Comptes Rendus Mathématique

Camille Laurent

Matthieu Léautaud

On uniform controllability of 1D transport equations in the vanishing viscosity limit

Christoph Walker

Comptes Rendus
Mathématique
Christoph Walker
A Remark on a Nonlocal-in-Time Heat Equation
Volume 361 (2023), p. 825-831
Published online: 11 May 2023
https://doi.org/10.5802/crmath.443
This article is licensed under the
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http://creativecommons.org/licenses/by/4.0/
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e-ISSN : 1778-3569
Comptes Rendus
Mathématique
2023, Vol. 361, p. 825-831
https://doi.org/10.5802/crmath.443
Partial differential equations / Équations aux dérivées partielles
A Remark on a Nonlocal-in-Time Heat
Equation
ChristophWalkera
a Leibniz Universität Hannover, Institut für Angewandte Mathematik, Welfengarten 1,
30167 Hannover, Germany
E-mail: walker@ifam.uni-hannover.de
Abstract. Schauder’s fixed point theorem is used to derive the existence of solutions to a semilinear heat
equation. The equation features a nonlinear term that depends on the time-integral of the unknown on the
whole, a priori given, interval of existence.
Keywords. Semilinear heat equation, nonlocal in time, existence of global solutions.
Mathematical subject classification (2010). 35K58.
Manuscript received 12 October 2022, revised 22 December 2022, accepted 23 December 2022.
1. Introduction
This note is dedicated to the nonlocal problem
∂t u −div
(
d(x)∇u)+ϕ(∫ ∞
0
a(s, x)u(s, x)ds
)
u = f (t , x) , (t , x) ∈ (0,∞)×Ω , (1a)
u(0, · ) = u0 , u|∂Ω = 0 (1b)
on a bounded smooth domain Ω ⊂ Rn with d ∈ C 1(Ω) such that d(x) ≥ d0 > 0 for x ∈ Ω. The
potential ϕ, the weight a, the initial datum u0, and the right-hand side f are suitable given
functions.
Equation (1) is used in the modeling of a biological nanosensor in the chaotic dynamics of a
polymer chain in an aqueous solution and has been introduced and considered in [5–8]. It can
also be seen as a toy model for equilibrium states in age-structured diffusive populations (with t
referring to the age of individuals), see [9, 10] for instance.
Note that the unknown weighted time-integral
u =
∫ ∞
0
a(s)u(s)ds
depends on the whole, a priori given interval of existence (0,∞) (the case of a bounded time
interval (0,T ) is included in (1), of course). Hence, only global solutions are of interest. Moreover,
(1) is no usual evolution problem satisfying a Volterra property since solutions at a time instant
depend also on future time instants. For the homogeneous version of (1) on a bounded interval
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
826 Christoph Walker
time (0,T ) with vanishing right-hand side f and without weight a, existence of weak solutions
was derived in [5, 6] and strong solutions in [11]. We also refer to [4] for existence of weak
solutions when the Laplacian is replaced by an integrodifferential operator of Lévy type. The
non-homogeneous problem (1) on a bounded interval (0,T ) was investigated in [7] where, for
unbounded potentials ϕ, a truncation approach and weak compactness methods were used to
prove the existence of weak solutions under fairly general conditions.
The purpose of this work is to propose an alternative approach to (1) for deriving the existence
of mild and strong solutions under slightly different conditions. This approach has been used
in [11] and may also be a template for other nonlocal problems. More precisely, we shall use the
fact that solutions to (1) may be written as mild solutions in the form
u(t ) = e t A(u)u0 +
∫ t
0
e(t−s)A(u) f (s)ds , t ≥ 0, (2)
where (e t A(u))t≥0 is the contraction semigroup on Lp (Ω) generated by the operator
A(u) := [w 7→ div(d(x)∇w)−ϕ(u)w]
subject to Dirichlet boundary conditions (see below for details). Integrating the representation (2)
yields the equivalent fixed point equation
u =
∫ ∞
0
a(t )e t A(u)u0 dt +
∫ ∞
0
a(t )
∫ t
0
e(t−s)A(u) f (s)ds dt (3)
for u. We then shall focus on this fixed point equation and prove, in particular, that the right-
hand side of (3) enjoys suitable compactness properties with respect to u that allow us to apply
Schauder’s theorem leading to the following existence result:
Theorem 1. Let a ∈ L1(R+,L∞(Ω)), ϕ ∈ C (R,R+), and f ∈ L1(R+,L∞(Ω))∩L∞,loc (R+,Lp (Ω)) for
some p ∈ (n/2∨1,∞) and let u0 ∈ L∞(Ω). Then there is a mild solution u ∈C
(
R+,Lp (Ω)
)
to (1) such
that
∥u(t )∥∞ ≤ ∥u0∥∞+∥ f ∥L1((0,t ),L∞(Ω)) , t ∈R+ .
If f ∈Cθ(R+,Lq (Ω))+C (R+,W θq (Ω)) with θ > 0 and q ∈ (1,∞), then u is a strong solution with
u ∈C(R+,Lq (Ω))∩C 1(Ṙ+,Lq (Ω))∩C(Ṙ+,W 2q (Ω)) ,
with Ṙ+ := (0,∞).
Some of the assumptions may be weaken, e.g. for linearly bounded ϕ or smoother a.
In Section 2 we prove Theorem 1. The crucial compactness properties of the integral terms
appearing on the right-hand side of (3) are postponed to Section 3. The proofs there are inspired
by the works [2, 3] and may be extended to more general frameworks than the one considered
herein for (1), e.g. to other semilinear and possibly quasilinear equations (see Remark 5 in this
regard).
2. Proof of Theorem 1
Notation and Preliminaries
We use the notation
W αp,D (Ω) :=
{
{u ∈W αp (Ω) ; u = 0 on ∂Ω} if 1p <α≤ 2,
W αp (Ω) if 0 ≤α< 1p ,
and we write A ∈H (W 2p,D (Ω),Lp (Ω)) if A ∈L (W 2p,D (Ω),Lp (Ω)) generates an analytic semigroup
(e tA )t≥0 on Lp (Ω). Recall that[
w 7→ div(d(x)∇w)] ∈H (W 2p,D (Ω),Lp (Ω))
Christoph Walker 827
provided d ∈C 1(Ω)) with d(x) ≥ d0 > 0 for x ∈Ω.
Since ϕ is uniformly continuous and bounded on bounded sets, it follows that (considered as
Nemytskii operator)
ϕ ∈C(L∞(Ω),L∞(Ω)) is bounded on bounded sets . (4)
Given R0 > 0 denote by
X :=BL∞(Ω)(0,R0)
the closed ball in L∞(Ω) of radius R0 centered at the origin. Recall that p ∈ (n/2∨1,∞) and note
that, given any u ∈ X , the mapping ϕ(u) := [w 7→ϕ(u)w] ∈L (Lp (Ω)) satisfies
∥ϕ(u)∥L (Lp (Ω)) ≤ ∥ϕ(u)∥∞ ≤ max[−R0,R0]ϕ , u ∈ X .
Setting
A(u)w := div(d∇w)−ϕ(u)w , w ∈W 2p,D (Ω) ,
it then follows from standard perturbation results that
A(u) ∈H (W 2p,D (Ω),Lp (Ω)) , u ∈ X .
In fact, since ϕ is nonnegative, (e t A(u))t≥0 is a positive contraction semigroup on each Lq (Ω) for
q ∈ (1,∞] (which, however, is not strongly continuous for q =∞), hence
∥e t A(u)∥L (Lq (Ω)) ≤ 1, t ≥ 0, q ∈ (1,∞] . (5)
Moreover, we have s(A(u)) ≤ s0 < 0 for its spectral bound with s0 denoting the spectral bound of
the operator
[
w 7→ div(d∇w)]. It then follows from [1, II.Lemma 5.1.3] that there is ν > 0 and,
given 2θ ∈ [0,2] \ {1/p}, there is M(R0) ≥ 1 such that
∥e t A(u)∥L (Lp (Ω),W 2θp,D (Ω)) ≤ M(R0)e
−νt t−θ , t > 0, u ∈ X . (6)
In the following we fix 2θ ∈ (n/p,2) and note the compact embedding
W 2θp,D (Ω)
c
,→ L∞(Ω) ,→ Lp (Ω) . (7)
Let us also observe that, given t > 0 and u, v ∈ X , we have
e t A(u) −e t A(v) =−
∫ t
0
d
ds
e(t−s)A(u)e s A(v) ds =−
∫ t
0
e(t−s)A(u)
(
ϕ(u)−ϕ(v))e s A(v) ds . (8)
We then use (6) and (7) to get
∥e t A(u) −e t A(v)∥L (Lp (Ω),L∞(Ω)) ≤ c ∥e t A(u) −e t A(v)∥L (Lp (Ω),W 2θp,D (Ω))
≤
∫ t
0
∥e(t−s)A(u)∥L (Lp (Ω),W 2θp,D (Ω)) ∥ϕ(u)−ϕ(v)∥L (Lp (Ω)) ∥e
s A(v)∥L (Lp (Ω)) ds
≤ c(R0)e−νt t 1−θ ∥ϕ(u)−ϕ(v)∥∞ . (9)
We are now in a position to provide the proof of Theorem 1.
Proof of Theorem 1. We show that the mapping
Φ(u) :=
∫ ∞
0
a(t )e t A(u)u0 dt +
∫ ∞
0
a(t )
∫ t
0
e(t−s)A(u) f (s)ds dt , u ∈ X =BL∞(Ω)(0,R0) ,
with
R0 := ∥a∥L1(R+,L∞(Ω))
(∥u0∥∞+∥ f ∥L1(R+,L∞(Ω)))
828 Christoph Walker
has a fixed point. Note first from (5) that ∥Φ(u)∥∞ ≤ R0 for u ∈ X , hence Φ : X → X . Next, due
to (9) and (7) we obtain, for u, v ∈ X ,
∥Φ(u)−Φ(v)∥∞ ≤
∫ T
0
∥a(t )∥∞ ∥e t A(u) −e t A(v)∥L (Lp (Ω),L∞(Ω)) ∥u0∥p dt
+
∫ T
0
∥a(t )∥∞
∫ t
0
∥e(t−s)A(u) −e(t−s)A(v)∥L (Lp (Ω),L∞(Ω)) ∥ f (s)∥p ds dt
≤ c(R0)
∫ ∞
0
∥a(t )∥∞ e−νt t 1−θ dt ∥u0∥p ∥ϕ(u)−ϕ(v)∥∞
+ c(R0)
∫ ∞
0
∥a(t )∥∞
∫ t
0
e−ν(t−s) (t − s)1−θ ∥ f (s)∥p ds dt ∥ϕ(u)−ϕ(v)∥∞
and hence, using ν> 0,
∥Φ(u)−Φ(v)∥∞ ≤ c(R0)∥a∥L1(R+,L∞(Ω))
(∥u0∥p +∥ f ||L1(R+,Lp (Ω)))∥ϕ(u)−ϕ(v)∥∞ . (10)
Therefore, Φ ∈ C (X , X ) according to (4). From Proposition 3 and Proposition 4 we deduce that
Φ ∈C (X , X ) has precompact image so that Schauder’s fixed point theorem yields u ∈ X such that
u =Φ(u). We may then define u by (2) in order to obtain a mild solution to (1) which belongs to
C (R+,Lp (Ω)) due to [1, II.Theorem 5.3.1] since f ∈ L∞,loc (R+,Lp (Ω)).
Finally, if f ∈ Cθ(R+,Lq (Ω)) + C (R+,W θq (Ω)) for some θ > 0 and q ∈ (1,∞), then u is a
strong solution with the regularity properties stated in Theorem 1, see [1, II.Theorem 1.2.1,
II.Theorem 1.2.2]. This proves Theorem 1. □
Remark 2. If ϕ is locally Lipschitz continuous, then one may derive the existence and unique-
ness of a solution using Banach’s fixed point argument for Φ : X → X provided that
R0 = ∥a∥L1(R+,L∞(Ω))
(∥u0∥∞+∥ f ∥L1(R+,L∞(Ω)))
is small enough.
3. Compactness Properties
We provide the compactness results used in the proof of Theorem 1. This section relies on the
papers [2, 3] and adapts these ideas to our setting.
We first consider the non-homogeneous part.
Proposition 3. Let a ∈ L1(R+,L∞(Ω)) and f ∈ L1(R+,L∞(Ω)) ∩ L∞,loc (R+,Lp (Ω)) for some p ∈
(n/2∨1,∞). Define
F (u) :=
∫ ∞
0
a(t )
∫ t
0
e(t−s)A(u) f (s)ds dt , u ∈ X .
Then the set
{
F (u) ; u ∈ X } is precompact in L∞(Ω).
Proof. Given T > 0 introduce
XT :=C ([0,T ],L∞(Ω))
and
F (u)(t ) :=
∫ t
0
e(t−s)A(u) f (s)ds , t ∈ [0,T ] , u ∈ X =BL∞(Ω)(0,R0) .
It suffices to prove that F ∈ C (X ,XT ) is compact for every T > 0 since the assertion then follows
by a diagonal sequence argument and the assumption a ∈ L1(R+,L∞(Ω)).
(i). Since f ∈ L∞,loc (R+,Lp (Ω)), we infer from [1, II.Theorem 5.3.1] that F (u) ∈ C ([0,T ],W 2θp (Ω))
and hence F (u) ∈XT for u ∈ X by (7). Moreover, given u, v ∈ X and t ∈ [0,T ], we have, as in (10),
∥F (u)(t )−F (v)(t )∥∞ ≤ c(R0)∥ f ∥L1(R+,Lp (Ω)) ∥ϕ(u)−ϕ(v)∥∞ .
Therefore, F ∈C (X ,XT ) owing to (4).
Christoph Walker 829
(ii). In order to prove that F ∈C (X ,XT ) has precompact image we use an idea inspired by [2, 3]:
for fixed λ> 0 we first show that{
eλA(u)F (u) ; u ∈ X
}
is precompact in XT . (11)
To this end, we note from (6) and (5) that, for u ∈ X and t ∈ [0,T ],
∥eλA(u)F (u)(t )∥W 2θp,D (Ω) ≤ c(R0)λ
−θ ∥F (u)(t )∥p ≤ c(R0)λ−θ∥ f ∥L1(R+,Lp (Ω)) . (12)
That is, invoking (7), the set{
eλA(u)F (u)(t ) ; u ∈ X
}
is precompact in L∞(Ω) (13)
for every t ∈ [0,T ]. Before proceeding let us note that, given u ∈ X , δ> 0 and h ≥ 0,
e(δ+h)A(u) −eδA(u) =
∫ h
0
d
ds
e(δ+s)A(u) ds =
∫ h
0
e(
δ
2 +s)A(u) A(u)e
δ
2 A(u) ds
so that, using (6),
∥e(δ+h)A(u) −eδA(u)∥L (Lp (Ω),W 2θp,D (Ω))
≤
∫ h
0
∥e( δ2 +s)A(u)∥L (Lp (Ω),W 2θp,D (Ω)) ∥A(u)∥L (W 2p,D (Ω),Lp (Ω)) ∥e
δ
2 A(u)∥L (Lp (Ω),W 2p,D (Ω)) ds
≤ c(R0)e−νδ
(
δ
2
)−1 ∫ h
0
(
δ
2
+ s
)−θ
ds .
Consequently, invoking (7),
∥e(δ+h)A(u) −eδA(u)∥L (Lp (Ω),L∞(Ω)) ≤ c(R0)e−νδδ−1−θh , h ≥ 0, δ> 0, u ∈ X . (14)
Next, given u ∈ X and 0 ≤ t ≤ t +h ≤ T we have, using (7),
∥eλA(u)F (u)(t +h)−eλA(u)F (u)(t )∥∞
≤ c
∫ t+h
t
∥e(λ+t+h−s)A(u)∥L (Lp (Ω),L∞(Ω)) ∥ f (s)∥p ds
+ c
∫ t
0
∥e(λ+t+h−s)A(u) −e(λ+t−s)A(u)∥L (Lp (Ω),L∞(Ω)) ∥ f (s)∥p ds .
We use (6)-(7) once more along with (14) to get
∥eλA(u)F (u)(t +h)−eλA(u)F (u)(t )∥∞
≤ c(R0)
∫ t+h
t
(λ+ t +h − s)−θ ∥ f (s)∥p ds + c(R0)h
∫ t
0
(λ+ t − s)−1−θ ∥ f (s)∥p ds
≤ c(R0)λ−θ
∫ t+h
t
∥ f (s)∥p ds + c(R0)hλ−1−θ ∥ f ∥L1(R+,Lp (Ω)) .
Therefore, since f ∈ L1(R+,Lp (Ω)) we deduce that
lim
h→0
sup
u∈X
∥eλA(u)F (u)(·+h)−eλA(u)F (u)∥XT = 0, λ> 0. (15)
Gathering (15) and (13) we conclude that
{
eλA(u)F (u) ; u ∈ X } is indeed precompact in XT due to
the Arzelà-Ascoli Theorem.
(iii). Next, we claim that
lim
λ→0
sup
u∈X
∥eλA(u)F (u)−F (u)∥XT = 0. (16)
830 Christoph Walker
Let δ ∈ (0,T ). Using (5) we have, for 0 ≤ t ≤ δ,
∥eλA(u)F (u)(t )−F (u)(t )∥∞ ≤ 2∥F (u)(t )∥∞ ≤ 2
∫ δ
0
∥ f (s)∥∞ ds . (17)
On the other hand, for δ≤ t ≤ T , we use (5) to get
∥eλA(u)F (u)(t )−F (u)(t )∥∞ ≤ ∥eλA(u)F (u)(t )−e(λ+δ)A(u)F (u)(t −δ)∥∞
+∥e(λ+δ)A(u)F (u)(t −δ)−eδA(u)F (u)(t −δ)∥∞
+∥eδA(u)F (u)(t −δ)−F (u)(t )∥∞
≤ 2∥eδA(u)F (u)(t −δ)−F (u)(t )∥∞
+∥e(λ+δ)A(u)F (u)(t −δ)−eδA(u)F (u)(t −δ)∥∞ .
For the first term on the right-hand side we use (5) to estimate
∥eδA(u)F (u)(t −δ)−F (u)(t )∥∞ =
∥∥∥∥∫ t
t−δ
e(t−s)A(u) f (s)ds
∥∥∥∥∞ ≤
∫ t
t−δ
∥ f (s)∥∞ ds
while we use (14) and (12) for the second term to obtain
∥e(λ+δ)A(u)F (u)(t −δ)−eδA(u)F (u)(t −δ)∥∞ ≤ c(R0)δ−1−θλ∥F (u)(t −δ)∥p
≤ c(R0)δ−1−θλ∥ f ∥L1(R+,Lp (Ω)) .
Gathering these estimates we derive, for δ≤ t ≤ T ,
∥eλA(u)F (u)(t )−F (u)(t )∥∞ ≤ 2
∫ t
t−δ
∥ f (s)∥∞ ds + c(R0)δ−1−θλ∥ f ∥L1(R+,Lp (Ω)) . (18)
Since f ∈ L1(R+,L∞(Ω)) we may first choose δ > 0 small enough and then let λ tend to zero to
conclude from (17) and (18) that (16) indeed holds true.
(iv). Let ε> 0 be arbitrary. Then, due to (16), there is λ0 > 0 such that
∥eλ0 A(u)F (u)−F (u)∥XT ≤
ε
3
, u ∈ X , (19)
while (11) yields finitely many u1, . . . ,uN ∈ X such that for every u ∈ X there exists k ∈ 1, . . . , N
such that
∥eλ0 A(u)F (u)−eλ0 A(uk )F (uk )∥XT ≤
ε
3
. (20)
Hence ∥F (u) − F (uk )∥XT ≤ ε so that
{
F (u) ; u ∈ X } is totally bounded in XT . This proves the
assertion. □
We prove a compactness result for the part involving the initial condition:
Proposition 4. Let a ∈ L1(R+,L∞(Ω)). Given u0 ∈ L∞(Ω) define
G (u) :=
∫ ∞
0
a(t )e t A(u) u0 dt , u ∈ X =BL∞(Ω)(0,R0) .
Then the set
{
G (u) ; u ∈ X } is precompact in L∞(Ω).
Proof. Let λ> 0 and set
Gλ(u) :=
∫ ∞
0
a(t )e(λ+t )A(u) u0 dt , u ∈ X .
Similarly as in Proposition 3 we infer that{
Gλ(u) ; u ∈ X
}
is precompact in L∞(Ω) . (21)
Christoph Walker 831
Taking δ> 0 and using (5), (7), and (14) we then get
∥Gλ(u)−G (u)∥∞ ≤ 2
∫ δ
0
∥a(t )∥∞ dt ∥u0∥∞+
∫ ∞
δ
∥a(t )∥∞
∥∥e(λ+t )A(u)u0 −e t A(u)u0∥∞ dt
≤ 2
∫ δ
0
∥a(t )∥∞ dt ∥u0∥∞+ c(R0)δ−1−θ ∥a∥L1(R+,L∞(Ω)) ∥u0∥p λ .
Since a ∈ L1(R+,L∞(Ω)) we may choose δ> 0 small to make the first term small and then let λ go
to zero to conclude
lim
λ→0
sup
u∈X
∥Gλ(u)−G (u)∥∞ = 0. (22)
Combining (21) and (22) we deduce that
{
G (u) ; u ∈ X } is precompact in L∞(Ω). □
Remark 5. The compactness results of this section rely on properties (5), (6), and (7) and thus
may also be derived for truly quasilinear operators A(u) based on the stability estimates of [1,
II.Section 5]. One has, however, to replace the set X ⊂ L∞(Ω) by a subset of W 2θp,D (Ω) with
θ ∈ (0,2) \ {1/p} and to impose suitable assumptions on the data (e.g. the weight a has to be
sufficiently smooth in x).
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A Remark on a Nonlocal-in-Time Heat Equation

Miaomiao Liu

Bin Guo

Comptes Rendus
Mathématique
Miaomiao Liu and Bin Guo
Blow-up solutions to the semilinear wave equation with overdamping
term
Volume 361 (2023), p. 667-672
Published online: 2 March 2023
https://doi.org/10.5802/crmath.432
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Comptes Rendus
Mathématique
2023, Vol. 361, p. 667-672
https://doi.org/10.5802/crmath.432
Partial differential equations / Équations aux dérivées partielles
Blow-up solutions to the semilinear wave
equation with overdamping term
Miaomiao Liua and Bin Guo∗, a
a School of Mathematics, Jilin University, Changchun 130012, PR China
E-mail: bguo@jlu.edu.cn
Abstract. This article deals with the Cauchy problem to the following damped wave equation{
ut t −∆u +b(t )ut = M(u), (t , x) ∈ R+×RN ,
u(0, x) = u0(x), ut (0, x) = u1(x), x ∈ RN ,
(CP)
with the focusing nonlinearity M(u) = |u|p−1u, p > 1. For the focusing nonlinearity M(u) = ±|u|p , p > 1,
Ikeda and Wakasugi in [8] have showed that the solution to Problem (CP) exists globally for small data and
fails to exist globally for large data. Meanwhile, they also proposed an open problem [8, Remark 1.3]. In this
note, we give a positive answer to this open problem by using a method different from the test-function
method. In addition, an inverse Hölder inequality associated with the solution and a differential inequality
argument are used to establish a lower bound for the blow-up time.
Funding. The first author’s research was supported by NSFC (11301211) and NSF of Jilin Prov.
(201500520056JH).
Manuscript received 14 April 2022, revised 22 September 2022, accepted 27 September 2022.
1. Introduction
In this paper, we consider Cauchy problem for the following wave equation{
ut t −△u +b(t )ut = M(u), (t , x) ∈ R+×RN ,
u(0, x) = u0(x), ut (0, x) = u1(x), x ∈ RN ,
(1)
where the coefficient of damping satisfies
b−1(t ) ∈ L1(0,∞), i.e.,
∫ ∞
0
b−1(t )dt <∞,
b(t ) > 0, t ≥ 0,b(t ) ∈C 1[0,+∞),
(2)
and the following technical condition
4(1+ t )b′(t )+N (p −1)b(t ) ≥ 0, t ≥ 0. (3)
∗Corresponding author.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
668 Miaomiao Liu and Bin Guo
The nonlinear term M(u) satisfies the following growth condition
M(0) = 0, and |M(u)−M(v)| ≤CM (1+|u|+ |v |)p−1|u − v |, ∀ u, v ∈ R, (4)
where CM is a positive constant and
1 < p <∞ (N = 1,2), 1 < p < N
N −2 (N ≥ 3). (5)
It is well known that damped wave equations may model real physical problems such as the
voltage and the current on an electrical transmission line with a resistance. Generally speaking,
the damping term prevents the motion of the wave and reduces its energy, while the presence
of the source term drives the equation towards instability. Thus, when both of damping and
source terms are present, the analysis of their interaction becomes more interesting. For example,
Ikeda–Wakasugi in [8] discussed the existence and nonexistence of global solutions to (1) when
the coefficient b(t ) satisfies (2), in which case the damping term is called as the overdamping.
In the opposite case b−1(t ) ̸∈ L1(0,+∞), the damping term is effective or noneffective, there
is also a large amount of literature about global existence, asymptotic behavior and blow-up
phenomena [3,11] and the references therein. In addition, for semilinear wave equations, another
interesting issue is to determine the critical exponent pc . That is, if p > pc , then Problem (1)
admits a global-in-time solution for small data, while, for p ≤ pc , the solution will blow up in
finite time for large data. Some related references may be found in [5,14]. However, this is without
the scope of the paper because the main aim of this note is to discuss how the initial data and
the coefficient b(t ) have an effect on the behavior of solutions. In the following, we firstly state
the local existence theorem and useful lemmas without their proofs because these proofs are
standard, the interested readers may refer to [8].
Theorem 1 ([8]). Let (u0,u1) ∈ H 1(RN )× L2(RN ) and (2)–(5) hold. Then there exists a solution
u ∈C ([0,T ); H 1(RN ))∩C 1([0,T );L2(RN )) to (1) for some T .
Further, if supp(u0,u1) ⊂ {x ∈ RN : |x| ≤ L} := BL(L > 1), then
suppu(t , · ) ⊂ {x ∈ RN : |x| ≤ L+ t} := BL+t .
For the sake of simplicity, we define
E(t ) = 1
2
∥ut∥2L2(RN ) +
1
2
∥∇u(t )∥2
L2(RN )
−
∫
RN
∫ u
0
M(s)dsdx.
Then
Lemma 2 ([8]). Under all the assumptions of Theorem 1, the energy functional E(t ) is non-
increasing with respect to time variable. Namely
E ′(t ) =−b(t )∥ut∥2L2(RN ), for t > 0,
which implies that
E(t )−E(0) =−
∫ t
0
b(s)∥us∥2L2(RN )ds. (6)
Remark 3. Actually, it is easy to see (6) shows that the set{
(u,ut ) ∈ H 1(RN )×L2(RN ) : E(t ) ≤ 0, t ≥ 0
}
is invariant under the semi-flow of Problem (1) which will come into play to finish our proof.
Ikeda–Wakasugi in [8] applied the test-function method to give some sufficient conditions
of blow-up solutions for the focusing nonlinearity M(u) = ±|u|p . However, for another case
M(u) = |u|p−1u, Ikeda–Wakasugi proposed an open problem:
“Remark 1.3. For another focusing nonlinearity M(z) = |z|p−1z, we expect
similar results as Theorem 1.4 and 1.5 hold, while they still remain open.”
Miaomiao Liu and Bin Guo 669
In fact, when we check their proofs step by step, it seems that some methods used in [8] can
not be applied to our problem because the nonlinearity M(u) may change the sign. In this note,
our strategy for getting around this difficulty is to construct an explicit sub-solution and apply
comparison principle of second order differential inequalities [11, 15] to end our proof. More
precisely, our first main result is as follows:
Theorem 4. Suppose that all the conditions of Theorem 1, (3) and (14) are fulfilled and M(u) =
|u|p u, then the solution of Problem (1) blows up at some finite time T ∗ in the sense that
lim
t→T ∗−
∥u∥L2(RN ) =+∞,
if the following condition holds
E(0) ≤ 0,
∫
RN
u0(x)u1(x)dx > 0.
Example 5. The hypotheses of Theorem 4 hold for the following function b(t ) on the interval
[0,∞)
(i) b(t ) = t 2 sin t + 2t
3
6+ (p −1)N +
(6+pN −N )2
27
, t ≥ 0;
(ii) b(t ) = b0(1+ t )−β, β<−1, b0 > 0;
(iii) b(t ) = t (1+ t )
k
ln(1+ t ) , k > 1.
Remark 6. Actually, if b(t ) satisfies (ii), Nishihara in [12] also obtained similar results. Therefore,
we find that the nonnegativity of b′(t ) plays an important role in his corresponding proof.
Naturally, a question arises: are similar results valid if the derivative of b(t ) changes the sign?
To give a positive or negative answer to this problem, one has to invent some methods.
Another problem is whether there exists a function b(t ) satisfying (2) and (3) as well as the
uncertainty of the sign of b′(t ) or not. In fact, the following remark answers this problem.
Remark 7. If b(t ) is chosen as (i) of Example 5, then it is not difficult to verify that (2) and (3) are
true, but the sign of
b′(t ) = 2t sin t + t 2 cos t + 6t
2
6+N p −N
is uncertain. In this note, we give more general conditions of b(t ) which guarantee that the
solution blows up in finite time.
The second theorem is devoted to giving the lower bound for blow-up time T ∗.
Theorem 8. If all the conditions of Theorem 4 and the condition
1 < p <
{
1+ 4N , N = 3;
1+ 2N−2 , N ≥ 4,
hold, then the blow-up time T ∗ satisfies
T ∗ ≥
∫ ∞
ψ(0)
1
C1ξσ+ξ
dξ, σ= 4(N +2p −N p)
N +4−N p > 1,
where ψ(0), C1 are defined in (19).
670 Miaomiao Liu and Bin Guo
2. Proofs of our main results
In this section, we borrow comparison principle of second-order differential inequality and
energy estimate method to complete our proof.
Proof of Theorem 1. We will divide this proof into some steps.
Step 1. Establish second-order differential inequality. Define ψ(t ) = 12
∫
RN |u(x, t )|2dx. Then
u ∈C (0,T ;L2), ut ∈C 1(0,T ;L2) allows us to make a direct computation
ψ′(t ) =
∫
RN
u(x, t )ut (x, t )dx, (7)
ψ′′(t ) =
∫
RN
u(x, t )ut t (x, t )dx +
∫
RN
|ut |2dx
=
∫
RN
|ut |2dx −
∫
RN
|∇u|2dx −b(t )
∫
RN
ut udx +
∫
RN
|u|p+1dx
= 2
∫
RN
|ut |2dx −2E(t )+ p −1
p +1
∫
RN
|u|p+1dx −b(t )
∫
RN
ut udx.
This becomes
ψ′′(t )+b(t )ψ′(t ) = 2
∫
RN
|ut |2dx −2E(t )+ p −1
p +1
∫
RN
|u|p+1dx. (8)
In addition, with the help of Hölder inequality and Theorem 1, we have∫
Bt+L
|u|p+1dx ≥ L− (p−1)N2 (1+ t )− (p−1)N2 ψ p+12 (t ). (9)
Moreover, together with (9) and the fact E(t ) ≤ E(0) ≤ 0, (8) may be rewritten as the following
b−1(t )ψ′′(t )+ψ′(t ) ≥C1h(t )ψ
p+1
2 (t ), (10)
where
C1 = p −1
p +1 L
− (p−1)N2 , h(t ) = (1+ t )− (p−1)N2 b−1(t ). (11)
Step 2. Construct blow-up sub-solution. Due to the nonlinearity of differential inequality, in
general, an explicit solution to the differential inequality (10) is hardly solved. To this end, we
construct an explicit sub-solution to compare with the solution to (10). Set
ϕ(t ) =
(
ψ−
p−1
4 (0)−ε
∫ t
0
h(τ)dτ
)− 4p−1
, 0 < t ≤ T ∗,
where the positive constants ε, T ∗ will be determined later to make the function ϕ(t ) be well-
defined. Indeed, a simple computation shows that
b−1(t )ϕ′′(t )+ϕ′(t ) ≤ 4ε
2(p +3)
(p −1)2
h2(t )
b(t )
ϕ
p+1
2 (t )+ 4ε
p −1
h′(t )
b(t )
ϕ
p+3
4 (t )+ 4εh(t )
p −1 ϕ
p+3
4 (t )
=
[
4ε2(p +3)
(p −1)2
h2(t )
b(t )
+ 4εh(t )
p −1 ϕ
1−p
4 (t )
]
ϕ
p+1
2 (t )+ 4ε
p −1
h′(t )
b(t )
ϕ
p+3
4 (t )
≤
[
4ε2(p +3)
(p −1)2b2(0) +
4ε
p −1ϕ
1−p
4 (0)
]
h(t )ϕ
p+1
2 (t ), for 0 < t < T ∗. (12)
Here we have used the condition (3) and the following facts
h′(t )
b(t )
= −h(t )
b(t )
[
b′(t )
b(t )
+ (p −1)N
2(1+ t )
]
≤ 0,
h(t )
b(t )
≤ 1
b2(0)
, ϕ(t ) ≥ϕ(0) ≥ψ(0), for t > 0.
Miaomiao Liu and Bin Guo 671
Step 3. Establish the constants ε, T ∗. First, to choose
ε= ε0 = min
{
1,
2
p−7
4 (p −1)b(0)∫RN u0u1dx
(
∫
RN |u0|2dx)
p+3
4
,
2
p−7
4 (p −1)3L− (p−1)N2
(p +3)(p +1)+ (p2 −1)(∫RN |u0|2dx) 1−p4
}
,
and then to apply (10) and (12) yield
b−1(t )ϕ′′(t )+ϕ′(t ) ≤C1h(t )ϕ
p+1
2 (t ), for 0 < t < T ∗;
b−1(t )ψ′′(t )+ψ′(t ) ≥C1h(t )ψ
p+1
2 (t ), for 0 < t < T ∗;
ϕ(0) =ψ(0) = 12
∫
RN |u0(x)|2dx, ϕ′(0) <ψ′(0) =
∫
RN u0(x)u1(x)dx.
(13)
Next, we estimate the value of T ∗ such that the function ϕ(t ) is well defined. Set
F (t ) =ψ− p−14 (0)−ε
∫ t
0
h(τ)dτ.
On the one hand, it is not hard to verify that F (t ) satisfies
F (0) =ψ(0) > 0, F ′(t ) =−εh(t ) ≤ 0.
On the other hand, if we assume that[∫ ∞
0
(1+ t )− (p−1)N2 b−1(t )dt
]−1
<ψ p−14 (0)ε0, (14)
then limt→+∞ F (t ) is negative. So, with the aid of the intermediate value theorem, we know that
there exists a unique constant 0 < T ∗ <+∞ satisfying F (T ∗) = 0, which indicates that the function
ϕ(t ) works well.
Step 4. Blow-up in finite time. Now, we prove that limt→T ∗−ψ(t ) = +∞. In fact, according to
comparison principle of Lemma 3.1 in [11], we have ψ(t ) ≥ ϕ(t ), for 0 < t < T ∗. This completes
the proof of Theorem 4. □
At last, we complete the proof of Theorem 8.
Proof of Theorem 8. To show this, we need to build up first order differential inequality associ-
ated with the functionalψ(t ). In fact, to follow the proof of Inequality (7) and use the well-known
inequality ab ≤ a22 + b
2
2 and the definition of E(t ) as well as (6) yield
ψ′(t ) ≤
∫
RN
|ut |2dx +
∫
RN
|u|2dx ≤
∫
RN
|u|p+1dx +
∫
RN
|u|2dx. (15)
Subsequently, we need to use ψ(t ) to control the term
∫
RN |u|p+1dx. Therefore, due to the failure
of the embedding inequality ∥u∥Lp+1(RN ) ≤ C∥∇u∥L2(RN )(C > 0) for p satisfying (5), it is not
possible to apply some methods used in [13] to obtain similar controlled relations between ψ(t )
and the source. To show this, we will make use of interpolation inequality and (6) to establish
an inverse Hölder inequality. Actually, the well-known Gagliardo–Nirenberg inequality in [1,
Corollary 9.14] or [4, Theorem 7.10] shows that there exists a positive constant C depending on
|Ω|, p, N such that ∫
RN
|u|p+1dx ÉC
(∫
RN
|u|2dx
) (1−θ)(p+1)
2
(∫
RN
|∇u|2dx
) θ(p+1)
2
, (16)
where
1
p +1 =
1−θ
2
+ θ
2∗
, 2∗ = 2N
N −2 (N Ê 3).
672 Miaomiao Liu and Bin Guo
Keeping 0 < θ(p+1)2 < 1 in mind and collecting Young inequality and (16), we have∫
RN
|u|p+1dx ≤Cε 2θp−θ−2
(∫
RN
|u|2dx
) (1−θ)(p+1)
(2−θp−θ) +Cε 2θp+θ
∫
RN
|∇u|2dx.
Further, choosing Cε
2
θp+θ = 1 and using energy identity (6), we obtain the following inverse Hölder
inequality ∫
RN
|u|p+1dx ≤
(∫
RN
|u|2dx
) 4(N+2p−N p)
N+4−N p
. (17)
Finally, together with (15) and (17), we have
T ∗ ≥
∫ ∞
ψ(0)
1
C1ξσ+ξ
dξ, (18)
where
σ= 4(N +2p −N p)
N +4−N p > 1, C1 =
p −1
p +1C
4
4+N−N p , ψ(0) = 1
2
∫
RN
|u0|2dx. (19)
□
References
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xiii+599 pages.
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(2015), no. 6, p. 1032-1045.
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Comptes Rendus
Mathématique
Mohammad Al Haj and Régis Monneau
The velocity diagram for traveling waves
Volume 361 (2023), p. 777-782
Published online: 11 May 2023
https://doi.org/10.5802/crmath.433
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
Les Comptes Rendus. Mathématique sont membres du
Centre Mersenne pour l’édition scientifique ouverte
www.centre-mersenne.org
e-ISSN : 1778-3569
Comptes Rendus
Mathématique
2023, Vol. 361, p. 777-782
https://doi.org/10.5802/crmath.433
Partial differential equations / Équations aux dérivées partielles
The velocity diagram for traveling waves
Mohammad Al Haja and Régis Monneau∗, b, c
a Lebanese University, Faculty of Science (section 5), Nabatiye 1700, Lebanon
b CERMICS, Ecole des Ponts ParisTech, Université Paris-Est, 6 et 8 avenue Blaise
Pascal, Cité Descartes, Champs-sur-Marne, 77455 Marne-la-Vallée Cedex 2, France
c CEREMADE, Université Paris-Dauphine, Place du Maréchal De Lattre De Tassigny,
75775 Paris Cedex 16, France
Abstract. In this Note, we consider traveling waves in a reaction-diffusion equation in dimension one.
Motivated by the motion of dislocations in crystals, we introduce an additive parameter σ in the reaction
term, which may be interpreted as an exterior force applied on the crystal. Under certain natural assumptions
and for every value of σ ∈ [σ−,σ+], we show the existence of traveling waves φ of velocity c. The range
σ ∈ (σ−,σ+) corresponds to bistable cases with a unique velocity c = c(σ). On the contrary, the case σ=σ+ is
positively monostable with a branch of velocities c ≥ c+, while the case σ = σ− is negatively monostable
with a branch of velocities c ≤ c−. This study gives rise to a natural connection between bistable cases
and monostable cases in a single velocity diagram. We also give some qualitative properties of the velocity
function σ 7→ c(σ).
Résumé. Dans cette Note, nous considérons des ondes progressives pour une équation de réaction-diffusion
en dimension un. Motivés par le mouvement de dislocations dans les cristaux, nous introduisons un para-
mètre additif σ dans le terme de réaction, qui peut être interprété comme une force extérieure appliquée au
cristal. Sous certaines hypothèses naturelles et pour chaque valeur de σ ∈ [σ−,σ+], nous montrons l’exis-
tence d’ondes progressives φ se déplaçant à la vitesse c. Le domaine σ ∈ (σ−,σ+) correspond aux cas bis-
tables avec une unique vitesse c = c(σ). Au contraire, le cas σ = σ+ est positivement monostable avec une
branche de vitesses c ≥ c+, et le cas σ=σ− est négativement monostable avec une branche de vitesse c ≤ c−.
Cette étude met en évidence un lien naturel entre les cas bistables et les cas monostables au sein d’un unique
diagramme en vitesse. On donne aussi des propriétés qualitatives de la fonction vitesse σ 7→ c(σ).
Manuscript received 2 August 2022, revised and accepted 28 September 2022.
1. Introduction
In this Note, we consider particular solutions u(t , x) to the standard reaction-diffusion equation
with an additional exterior parameter σ ∈R
u(t , x) =φ(x + ct ) satisfying ut = uxx + f (u)+σ for (t , x) ∈R×R (1)
where φ is a traveling wave moving with velocity c. Such a model is for instance inspired from
the dynamics of a dislocation defect in a crystal where σ is the exterior shear stress applied on
∗Corresponding author.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
778 Mohammad Al Haj and Régis Monneau
the crystal. From this point of view, equation (1) can be seen as an approximation of the fully
overdamped Frenkel–Kontorova model (see [8]). We assume that the function f satisfies
(Regularity) f is Lipschitz-continuous
(Periodicity) f (v +1) = f (v) for all v ∈R
(Decreasing) f decreasing on (θ,1) for θ ∈ (0,1)
(Stricly increasing) f ′(v) ≥ g (v) > 0
for almost every v ∈ (0,θ) with g : (0,θ) →R continuous
(2)
The graph of f is represented on Figure 1.
0 θ 1
Figure 1. Nonlinearity f
Setting
σ+ :=−min f , σ− :=−max f
we define for each σ ∈ [σ−,σ+], the unique roots mσ and bσ solutions of
f (mσ)+σ= 0, mσ ∈ [θ−1,0] and f (bσ)+σ= 0, bσ ∈ [0,θ]
Here the nonlinearity f σ := f +σ falls in one the following three cases
bistable case σ ∈ (σ−,σ+), f σ < 0 on (mσ,bσ) and f σ > 0 on (bσ,1+mσ)
positive monostable case σ=σ+, f σ+ > 0 on (mσ+ ,1+mσ+ ) with bσ+ = mσ+ = 0
negative monostable case σ=σ−, f σ− < 0 on (mσ− ,1+mσ− ) with bσ− = 1+mσ− = θ
Our goal is to connect bistable and monostable types as the parameter σ varies, as it is very
natural from the point of view of the motion of a dislocation defect under the exterior force σ. To
this end, we now consider traveling wavesφ of velocity c associated to the parameterσ ∈ [σ−,σ+],
solutions of
cφ′ =φ′′+ f (φ)+σ on R with φ(−∞) = mσ, φ(+∞) = 1+mσ (3)
Recall here some known results. When f ∈ C 1(R) with f ′(mσ) < 0, the uniqueness of c is known
in [6] in the bistable case. In the positive monostable case, when f ∈C 1([0,1]) with f ′(0) > 0, the
existence of a branch of velocities c ≥ c+ is also known (see [3, 7, 9]).
Then our result is the following velocity diagram.
Theorem 1 (Velocity diagram). Assume that f satisfies (2).
(i) (Bistable case σ ∈ (σ−,σ+)) For every σ ∈ (σ−,σ+), there exists a unique velocity c = c(σ)
and a unique (up to translations) traveling waveφ solution of (3). Moreoverφ is increasing
and the map σ 7→ c(σ) is continuous increasing for σ ∈ (σ−,σ+) with finite limits
c− := lim
σ−<σ→σ− c(σ), c
+ := lim
σ+>σ→σ+
c(σ)
There exists also δ> 0 such that 0 < δ≤ c(σ2)−c(σ1)σ2−σ1 for all σ1,σ2 ∈ (σ−,σ+) with σ1 ̸=σ2.
If moreover f ∈C 1,1(θ,1) with f ′ < 0 on (θ,1), then the mapσ 7→ c(σ) is locally Lipschitz-
continuous inside the interval (σ−,σ+).
(ii) (Positive monostable case σ = σ+) For every c ≥ c+, there exists a unique (up to transla-
tions) traveling wave φ solution of (3). Moreover φ is increasing. For all c < c+, there are
no traveling waves solutions of (3).
Mohammad Al Haj and Régis Monneau 779
(iii) (Negative monostable case σ = σ−) For every c ≤ c−, there exists a unique (up to transla-
tions) traveling wave φ solution of (3). Moreover φ is increasing. For all c > c−, there are
no traveling waves solutions of (3).
Figure 2. Schematic velocity diagram: the velocity function c(σ) with vertical branches at
σ=σ±.
The result of Theorem 1 is presented in the velocity diagram in Figure 2, and there are cases
where dcdσ blows up at σ = σ± (see Remark 2). This result allows to understand the branch of
solutions c ≥ c+ in the positive monostable case as a sort of continuation of the increasing
velocity σ 7→ c(σ) for σ ∈ (σ−,σ+). This phenomenon is also related to the fact that for σ ∈
R \ [σ−,σ+], there are no roots of f +σ = 0 and then no possible traveling waves. We see that
Theorem 1 gives the whole picture of the velocity diagram which connects bistable to monostable
cases.
Let us also mention that part of the results stated in Theorem 1 stays true (loosing possibly
the strict monotonicity of the velocity function σ 7→ c(σ) and loosing possibly the uniqueness of
the profile (up to translations)) for traveling waves of discrete reaction-diffusion equations of the
type
u(t , x) =φ(x + ct ) satisfying ut = F (u(t , x + r0), . . . ,u(t , x + rN ))
where r0 = 0 and the ri ∈ R are discrete shifts for i = 1, . . . , N with N ≥ 1, under certain period-
icity and monotonicity assumptions on F (in order to insure a comparison principle, but pos-
sibly loosing the strong comparison principle) joint to the previous assumption (2) on f (v) :=
F (v, . . . , v). The results are given in [2].
2. Sketch of the proof of the theorem
The originality of this Note is probably more in the statement of the theorem than in the proof
itself. Part of the arguments of the proof are classical. Some of those arguments are for instance
780 Mohammad Al Haj and Régis Monneau
developed in [2] for discrete reaction-diffusion equations, and are simply adapted here. For those
reasons, we only indicate here the sketch of the proof of the theorem.
Step 1: existence and uniqueness for σ ∈ (σ−,σ+). A standard way to build a solution (c,φ) con-
sists to use the method of sub/supersolutions to build a solution on a large finite interval [−R,R]
and to adjust the velocity c in such a way that the transition arises in the middle of the interval for
instance, and finally to pass to the limit as R →+∞ (like in [4] for ignition type nonlinearities).
A different method is used in [1] and is more generally useful to get results on the velocity in the
case of degenerate equations without strong maximum principle. This method consists to solve
the cell problem for homogenization (similarly to what is done in [8]) finding solutions (λp ,hp )
with λp ∈R, associated to a fixed slope p > 0
λp h
′
p = h′′p + f (hp )+σ, hp
(
x + 1
p
)
= hp (x)+1, h′p ≥ 0
and to define (c,φ) as a limit of (λp ,hp ) as p → 0 (similarly to what is done in [1, 2]). The
strict increasing property of assumption (2) on f insures that the root bσ is unstable, and then
that the limit profile φ does not split in two profiles, one in the range (mσ,bσ) and one in the
range (bσ,1+mσ). This insures that the profile φ is a true transition in the range (mσ,1+mσ).
The decreasing property of assumption (2) insures the comparison principle at infinity. This
allows to prove uniqueness of the velocity denoted c = c(σ). Using moreover the standard sliding
method [5] (based on the strong maximum principle), this allows to get uniqueness (up to
translations) and the monotonicity of the profile φ=:φσ.
Step 2: continuity and monotonicity of c(σ) for σ ∈ (σ−,σ+). The continuity of the map σ 7→
c(σ) follows from the uniqueness of the velocity. The fact that the maps σ 7→ mσ and σ 7→ bσ are
increasing, insures the comparison at infinity of the associated profiles for two values σ1 < σ2,
and this shows that the map σ 7→ c(σ) is nondecreasing. Moreover the strong comparison princi-
ple implies that the map σ 7→ c(σ) is increasing.
Step 3: vertical branch of solutions for large velocities c >> 1 for σ=σ+. Using the solution g
to the ODE g ′ = f (g ) +σ+ ≥ 0, say with g (0) = δ > 0, and using the Lipschitz regularity of f ,
we deduce that g ε(x) := g (εx) satisfies the supersolution inequality for all ε> 0
cεg
′
ε ≥ g ′′ε + f (g ε)+σ+ with cε =
1
ε
+ε| f ′|L∞(R)
Using the fact that g
ε
:= δ is a subsolution on (0,+∞), we can then build a solution gε,δ on (0,+∞).
Up to translate gε,δ, we can pass to the limit δ→ 0 and get a solution φ := gε associated to the
velocity c := cε, which works for all velocities c ≥ 2
√| f ′|L∞(R).
Step 4: definition of c±. If (c,φ) is a solution for the parameter σ+, then we can compare it to a
solution of velocity c(σ) for σ ∈ (σ−,σ+), and get c(σ) ≤ c. This implies that limσ+>σ→σ+ c(σ) =:
c+ ≤ c. In particular from Step 3, we deduce that c+ < +∞, and that there are no solutions (c,φ)
with c < c+ for the parameter σ+. We get similar results for c−.
Step 5: full vertical branch of solutions for c ≥ c+. Using (c(σ),φσ) the solution given in Step 1
for σ ∈ (σ−,σ+), we can pass to the limit and get φσ→φ+ as σ→σ+, which is a traveling wave of
velocity c+ for σ=σ+. Then for every c ≥ c+, we see that (c,φ+) is a supersolution of the equation
for σ = σ+. Then using a construction as in Step 3, we get the existence of a solution (c,φ) for
σ = σ+, for each c ≥ c+. Finally the decreasing property of f in (2) implies the comparison
principle at x =+∞. Let us consider intervals (aδ,+∞) where φ>φ(aδ) = δ> 0. Using the sliding
method for all δ > 0 small enough with mσ+ = 0, we can easily deduce the monotonicity and
uniqueness of the profile φ (up to translations).
Mohammad Al Haj and Régis Monneau 781
Step 6: full vertical branch of solutions for c ≤ c−. Proceeding as in Step 5, we get a full branch
of solutions (c,φ) for all c ≤ c−, and no solutions for c > c−.
Step 7: bounds on dcdσ . Recall that for any σ ∈ (σ−,σ+), we have c− ≤ c(σ) ≤ c+, and the interior
regularity theory for the elliptic equation satisfied by the profile φ=φσ gives a bound |φ′σ|L∞(R) ≤
δ−1 uniformly in σ ∈ (σ−,σ+), for some δ > 0. Consider σ1,σ2 ∈ (σ−,σ+) with σ1 < σ2, and the
associated solutions (ci ,φi ) for i = 1,2 with ci = c(σi ), φi =φσi and mi = mσi .
Step 7.1: bound from below. We deduce that
c2φ
′
1 ≤φ′′1 + f (φ1)+σ2 with c2 := c1 +δ(σ2 −σ1)
The comparison implies (up to an initial shift of the profiles) that φ2(x + c2t ) ≥ φ1(x + c2t ) and
then the fact that φ2(−∞) < φ1(+∞) implies that c2 ≥ c1 +δ(σ2 −σ1) which implies the bound
from below on dcdσ .
Step 7.2: bound from above. For c = c(σ), let us consider the equation cζ′ = ζ′′+ f ′(φσ)ζ satisfied
by ζ := φ′σ. Using the fact that f ′ < −µ < 0 in a neighborhood of mσ, we can introduce the roots
of λ2 − cλ−µ = 0 which are λ± := c±
p
c2+4µ
2 and show by comparison that ζ(y) ≤ ζ(x)eλ+(y−x)
for y ≤ x and ζ(x) small enough. This gives by integration for φσ(x) − mσ =
∫ x
−∞ dy ζ(y) that
φ′σ ≥λ+(φσ−mσ) whereφσ−mσ is small enough (using for instance interior estimates for elliptic
equations). Now forσ1 <σ2, we setφ2 :=ψ+m2 withψ :=φ1−m1. Using the fact that f (m2+ψ)−
f (m2)−
{
f (m1 +ψ)− f (m1)
}= ∫ 10 dt ∫ 10 ds { f ′′(tψ+m1 + s(m2 −m1))− f ′′(tψ+m1)}·ψ(m2−m1)
and f ′(mσ) dmσdσ =−1 with f ′(mσ) ≤−µ< 0 and λ+ ≥ λ+ > 0 for σ ∈ [σ1,σ2], we get on an interval
(−∞,−R] where ψ is small enough
c+2φ
′
2 ≥φ
′′
2 + f (φ2)+σ2 on (−∞,−R], with c+2 := c1 +
{
| f ′′|L∞(R)
µλ+
}
(σ2 −σ1)
We get a similar inequality on [R,+∞) where 1−ψ is small enough, with some velocity c−2 . Using
Harnack inequality, we can show that there exists some KR > 0 such that φ′1 ≥ (KR )−1 on [−R,R],
which implies
c02φ
′
2 ≥φ
′′
2 + f (φ2)+σ2 on [R,R], with c02 := c1 +KR
{
1+ | f
′|L∞(R)
µ
}
(σ2 −σ1)
Using the fact that all the constants can be taken uniformly forσ ∈ [σ1,σ2] ⊂⊂ [σ−,σ+], we deduce
that c2 ≤ max(c−2 ,c02,c+2 ) which implies the bound from above on dcdσ . This ends the sketch of the
proof of the theorem. □
Remark 2. Choosing a normalization like for instance φσ(0) = θ, we can show that ψ := ddσ (φσ−
mσ) formally satisfies an equation that we can multiply by e−cxφ′σ(x) and integrate by parts to get
with I1 :=
∫
R e
−cxφ′2σdx =
∫
R e
−cx (φσ−mσ)( f (φσ)− f (mσ))dx and I2 =
∫
R e
−cx
{
1− f ′(φσ)f ′(mσ)
}
φ′σdx,
the relation(
dc
dσ
)
· I1 = I2
with I2 :=
{
c
− f ′(mσ)
∫
R e
−cx { f (φσ)− f (mσ)− (φσ−φσ(∓∞)) f ′(mσ)}dx if ± c > 0
1 if c = 0
When f ∈ C 1,1(θ,1) and f ′ < 0 on (θ,1), then we can justify the above computation at least at
every point of differentiability of c. This is the case for instance when we consider the 1-periodic
function f defined by f (v) = 12 −|v − 12 | for v ∈ [0,1]. Then θ = 12 and c+ = 2
√
f ′(0+). Moreover we
can then show that the integral I2 blows up much faster than I1 as σ→ σ+, which implies that
dc
dσ is not bounded as σ→ σ+ (and by symmetry also as σ→ σ−). For general nonlinearities f ,
782 Mohammad Al Haj and Régis Monneau
it would be interesting to refine the analysis on the behaviour of c at σ = σ± which is out of the
scope of this Note.
References
[1] M. Al Haj, N. Forcadel, R. Monneau, “Existence and uniqueness of traveling waves for fully overdamped Frenkel-
Kontorova models”, Arch. Ration. Mech. Anal. 210 (2013), no. 1, p. 45-99.
[2] M. Al Haj, R. Monneau, “The velocity diagram of traveling waves for discrete reaction-diffusion equations”, https:
//hal.science/hal-03777384, 2022.
[3] D. G. Aronson, H. F. Weinberger, “Multidimensional nonlinear diffusion arising in population genetics”, Adv. Math. 30
(1978), p. 33-76.
[4] H. Berestycki, B. Nicolaenko, B. Scheurer, “Traveling wave solutions to combustion models and their singular limits”,
SIAM J. Math. Anal. 16 (1985), no. 6, p. 1207-1242.
[5] H. Berestycki, L. Nirenberg, “On the method of moving planes and the sliding method”, Bol. Soc. Bras. Mat. 22 (1991),
no. 1, p. 1-37.
[6] P. C. Fife, Dynamics of internal layers and diffusive interfaces, CBMS-NSF Regional Conference Series in Applied
Mathematics, vol. 53, Society for Industrial and Applied Mathematics, 1988.
[7] R. A. Fisher, “The wave of advance of advantageous genes”, Ann. Eugenics, London 7 (1937), p. 335-369.
[8] N. Forcadel, C. Imbert, R. Monneau, “Homogenization of fully overdamped Frenkel-Kontorova models”, J. Differ.
Equations 246 (2009), no. 3, p. 1057-1097.
[9] A. N. Kolmogorov, I. G. Petrovsky, N. S. Piskunov, “Etude de l’équation de la diffusion avec croissance de la quantité
de matière et son application à un problème biologique”, Bull. Univ. État Moscou, Sér. Int., Sect. A: Math. et Mécan. 1
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