φ*

_{M}*(u*

_{ε}*)*is for each

*(M, T )*precompact in

*L*

^{1}

*((0, T )*×

**R**

^{n}*). Thus, by a diagonal sequence argument, we may take a sequenceε*

*→0 such that*

_{m}*Φ*

_{M}*(u*

_{ε}

_{m}*)*→

*z*

*a.e. in*

_{M}*(0,*+∞

*)*×

**R**

*as*

^{n}*m*→ ∞, for every

*M*∈

**N.**At a.e. point of the set{

*z*

_{M}*< M*−1}

*, u*

_{ε}*con- verges to*

_{m}*z*

_{M}*.*At each point

*(t, x)*of the remainder

*M*∈**N**{*z*_{M}*M*−1}*,* the value *u*_{ε}_{m}*(t, x)* must for large *m*
(depending on *(M, t, x)) be larger than* *M*−2. But that means that on the set

*M*∈**N**{*z*_{M}*M*−1}*,* the se-
quence *(u*_{ε}_{m}*)*_{m}_{∈}** _{N}** converges a.e. to+∞

*.*It follows that

*(u*

_{ε}

_{m}*)*

_{m}_{∈}

**converges a.e. in**

_{N}*(0,*+∞

*)*×

*Ω*to a function

*z*:

*(0,*+∞

*)*×

*Ω*→

**R**∪ {+∞}. But then, as

*(u*

_{ε}

_{m}*)*

_{m}_{∈}

**is for each**

_{N}*T*∈

*(0,*+∞

*)*bounded in

*L*

^{2}

*((0, T )*×

*Ω),(u*

_{ε}

_{m}*)*

_{m}_{∈}

**converges by Vitali’s theorem (stating that a.e. convergence and a non-concentration condition in**

_{N}*L*

*imply in bounded domains*

^{p}*L*

*-convergence) for each*

^{p}*p*∈ [1,2)in

*L*

^{p}*((0, T )*×

*Ω)*to the weak

*L*

^{2}-limit

*u*of

*(u*

_{ε}

_{m}*)*

_{m}_{∈}

**. It follows that**

_{N}*L*

^{n}^{+}

^{1}

*(*

*M*∈**N**{*z*_{M}*M*−1}*)*=*L*^{n}^{+}^{1}*(*{*u*= +∞}*)*=0.

*Step 4 (Identification of the limit equation in*esssup_{(0,t )}*u >*0): Let us consider*(t, x)*∈*(0,*+∞*)*×*Ω* such that
*u*_{ε}_{m}*(s, x)*→*u(s, x)*for a.e.*s*∈*(0, t )*and*u(*·*, x)*∈*L*^{2}*((0, t )). In the case esssup*_{(0,t )}*u(*·*, x) >*0, we obtain by Egorov’s
theorem and assumption (2) that exp(−_{ε}^{1}_{m}_{t}

0*g*_{ε}_{m}*(u*_{ε}_{m}*(s, x)) ds)*→0 as*m*→ ∞.

*Step 5 (The case∂**t**u**ε*0): Let*(t, x)*be such that*u**ε*_{m}*(t, x)*→*u(t, x)*=*λ <*0: Then by assumption (1),
exp

−1
*ε*_{m}

*t*
0

*g*_{ε}_{m}

*u*_{ε}_{m}*(s, x)*

*ds* exp

−*t*max_{[}*u*_{min}*,λ/2*]*g**ε*_{m}

*ε*_{m}

→1 as*m*→ ∞*.* 2

**5. Complete characterization of the limit equation in the case of one space dimension**

The aim of this main section is the following theorem:

**Theorem 5.1.***Suppose in addition to the assumptions at the beginning of Section*4*that the space dimensionn*=1*and*
*that the initial datau*^{0}_{ε}*converge inC*^{1}*to a functionu*^{0}*satisfying*∇*u*^{0}=0*on*{*u*^{0}=0}. Then the family*(u*_{ε}*)*_{ε}_{∈}_{(0,1)}*is*
*for eachT* ∈*(0,*+∞*)precompact inL*^{1}*((0, T )*×*Ω), and each limituof(u*_{ε}*)*_{ε}_{∈}_{(0,1)}*as a sequenceε** _{m}*→0,

*satisfies*

*in the sense of distributions the initial-boundary value problem*

Fig. 1. Clearing out.

*∂**t**u*−*v*^{0}*∂**t**χ*=*u* *in(0,*+∞*)*×*Ω,*
*u(0,*·*)*=*u*^{0}+*v*^{0}*H*

*u*^{0}

*inΩ,* ∇*u*·*ν*=0 *on(0,*+∞*)*×*∂Ω,* (7)

*whereH* *is the maximal monotone graph*
*H (z)*

=0, *z <*0,

∈ [0,1]*,* *z*=0,

=1, *z >*0
*and*

*χ (t, x)*=*H*

esssup

*(0,t )*

*u(*·*, x)*
⎧

⎨

⎩

=0, esssup_{(0,t )}*u(*·*, x) <*0,

∈ [0,1]*,* esssup_{(0,t )}*u(*·*, x)*=0,

=1, esssup_{(0,t )}*u(*·*, x) >*0.

Although we assume the space dimension from now on to be 1, we keep the multi-dimensional notation for the
sake of convenience. Moreover we extend*u** _{ε}*by even reflection at the lateral boundary to a space-periodic solution on
[0,+∞

*)*×

**R.**

We start out with some elementary lemmata:

**Lemma 5.2***(Clearing out). There exists a continuous increasing functionω*:[0,1)→ [0,+∞*)such thatω(0)*=0*and*
*the following holds:suppose thatκ <*0, that*εω(*|*κ*|*), thatδ*∈*(0,*1)*and thatu*_{ε}*(1*+*ω(δ))κon the parabolic*
*boundary of the domain* *Q(t*_{0}*, δ, φ*_{1}*, φ*_{2}*)*:= {*(t, x): 0t*_{0}−2δ < t < t0*, φ*_{1}*(t ) < x < φ*_{2}*(t )*}, where *φ*_{1}*< φ*_{2} *are*
*C*^{1}*-functions. Thenu*_{ε}*κinQ(t*_{0}*, δ, φ*_{1}*, φ*_{2}*)*(cf. Fig.1).

**Proof.** Comparing*u**ε*in*Q(t*0*, δ, φ*1*, φ*2*)*to the solution of the ODE
*y*^{}*(t )*=*Cg**ε**(y)/ε, y(t*0−2δ)=

1+*ω(δ)*
*κ*
we obtain the statement of the lemma. 2

**Lemma 5.3.***For almost allκ <*0*the level set((0,*+∞*)*×*Ω)*∩ {*u** _{ε}*=

*κ*}

*is a locally finite union ofC*

^{1}

*-curves. For*

*suchκwe define the set*

*S** _{κ,ε}*:=

*(t, x)*∈*(0,*+∞*)*×*Ω):* *u*_{ε}*(t, x) > κ*

*and there is no(t*_{0}*, δ, φ*_{1}*, φ*_{2}*)*∈ [0,+∞*)*×*(0,*1)×*C*^{1}×*C*^{1}

*such thatu*_{ε}*κon the parabolic boundary of the domainQ(t*_{0}*, δ, φ*_{1}*, φ*_{2}*)*
(cf. Fig.2). Then*∂S** _{κ,ε}*=

*N*

*κ,ε*

*j*=1 *graph(g*_{j,κ,ε}*)whereg** _{j,κ,ε}*:[0, T

*j,κ,ε*] →

**R**

*are piecewiseC*

^{1}

*-functions andN*

_{κ,ε}*is*

*for smallεbounded by a constant depending only on the limitu*

^{0}

*of the initial data.*

Fig. 2. The set*S**κ,ε*.

**Remark 5.4.**For illustration of the definition of*S** _{κ,ε}*, imagine the set{

*u*

_{ε}*> κ*}filled with water in a

*(t, x)-plane where*

*t*represents the height. Our modification of{

*u*

_{ε}*> κ*}means then that the water is now allowed to flow out through the

“bottom”{*t*=0}.

**Proof of Lemma 5.3.** By the definition of*S** _{κ,ε}*and by the fact that

*u*

*is supercaloric, each connected component of*

_{ε}*∂S** _{κ,ε}*is a piecewise

*C*

^{1}-curve and touches{

*t*=0}. Therefore the number of connected components is for small

*ε >*0 bounded by a constant

*N*˜ depending only on the limit

*u*

^{0}of the initial data.

Let us consider one connected component*γ*of*∂S** _{κ,ε}*. By the definition of

*S*

*and by the fact that*

_{κ,ε}*u*

*is supercaloric, the derivative of the time-component of the piecewise*

_{ε}*C*

^{1}-curve

*γ*can change its sign at most once! Thus we can define for each curve

*γ*one or two piecewise

*C*

^{1}-functions of time such that

*γ*is the union of the graphs of the two functions.

The total number of graphs*N** _{κ,ε}*is therefore bounded by 2

*N*˜. 2

**Proof of Theorem 5.1.** By Theorem 4.1 we only have to prove that *χ*=0 in the set{esssup*(0,t )**u(*·*, x) <*0}. The
main problem is to exclude “peaking” of the solution*u** _{ε}*, i.e. tiny sets where

*u*

_{ε}*> κ. Here we show that in the case of*one space dimension, “peaking” is not possible. More precisely, if the measure of the set

*u*

_{ε}*> κ*is small in a parabolic cube, then

*u*

*ε*is strictly negative in the cube of half the radius, uniformly in

*ε. The proof is carried out in two steps:*

*Step 1: Let(ε**m**)**m*∈**N**be the subsequence in the proof of Theorem 4.1. As a.e. point*(t, x)*∈*((0,*+∞*)*×R)∩{*u <*0}

is a Lebesgue point of the set{*u <*0}, we may assume that there exists*κ <*0 such that for any*θ*∈*(0,*1), sufficiently
small*r*0*>*0 and every*ε**m*∈*(0, ε*0*),*

*L*^{2}

*(t*−2r0*, t )*×*B*2r_{0}*(x)*

∩ {*u**ε*_{m}*<*2κ}

*θL*^{2}

*(t*−2r0*, t )*×*B*2r_{0}*(x)*
*.*

*Step 2: Suppose now that((t*−*r*0*, t )*×*B**r*_{0}*(x))*∩ {*u**ε*_{m}*> κ*} = ∅(where*κ*is chosen such that{*u**ε** _{m}*=

*κ*}and{

*u*

*ε*

*= 2κ}are locally finite unions of*

_{m}*C*

^{1}-curves): then

*((t*−

*r*0

*, t )*×

*B*

*r*

_{0}

*(x))*∩

*∂S*

*κ,ε*

*and*

_{m}*((t*−

*r*0

*, t )*×

*B*

*r*

_{0}

*(x))*∩

*∂S*2κ,ε

_{m}must by Lemma 5.3 be connected to the parabolic boundary of*(t*−2r0*, t )*×*B*2r_{0}*(x). TheL*^{2}*((0, T )*×*Ω)-bound for*

∇min(u*ε**, M), the fact thatL*^{2}*(((t*−2r0*, t )*×*B*_{2r}_{0}*(x))*∩ {*u*_{ε}_{m}*<*2κ}*)θL*^{2}*(((t*−2r0*, t )*×*B*_{2r}_{0}*(x)))*and Lemma 5.2
imply now (see Fig. 3) that there must be an “almost horizontal” component of*∂S*_{κ,ε}* _{m}* (cf. Fig. 4) with the following
properties: for any

*δ*∈

*(0,*1), there are

*t*−

*r*

_{0}

*< t*

_{1}

*< t*

_{2}

*< t*

_{3}

*< t*such that (see Fig. 5)

*t*

_{3}−

*t*

_{1}→0 as

*ε*

*→0, for some*

_{m}*j*

g_{j,κ,ε}_{m}*(t*_{2}*)*−*g*_{j,κ,ε}_{m}*(t*_{1}*)c*_{1}*>*0,
and

*L*^{1}

*y*∈*B**r*_{0}*(x):* *u**ε*_{m}*(t*3*, y) >*2κ

*δ,*

*B**r*0*(x)*∩{*u*_{εm}*(t*_{3}*,*·*)>2κ*}

u_{ε}_{m}*(t*_{3}*, y)dyδ.*

Fig. 3. Situation excluded by the*L*^{2}*(W*^{1,2}*)-estimate.* Fig. 4. The main task is to exclude almost horizontal propa-
gation.

Fig. 5. The set*D**ε**m*.

We may assume that*c*_{1}*< r*_{0}, that*g*_{j,κ,ε}_{m}*(t*_{2}*)*=sup_{(t}_{1}_{,t}_{2}_{)}*g*_{j,κ,ε}* _{m}*, that

*g*

_{j,κ,ε}

_{m}*(t*

_{2}

*) > g*

_{j,κ,ε}

_{m}*(t*

_{1}

*)*and that

*u*

_{ε}

_{m}*(s, y) > κ*for some

*d >*0 and

*(s, y)*∈

*(t*

_{1}

*, t*

_{2}

*)*×

*B*

_{r}_{0}

*(x)*such that

*g*

_{j,κ,ε}

_{m}*(s) < y < d*+

*g*

_{j,κ,ε}

_{m}*(s). We define the set*

*D*

_{ε}*:=*

_{m}{*(s, y):* *t*_{1}*< s < t*_{3}*, y < g*_{j,κ,ε}_{m}*(s)* for *s*∈*(t*_{1}*, t*_{2}*)*and *y < g*_{j,κ,ε}_{m}*(t*_{2}*)*for *s*∈ [*t*_{2}*, t*_{3}*)*}(cf. Fig. 5) and the cut-off
function*φ(y)*:=max(0,min(y−*g*_{j,κ,ε}_{m}*(t*_{1}*), g*_{j,κ,ε}_{m}*(t*_{2}*)*−*y)). It follows that*

*c*^{2}_{1}*κ/4*+2δ+o(1)o(1)+

*g*_{j,κ,εm}*(t*_{2}*)*
*g*_{j,κ,εm}*(t*_{1}*)*

*φ(y)*

*u*_{ε}_{m}*(t*_{3}*, y)*−*κ*
*dy*

o(1)+

*g*_{j,κ,εm}*(t*_{2}*)*
*g*_{j,κ,εm}*(t*_{1}*)*

*φ(y)u*_{ε}_{m}*(t*_{3}*, y) dy*−*κ*

*t*2

*t*1

*φ*

*g*_{j,κ,ε}_{m}*(s)*
*g*_{j,κ,ε}^{}

*m**(s) ds*

*g*_{j,κ,εm}*(t*_{2}*)*
*g*_{j,κ,εm}*(t*_{1}*)*

*φ(y)u**ε*_{m}*(t*3*, y) dy*−

*t*_{2}

*t*1

*φ*

*g**j,κ,ε*_{m}*(s)*
*u**ε*_{m}

*s, g**j,κ,ε*_{m}*(s)*
*g*^{}_{j,κ,ε}

*m**(s) ds*

*D*_{εm}

*φ∂*_{t}*u*_{ε}_{m}

*D*_{εm}

*φu*_{ε}_{m}

= −

*D*_{εm}

∇*φ*· ∇*u*_{ε}* _{m}*+

*t*1

*φ*

*s, g*_{j,κ,ε}_{m}*(s)*

*∂*_{x}*u*_{ε}_{m}

*s, g*_{j,κ,ε}_{m}*(s)*

*χ*_{{}_{u}_{εm}_{(s,g}_{j,κ,εm}_{(s))}_{=}_{κ}_{}}*ds*

−

*D*_{εm}

∇*φ*· ∇*u*_{ε}* _{m}*→0 as

*ε*

*→0,*

_{m}a contradiction for small*ε** _{m}*provided that

*δ*has been chosen small enough; in the third inequality we used Lemma 5.2, and the convergence to 0 is due to the uniform

*L*

^{2}

*(W*

^{1,2}

*)-bound.*2

**6. Applications**

Here we mention two examples of different systems leading to the same limit.

For the convergence results below we assume that the space dimension is 1.

*6.1. The Matkowsky–Sivashinsky scaling*

We apply our result to the scaling in [8, Eq. (2)], i.e. the following system of solid combustion

*∂*_{t}*u** _{N}*−

*u*

*=*

_{N}*(1*−

*σ*

_{N}*)N v*

*exp*

_{N}*N (1*−1/u*N**)*
*,*

*∂*_{t}*v** _{N}*= −

*N v*

*exp*

_{N}*N (1*−1/u*N**)*

*,* (8)

where the normalized temperature*u** _{N}* and the normalized concentration

*v*

*are non-negative,*

_{N}*(σ*

_{N}*)*

_{N}_{∈}

**[0,1)(in the case**

_{N}*σ*

*↑1, N↑ ∞the limit equation in the scaling as it is would be the heat equation, but we could still apply our result to*

_{N}*u*

_{N}*/(1*−

*σ*

_{N}*)) and the activation energyN*→ ∞.

Setting*u*_{min}:= −1, ε:=1/N, u*ε*:=*u** _{N}*−1 and

*g*

_{ε}*(z)*:=

exp((1−1/(z+1))/ε), *z >*−1,

0, *z*−1

and integrating the equation for*v** _{N}* in time, we see that the assumptions of Theorem 5.1 are satisfied and we obtain
that each limit

*u*

_{∞}

*, σ*

_{∞}of

*u*

_{N}*, σ*

*satisfies*

_{N}*∂*_{t}*u*_{∞}−*(1*−*σ*_{∞}*)v*^{0}*∂*_{t}*H*

esssup

*(0,t )*

*u*_{∞}

=*u*_{∞} in*(0,*+∞*)*×*Ω,*

*u*_{∞}*(0,*·*)*=*u*^{0}+*v*^{0}*H*
*u*^{0}

in*Ω,* ∇*u*_{∞}·*ν*=0 on*(0,*+∞*)*×*∂Ω,* (9)

where*v*^{0}are the initial data of*v*_{∞}. Moreover,*χ*is increasing in time and*u*_{∞}is a supercaloric function.

*6.2. Another scaling with temperature threshold*

Here we consider (cf. [1, pp. 109–110]), i.e. the following system of solid combustion

*∂**t**θ**N*−*θ**N*=*(1*−*σ**N**)N Y**N*exp

*N (1*−*σ**N**)(θ**N*−1)
*/*

*σ**N*+*(1*−*σ**N**)θ**N*

*χ*_{{}_{θ}

*N**>¯**θ*}*,*

*∂**t**Y**N*= −*(1*−*σ**N**)N Y**N*exp

*N (1*−*σ**N**)(θ**N*−1)
*/*

*σ**N*+*(1*−*σ**N**)θ**N*

*χ*_{{}_{θ}

*N**>**θ*¯} (10)

where*N (1*−*σ*_{N}*)*1,*σ** _{N}*∈

*(0,*1)and the constant

*θ*¯∈

*(0,*1)is a threshold parameter at which the reaction sets in.

Setting*u*_{min}= −1,*ε*:=1/(N (1−*σ*_{N}*)),κ(ε)*:=1−*σ** _{N}*,

*u*

*:=*

_{ε}*θ*

*−1,*

_{N}*g*

_{ε}*(z)*:=

exp((z/(κ(ε)z+1))/ε), *z >θ*¯−1,

0, *zθ*¯−1

and integrating the equation for*Y** _{N}* in time, we see that the assumptions of Theorem 5.1 are satisfied and we obtain
that each limit

*u*

_{∞}of

*u*

*satisfies*

_{N}*∂*_{t}*u*_{∞}−*v*^{0}*∂*_{t}*H*

esssup

*(0,t )*

*u*_{∞}

=*u*_{∞} in*(0,*+∞*)*×*Ω,*

*u*_{∞}*(0,*·*)*=*u*^{0}+*v*^{0}*H*
*u*^{0}

in*Ω,* ∇*u*_{∞}·*ν*=0 on*(0,*+∞*)*×*∂Ω,* (11)

where*v*^{0}are the initial data of*v*_{∞}. Moreover,*χ* is increasing in time and*u*_{∞}is a supercaloric function.

**7. Existence of pulsating waves**

The aim of this section is to construct pulsating waves for the limit problem. For the sake of clarity we have chosen not to present the most general result in the following theorem. Moreover we confine ourselves to the one-phase case.

**Theorem 7.1***(Existence of pulsating waves). Let us consider a Hölder continuous functionv*^{0}*defined on* **R**^{n}*that*
*satisfies*

*v*^{0}*(x)*1 *and* *v*^{0}*(x*+*k)*=*v*^{0}*(x)* *for everyk*∈**Z**^{n}*, x*∈**R**^{n}*.*

*Given a unit vectore*∈**R**^{n}*and a velocityc >*0, there exists a solution*u(t, x)of the one-phase problem*

⎧⎪

⎪⎨

⎪⎪

⎩

*∂*_{t}*u*−*v*^{0}*∂*_{t}*χ*_{{}_{u}_{}_{0}_{}}=*u* *on***R**×**R**^{n}*,*

*∂*_{t}*u*0 *and* −*μ*_{0}:= −

[0,1)^{n}

*v*^{0}*u*0, (12)

*which satisfies*

⎧⎪

⎨

⎪⎩

*u(t, x*+*k)*=*u*

*t*−*e*·*k*
*c* *, x*

*for everyk*∈**Z**^{n}*,* *(t, x)*∈**R**×**R**^{n}*,*
*u(t, x)*=0 *forx*·*e*−*ct*0 *and* lim sup

*x*·*e*−*ct*→+∞*u(t, x)*= −*μ*_{0}*,* (13)

*where the last limit is uniform asx*·*e*−*ct* *tends to*+∞.

Let us transform the problem by the so-called Duvaut transform (see [10]), setting*w(t, x)*= −_{+∞}

*t* *u(s, x) ds.*

In this section we will prove the existence of a pulsating wave*w. More precisely, Theorem 7.1 is a corollary of the*
following result which will be proved later.

**Theorem 7.2** *(Pulsating waves for the obstacle problem). Under the assumptions of Theorem*7.1, there exists a
*functionw(t, x)solving the obstacle problem*

*∂*_{t}*w*=*w*−*v*^{0}*χ*_{{}_{w>0}_{}} *on***R**×**R**^{n}*,*

*w*0, −*μ*_{0}*∂*_{t}*w*0, *∂*_{t t}*w*0, (14)

*with the conditions*

⎧⎨

⎩

*w(t, x*+*k)*=*w*

*t*−*e*·*k*
*c* *, x*

*for everyk*∈**Z**^{n}*, (t, x)*∈**R**×**R**^{n}*,*

*w(t, x)*=0 *forx*·*e*−*ct*0 *and* *∂**t**w(t, x)*→ −*μ*0 *asx*·*e*−*ct*→ +∞*.*

(15)
*The convergence is uniform asx*·*e*−*ct* *tends to*+∞*.*

**Proof of Theorem 7.1.** Simply set*u(t, x)*:=*∂*_{t}*w(t, x)*with*w*given by Theorem 7.2, and use the fact that*χ*_{{}_{u<0}_{}}=
*χ*_{{}_{w>0}_{}}. To check this last property, it is sufficient to exclude the case where*w(t*_{0}*, x*_{0}*) >*0 and*∂*_{t}*w(t*_{0}*, x*_{0}*)*=0 at some
point*(t*_{0}*, x*_{0}*): using the fact that∂*_{t}*w*is caloric in{*w >*0}as well as the strong maximum principle, we deduce that

*∂*_{t}*w(t, x)*=0 for*t*∈*(*−∞*, t*_{0}]and*x*in a neighborhood of*x*_{0}. This contradicts the last line of (15). 2

**Proof of Theorem 7.2.** We will prove the existence of an unbounded solution*w*in six steps, approximating*w*by
bounded solutions of a truncated equation, for which we can apply the existence of pulsating fronts due to Berestycki
and Hamel [2].

*Step 1: Approximation by bounded solutions and estimates of the velocity. For any 0< A < M, let us start by*
approximating the function*χ*_{(0,}_{+∞}* _{)}* by the characteristic function

*g*=

*χ*

_{(0,A}_{]}. In that case we can compute explicitly the

*traveling wave(φ, c*

_{0}

*)*(unique up to translations of

*φ) of*

*c*_{0}*φ*^{}=*φ*^{}−*g(φ),* *φ*^{}0 on**R,** *φ(*−∞*)*=*M* and *φ(*+∞*)*=0. (16)

Let us define for*c*_{0}*>*0, M >0, s0∈*(*−∞*,*0)and*s*_{1}∈*(s*_{0}*,*0)

*φ(s)*=

⎧⎪

⎨

⎪⎩

*M(1*−*e*^{c}^{0}^{(s}^{−}^{s}^{0}^{)}*)* for*s*∈*(*−∞*, s*_{1}]*,*

1

*c*_{0}^{2}*(e*^{c}^{0}* ^{s}*−1−

*c*0

*s)*for

*s*∈ [

*s*1

*,*0]

*,*

0 for*s*∈ [0,+∞*).*

For any *A*∈*(0, M)* and for suitable *s*0*, s*1, we see that *φ* is continuous and satisfies*φ(s*1*)*=*A, which fixes the*
parameter*s*1as a function of*A. Moreover we see thatφ*is of class*C*^{1}if and only if*s*1= −*c*0*M*and

*M*−*A*= 1
*c*_{0}^{2}

1−*e*^{−}^{c}^{2}^{0}^{M}

*.* (17)

Thus*A*is determined in terms of the velocity*c*_{0}and*M. The above calculations show in particular thatφ(c*_{0}*t*−*e*·*x)*
is a good bounded approximation of the solution of (14), (15) in the case*v*^{0}=1, i.e. the traveling wave case.

In all that follows let*A*be given by (17).

Now, when the function*g* in (16) is replaced by a Lipschitz continuous function whose support is a compact
interval, there are known results on the existence of pulsating waves. For such*g, it is possible to apply Theorem 1.13 of*
Berestycki and Hamel [2], which states the existence (and uniqueness up to translation in time) of bounded pulsating
solutions traveling at a unique velocity. Bearing that in mind, we define *g** _{M}* as a Lipschitz regularization of the
characteristic function

*g*such that supp

*g*

*= [0, A]and – for later use –*

_{M}*g**M*=1 on[1/M, A/2]*,* 0*g**M*1 on**R,** and

*g*^{}* _{M}*0,

*g*

^{}

*0 on*

_{M}*(0, A/2).*(18)

Let us call*c*_{M}^{0} the unique velocity of the traveling wave equation (16) with*g*replaced by*g** _{M}*. As

*(φ, c*

_{0}

*)*can be shown to be unique up to translations of

*φ),c*

_{M}^{0}→

*c*0as

*g*

*→*

_{M}*g.*

Then there exists by Theorem 1.13 of [2] a bounded pulsating wave*w** _{M}* traveling at velocity

*c*

*such that*

_{M}⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎩

*∂*_{t}*w** _{M}*=

*w*

*−*

_{M}*v*

^{0}

*g*

_{M}*(w*

_{M}*)*on

**R**×

**R**

^{n}*,*

*∂**t**w**M*0 and lim sup

*x*·*e*−*c**M**t*→−∞*w**M**(t, x)*=0*w**M**M*= lim inf

*x*·*e*−*c*_{M}*t*→+∞*w**M**(t, x)* and
*w*_{M}*(t, x*+*k)*=*w*_{M}

*t*−*e*·*k*

*c*_{M}*, x*

for every*k*∈**Z**^{n}*, (t, x)*∈**R**×**R**^{n}*.*

From the assumption 1*v*^{0}and the comparison principle (see Lemma 3.2 and 3.4 of [2]) we infer that the velocities
satisfy the ordering*c*^{0}_{M}*c** _{M}*. Similarly, defining

*λ*=

*v*

^{0}

*L*

^{∞}and comparing to

*w*

_{M}*(λt,*√

*λx), we getc*_{M}*c*^{0}* _{M}*√

*λ.*

Furthermore the above comparison principles tell us that the velocity *c**M* (resp. *c*^{0}* _{M}*) is continuous and non-
decreasing in

*M.*

For all that follows let*c >*0 be an arbitrary but fixed velocity for which we want to construct the pulsating wave.

Then, for any*M >*0 we can adjust*A*∈*(0, M)*such that for*c*0defined by (17),
*c*=*c** _{M}*∈

*c*_{0}*/2,*2c0

v^{0}

*L*^{∞}

*.*

In order to pass to the limit as*M*→ +∞, we need to get some bounds on the solution first. To this end, rotating
space–time proves to be very convenient:

*Step 2: Space–time transformation and first estimates on the time derivatives. Let us introduce the functionw*˜* _{M}*
defined by

*w*˜

_{M}*(s, x)*=

*w*

_{M}*(*

^{s}^{+}

_{c}

^{e}^{·}

^{x}*, x)*which is periodic in

*x*and satisfies

*Lw*˜* _{M}*=

*v*

^{0}

*(x)g*

_{M}*(w*˜

_{M}*)*with

*Lw*˜

*=*

_{M}*w*˜

*+*

_{M}*∂*

_{ss}*w*˜

*−2∂*

_{M}*e,s*

*w*˜

*−*

_{M}*c∂*

_{s}*w*˜

_{M}