Blow-up rate for a semilinear wave equation with exponential nonlinearity in one space dimension

0.42 MB·

by  Asma Azaiez

Blow-up rate for a semilinear wave equation
with exponential nonlinearity in one space
dimension
6 Asma AZAIEZ∗
1
0 Universit´e de Cergy-Pontoise,
2
AGM, CNRS (UMR 8088), 95302, Cergy-Pontoise, France.
n
a Nader MASMOUDI†
J
1 Courant Institute of Mathematical Sciences.
2
Hatem ZAAG‡

] Universit´e Paris 13, Sorbonne Paris Cit´e
P
A
LAGA, CNRS (UMR 7539), F-93430, Villetaneuse, France.
.
h
t January 22, 2016
a
m
[

2
v Abstract
7
0 We consider in this paper blow-up solutions of the semilinear wave equation
0
in one space dimension, with an exponential source term. Assuming that initial
4
0 data are in H1 ×L2 or some times in W1,∞ ×L∞, we derive the blow-up rate
loc loc
1. near a non-characteristic point in the smaller space, and give some bounds near
0 other points. Our result generalize those proved by Godin under high regularity
6
assumptions on initial data.
1
:
v
i
X 1 Introduction
r
a
We consider the one dimensional semilinear wave equation:
(cid:26) ∂2u = ∂2u+eu,
t x (1)
u(0) = u and ∂ u(0) = u
0 t 1
where u(t) : x ∈ R → u(x,t) ∈ R,u ∈ H1 and u ∈ L2 . We may also add more
0 loc,u 1 loc,u
restriction on initial data by assuming that (u ,u ) ∈ W1,∞ ×L∞. The Cauchy problem
0 1
for equation (1) in the space H1 ×L2 follows from fixed point techniques (see Section
loc,u loc,u
2 below).
∗This author is supported by the ERC Advanced Grant no. 291214, BLOWDISOL.
†This author is partially supported by NSF grant DMS- 1211806.
‡This author is supported by the ERC Advanced Grant no. 291214, BLOWDISOL. and by ANR
project ANAE´ ref. ANR-13-BS01-0010-03.
1
If the solution is not global in time, we show in this paper that it blows up (see Theorems
1 and 2 below). For that reason, we call it a blow-up solution. The existence of blow-up
solutions is guaranteed by ODE techniques and the finite speed of propagation.
More blow-up results can be found in Kichenassamy and Littman [12], [13], where the
authors introduce a systematic procedure for reducing nonlinear wave equations to char-
acteristic problems of Fuchsian type and construct singular solutions of general semilinear
equations which blow up on a non characteristic surface, provided that the first term of
an expansion of such solutions can be found.
The case of the power nonlinearity has been understood completely in a series of
papers, in the real case (in one space dimension) by Merle and Zaag [16], [17], [20] and
[21] and in Coˆte and Zaag [6] (see also the note [18]), in the complex case by Azaiez
[3]. Some of those results have been extended to higher dimensions for conformal or
subconformal p:
4
1 < p ≤ p ≡ 1+ , (2)
c
N −1
under radial symmetry outside the origin in [19]. For non radial solutions, we would
like to mention [14] and [15] where the blow-up rate was obtained. We also mention the
recent contribution of [22] and [?] where the blow-up behavior is given, together with
some stability results.
In [5] and [4], Caffarelli and Friedman considered semilinear wave equations with a
nonlinearity of power type. If the space dimension N is at most 3, they showed in [5] the
existence of solutions of Cauchy problems which blow up on a C1 spacelike hypersurface.
If N = 1 and under suitable assumptions, they obtained in [4] a very general result which
shows that solutions of Cauchy problems either are global or blow up on a C1 spacelike
curve. In [11] and [10], Godin shows that the solutions of Cauchy problems either are
global or blow up on a C1 spacelike curve for the following mixed problem (γ (cid:54)= 1, |γ| ≥ 1)
(cid:26) ∂2u = ∂2u+eu, x > 0
t x (3)
∂ u+γ∂ u = 0 if x = 0.
x t
In [11], Godin gives sharp upper and lower bounds on the blow-up rate for initial data in
C4×C3. It happens that his proof can be extended for initial data (u ,u ) ∈ H1 ×L2
0 1 loc,u loc,u
(see Proposition 4.6 below).
Let us consider u a blow-up solution of (1). Our aim in this paper is to derive upper
and lower estimates on the blow-up rate of u(x,t). In particular, we first give general
results (see Theorem 1 below), then, considering only non-characteristic points, we give
better estimates in Theorem 2.
From Alinhac [1], we define a continuous curve Γ as the graph of a function x (cid:55)→ T(x)
such that the domain of definition of u (or the maximal influence domain of u) is
D = {(x,t)|0 ≤ t < T(x)}. (4)
From the finite speed of propagation, T is a 1-Lipschitz function. The graph Γ is called
the blow-up graph of u.
Let us introduce the following non-degeneracy condition for Γ. If we introduce for all
x ∈ R, t ≤ T(x) and δ > 0, the cone
C = {(ξ,τ) (cid:54)= (x,t)|0 ≤ τ ≤ t−δ|ξ −x|}, (5)
x,t,δ
2
then our non-degeneracy condition is the following: x is a non-characteristic point if
0
∃δ = δ (x ) ∈ (0,1) such that u is defined on C . (6)
0 0 0 x0,T(x0),δ0
If condition (6) is not true, then we call x a characteristic point. We denote by R ⊂ R
0
(resp. S ⊂ R) the set of non-characteristic (resp. characteristic) points.
We also introduce for each a ∈ R and T ≤ T(a) the following similarity variables:
x−a
w (y,s) = u(x,t)+2log(T −t), y = , s = −log(T −t). (7)
a,T
T −t
If T = T(a), we write w instead of w .
a a,T(a)
From equation (1), we see that w (or w for simplicity) satisfies, for all s ≥ −logT, and
a,T
y ∈ (−1,1),
∂2w−∂ ((1−y2)∂ w)−ew +2 = −∂ w−2y∂2 w. (8)
s y y s y,s
In the new set of variables (y,s), deriving the behavior of u as t → T is equivalent to
studying the behavior of w as s → +∞.
Our first result gives rough blow-up estimates. Introducing the following set
D ≡ {(x,t) ∈ (R,R ),|x| < R−t}, (9)
R +
where R > 0, we have the following result
Theorem 1. (Blow-up estimates near any point) We claim the following:
i) (Upper bound) For all R > 0 and a ∈ R such that (a,T(a)) ∈ D , it holds:
R
∀|y| < 1, ∀s ≥ −logT(a), w (y,s) ≤ −2log(1−|y|)+C(R),
a
C(R) C(R)
∀t ∈ [0,T(a)), eu(a,t) ≤ ≤ .
d((a,t),Γ)2 (T(a)−t)2
where d((x,t),Γ) is the (Euclidean) distance from (x,t) to Γ.
ii) (Lower bound) For all R > 0 and a ∈ R such that (a,T(a)) ∈ D , it holds
R
that
(cid:90)
1 (cid:112) (cid:112)
e−u(x,t)dx ≤ C(R) d((a,t),Γ) ≤ C(R) T(a)−t.
T(a)−t
I(a,t)
If in addition, (u ,u ) ∈ W1,∞ ×L∞ then
0 1
C(R) C(R)
∀t ∈ [0,T(a)),eu(a,t) ≥ ≥ .
d((a,t),Γ) T(a)−t
3
iii) (Lower bound on the local energy ”norm”) There exists (cid:15) > 0 such that
0
for all a ∈ R, and t ∈ [0,T(a)),
(cid:90)
1 (cid:15)
((u (x,t))2 +(u (x,t))2 +eu(x,t))dx ≥ 0 , (10)
T(a)−t t x (T(a)−t)2
I(a,t)
where I(a,t) = (a−(T(a)−t),a+(T(a)−t)).
Remark: The upper bound in item i) was already proved by Godin [11], for more regular
initial data. Here, we show that Godin’s strategy works even for less regular data. We
refer to the integral in (10) as the local energy ”norm”, since it is like the local energy
as in Shatah and Struwe, though with the ”+” sign in front of the nonlinear term. Note
that the lower bound in item iii) is given by the solution of the associated ODE u(cid:48)(cid:48) = eu.
However the lower bound in ii) doesn’t seem to be optimal, since it does not obey the
ODE behavior. Indeed, we expect the blow-up for equation (1) in the ”ODE style”, in
the sense that the solution is comparable to the solution of the ODE u” = eu at blow-up.
This is in fact the case with regular data, as shown by Godin [11].
If in addition a ∈ R, we have optimal blow-up estimates:
Theorem 2. (An optimal bound on the blow-up rate near a non-characteristic
point in a smaller space) Assume that (u ,u ) ∈ W1,∞×L∞. Then, for all R > 0, for
0 1
any a ∈ R such that (a,T(a)) ∈ D , we have the following
R
i) (Uniform bounds on w) For all s ≥ −logT(a)+1,
(cid:90) 1
(cid:0) (cid:1)
|w (y,s)|+ (∂ w (y,s))2 +(∂ w (y,s))2 dy ≤ C(R)
a s a y a
−1
where w is defined in (7).
a
ii) (Uniform bounds on u) For all t ∈ [0,T(a))
(cid:90)
|u(x,t)+2log(T(a)−t)|+(T(a)−t) (∂ u(x,t))2 +(∂ u(x,t))2dx ≤ C(R).
x t
I
In particular, we have
1
≤ eu(x,t)(T(a)−t)2 ≤ C(R).
C(R)
Remark: This result implies that the solution indeed blows up on the curve Γ.
Remark: Note that when a ∈ R, Theorem 1 already holds and directly follows from
Theorem 2. Accordingly, Theorem 1 is completely meaningful when a ∈ S.
Following Antonini, Merle and Zaag in [2] and [15], we would like to mention the existence
of a Lyapunov functional in similarity variables. More precisely, let us define
(cid:90) 1 (cid:18)1 1 (cid:19)
E(w(s)) = (∂ w)2 + (1−y2)(∂ w)2 −ew +2w dy. (11)
s y
2 2
−1
WeclaimthatthefunctionalE definedby(11)isadecreasingfunctionoftimeforsolutions
of (8) on (-1,1).
4
Proposition 3. (A Lyapunov functional for equation (1)) For all a ∈ R, T ≤
T(a), s ≥ s ≥ −logT, the following identities hold for w = w :
2 1 a,T
(cid:90) s2
E(w(s ))−E(w(s )) = − (∂ w(−1,s))2 +(∂ w(1,s))2ds.
2 1 s s
s1
Remark: The existence of such an energy in the context of the nonlinear heat equation
has been introduced by Giga and Kohn in [7], [8] and [9].
Remark: As for the semilinear wave equation with conformal power nonlinearity, the
dissipation of the energy E(w) degenerates to the boundary ±1.
This paper is organized as follows:
In Section 2, we solve the local in time Cauchy problem.
Section 3 is devoted to some energy estimates.
In Section 4, we give and prove upper and lower bounds, following the strategy of Godin
[11].
Finally, Section 5 is devoted to the proofs of Theorem 1, Theorem 2 and Proposition 3.
2 The local Cauchy problem
In this section, we solve the local Cauchy problem associated to (1) in the space H1 ×
loc,u
L2 . In order to do so, we will proceed in two steps :
loc,u
1) In Step 1, we solve the problem in H1 ×L2 , for some uniform T > 0 small enough.
loc,u loc,u
2) In Step 2, we consider x ∈ R, and use step 1 and a truncation to find a local solution
0
˜
defined in some cone C for some T(x ) > 0. Then, by a covering argument, the
x0,T˜(x0),1 0
maximal domain of definition is given by D = ∪ C .
x0∈R x0,T˜(x0),1
3) In Step 3, we consider some approximation of equation (1), and discuss the convergence
of the approximating sequence.
Step 1: The Cauchy problem in H1 ×L2
loc,u loc,u
In this step, we will solve the local Cauchy problem associated to (1) in the space H =
H1 ×L2 . In order to do so, we will apply a fixed point technique. We first introduce
loc,u loc,u
the wave group in one space dimension:
S(t) : H → H
(u ,u ) (cid:55)→ S(t)(u ,u )(x),
0 1 0 1
 1 1 (cid:90) x+t 
(u (x+t)+u (x−t))+ u
S(t)(u0,u1)(x) =  2 0 0 2 x−t 1 .
1(u(cid:48)(x+t)−u(cid:48)(x−t))+ 1(u (x+t)+u (x−t))
2 0 0 2 1 1
Clearly, S(t) is well defined in H, for all t ∈ R, and more precisely, there is a universal
constant C such that
0
||S(t)(u ,u )|| ≤ C (1+t)||(u ,u )|| . (12)
0 1 H 0 0 1 H
This is the aim of the step:
5
Lemma 2.1. (Cauchy problem in H1 × L2 ) For all (u ,u ) ∈ H, there exists
loc,u loc,u 0 1
T > 0 such that there exists a unique solution of the problem (1) in C([0,T],H).
Proof. Consider T > 0 to be chosen later small enough in terms of ||(u ,u )|| .
0 1 H
We first write the Duhamel formulation for our equation
(cid:90) t
u(t) = S(t)(u ,u )+ S(t−τ)(0,eu(τ))dτ. (13)
0 1
0
Introducing
R = 2C (1+T)||(u ,u )|| , (14)
0 0 1 H
we will work in the Banach space E = C([0,T],H) equipped with the norm ||u|| =
E
sup ||u|| . Then, we introduce
H
0≤t≤T
Φ : E → E
(cid:18)v(t)(cid:19) (cid:90) t
V(t) = (cid:55)→ S(t)(u ,u )+ S(t−τ)(0,ev(t))dτ
v (t) 0 1
1 0
and the ball B (0,R).
E
We will show that for T > 0 small enough, Φ has a unique fixed point in B (0,R). To
E
do so, we have to check 2 points:
1. Φ maps B (0,R) to itself.
E
2. Φ is k-Lipschitz with k < 1 for T small enough.
(cid:18) (cid:19)
v
• Proof of 1: Let V = ∈ B (0,R), this means that:
v E
1
∀t ∈ [0,T], v(t) ∈ H1 (R) ⊂ L∞(R)
loc,u
and that
||v(t)|| ≤ C R.
L∞(R) ∗
Therefore
||(0,ev)|| = sup ||ev(t)||
E L2
0≤t≤T loc,u
√
≤ eC∗R 2. (15)
This means that
∀τ ∈ [0,T] (0,ev(τ)) ∈ H,
hence S(t−τ)(0,ev(τ)) is well defined from (12) and so is its integral between 0 and
t. So Φ is well defined from E to E.
6
Let us compute ||Φ(v)|| :
E
Using (12), (14) and (15) we write for all t ∈ [0,T],
(cid:90) t
||Φ(v)(t)|| ≤ ||S(t)(u ,u )|| + ||S(t−τ)(0,ev(τ))|| dτ
H 0 1 H H
0
R (cid:90) T √
≤ + C (1+T) 2eC∗Rdτ
0
2
0
R √
≤ +C T(1+T) 2eC∗R. (16)
0
2
Choosing T small enough so that
R √
+C T(1+T) 2eC∗R ≤ R
0
2
or
Re−C∗R
T(1+T) ≤ √
2 2C
0
guarantees that Φ goes from B (0,R) to B (0,R).
E E
¯
• Proof of 2: Let V,V ∈ B (0,R) we have
E
(cid:90) T
Φ(V)−Φ(V¯) = S(t−τ)(0,ev(t) −ev¯(t))dτ.
0
Since ||v(t)|| ≤ C R and the same for ||v¯(t)|| , we write
L∞(R) ∗ L∞(R)
|ev(τ) −ev¯(τ)| ≤ eC∗R|v(τ)−v¯(τ)|,
hence
||ev(τ) −ev¯(τ)|| ≤ eC∗R||v(τ)−v¯(τ)||
L2 L2
loc,u loc,u
≤ eC∗R||V −V¯|| . (17)
E
Applying S(t−τ) we write from (12), for all 0 ≤ τ ≤ t ≤ τ ∈ T,
||S(t−τ)(0,ev(τ) −ev¯(τ))|| ≤ C (1+T)||(0,ev(τ) −ev¯(τ))||
H 0 H
≤ C (1+T)||ev(τ) −ev¯(τ)||
0 L2
loc,u
≤ C (1+T)eC∗R||V −V¯|| . (18)
0 E
Integrating we end-up with
||Φ(V)−Φ(V¯)|| ≤ C T(1+T)eC∗R||V −V¯|| . (19)
E 0 E
k = C T(1+T)eC∗R can be made < 1 if T is small.
0
7
Conclusion: From points 1 and 2, Φ has a unique fixed point u(t) in B (0,R). This
E
fixed point is the solution of the Duhamel formulation (13) and of our equation (1). This
concludes the proof of Lemma 2.1.
Step 2: The Cauchy problem in a larger region
Let (u ,u ) ∈ H1 × L2 initial data for the problem (1). Using the finite speed of
0 1 loc,u loc,u
propagation, we will localize the problem and reduces it to the case of initial data in
H1 × L2 already treated in Step 1. For (x ,t ) ∈ R × (0,+∞), we will check the
loc,u loc,u 0 0
existence of the solution in the cone C . In order to do so, we introduce χ a C∞
x0,t0,1
function with compact support such that χ(x) = 1 if |x − x | < t , let also (u¯ ,u¯ ) =
0 0 0 1
(u χ,u χ) (note that u¯ and u¯ depend on (x ,t ) but we omit this dependence in the
0 1 0 1 0 0
indices for simplicity). So, (u¯ ,u¯ ) ∈ H1 ×L2 . From Step 1, if u¯ is the corresponding
0 1 loc,u loc,u
solution of equation (1), then, by the finite speed of propagation, u = u¯ in the intersection
oftheirdomainsofdefinitionwiththeconeC . Asu¯isdefinedforall(x,t)inR×[0,T)
x0,t0,1
from Step 1 for some T = T(x ,t ), we get the existence of u locally in C ∩R×[0,T).
0 0 x0,t0,1
Varying (x ,t ) and covering R × (0,+∞[ by an infinite number of cones, we prove the
0 0
existence and the uniqueness of the solution in a union of backward light cones, which
is either the whole half-space R × (0,+∞), or the subgraph of a 1-Lipschitz function
x (cid:55)→ T(x). We have just proved the following:
Lemma 2.2. (The Cauchy problem in a larger region) Consider (u ,u ) ∈ H1 ×
0 1 loc,u
L2 . Then, there exists a unique solution defined in D, a subdomain of R×[0,+∞), such
loc,u
that for any (x ,t ) ∈ D,(u,∂ u) ∈ H1 ×L2 (D ), with D = {x ∈ R|(x,t ) ∈ D}.
0 0 t (t0) loc loc t0 t0 0
Moreover,
− either D = R×[0,+∞),
− or D = {(x,t)|0 ≤ t < T(x)} for some 1-Lipschitz function x (cid:55)→ T(x).
Step 3: Regular approximations for equation (1)
Consider (u ,u ) ∈ H1 ×L2 , u its solution constructed in Step 2, and assume that it
0 1 loc,u loc,u
is non global, hence defined under the graph of a 1-Lipschitz function x (cid:55)→ T(x). Consider
for any n ∈ N, a regularized increasing truncation of F satisfying
(cid:26) eu if u ≤ n
F (u) = (20)
n en if u ≥ n+1
and F (u) ≤ min(eu,en+1). Consider also a sequence (u ,u ) ∈ (C∞(R))2 such that
n 0,n 1,n
(u ,u ) → (u ,u ) in H1 ×L2(−R,R) as n → ∞, for any R > 0.
0,n 1,n 0 1
Then, we consider the problem
(cid:26) ∂2u = ∂2u +F (u ),
t n x n n n (21)
(u (0),∂ u (0)) = (u ,u ) ∈ H1 ×L2 .
n t n 0,n 1,n loc,u loc,u
Since Steps 1 and 2 clearly extend to locally Lipschitz nonlinearities, we get a unique
solution u defined in the half-space R × (0,+∞), or in the subgraph of a 1-Lipschitz
n
function. Since F (u) ≤ en+1, for all u ∈ R, it is easy to see that in fact: u is defined for
n n
8
all (x,t) ∈ R×[0,+∞). From the regularity of F , u and u , it is clear that u is a
n 0,n 1,n n
strong solution in C2(R,[0,∞)). Introducing the following sets:
K+(x,t) = {(y,s) ∈ (R,R ),|y −x| < s−t}, (22)
+
K−(x,t) = {(y,s) ∈ (R,R ),|y −x| < t−s},
+
and
K±(x,t) = K±(x,t)∩D .
R R
we claim the following
Lemma 2.3. (Uniform bounds on variations of u in cones) Consider R > 0, one
n
can find C(R) > 0 such that if (x,t) ∈ D∩D , then ∀n ∈ N:
R
u (y,s) ≥ u (x,t)−C(R), ∀(y,s) ∈ K+(x,t),
n n R
u (y,s) ≤ u (x,t)+C(R), ∀(y,s) ∈ K−(x,t).
n n
Remark: Of course C depends also on initial data, but we omit that dependence, since
we never change initial data in this setting. Note that since (x,t) ∈ D , it follows that
R
K−(x,t) = K−(x,t).
R
Proof. We will prove the first inequality, the second one can be proved by the same way.
For more details see [11] page 74.
Let R > 0, consider (x,t) fixed in D ∩D , and (y,s) in D ∩K+(x,t). We introduce the
R R
following change of variables:
ξ = (y −x)−(s−t), η = −(y −x)−(s−t), u¯ (ξ,η) = u (y,s). (23)
n n
From (21), we see that u¯ satisfies :
n
1
∂ u¯ (ξ,η) = F (u¯ ) ≥ 0. (24)
ξη n n n
4
¯ ¯
Let (ξ,η¯) the new coordinates of (y,s) in the new set of variables. Note that ξ ≤ 0 and
¯
η¯≤ 0. We note that there exists ξ ≥ 0 and η ≥ 0 such that the points (ξ ,η¯) and (ξ,η )
0 0 0 0
lay on the horizontal line {s = 0} and have as original coordinates respectively (y∗,0)
and (y˜,0) for some y∗ and y˜ in [−R,R]. We note also that in the new set of variables, we
have :
¯ ¯ ¯ ¯
u (y,s)−u (x,t) = u¯ (ξ,η¯)−u¯ (0,0) = u¯ (ξ,η¯)−u¯ (ξ,0)+u¯ (ξ,0)−u¯ (0,0)
n n n n n n n n
(cid:82)0 ¯ (cid:82)0
= − ∂ u¯ (ξ,η)dη − ∂ u¯ (ξ,0)dξ. (25)
η¯ η n ξ¯ ξ n
¯
From (24), ∂ u¯ is monotonic in ξ. So, for example for η = η¯, as ξ ≤ 0 ≤ ξ , we have :
η n 0
¯
∂ u¯ (ξ,η¯) ≤ ∂ u¯ (0,η¯) ≤ ∂ u¯ (ξ ,η¯).
η n η n η n 0
¯
Similarly, for any η ∈ (η¯,0), we can bound from above the function ∂ u¯ (ξ,η) by its
η n
value at the point (ξ∗(η),η), which is the projection of (ξ¯,η) on the axis{s = 0} in parallel
to the axis ξ (as ξ¯≤ 0 ≤ ξ∗(η)).
9
By the same way, from (24), ∂ u¯ is monotonic in η. As η¯≤ 0 ≤ η , we can bound, for
ξ n 0
ξ ∈ (ξ¯,0), ∂ u¯ (ξ,0) by its value at the point (ξ,η∗(ξ)), which is the projection of (ξ,0)
ξ n
on the axis {s = 0} in parallel to the axis η (0 < η∗(ξ)). So it follows that:
∂ u¯ (ξ¯,η) ≤ ∂ u¯ (ξ∗(η),η), ∀η ∈ (η¯,0)
η n η n (26)
∂ u¯ (ξ,0) ≤ ∂ u¯ (ξ,η∗(ξ)), ∀ξ ∈ (ξ¯,0)
ξ n ξ n
Byastraightforwardgeometricalconstruction, weseethatthecoordinatesof(ξ∗(η),η)
√
and (ξ,η∗(ξ)), in the original set of variables {y,s}, are respectively (x+t−η 2,0) and
√
(x−t+η 2,0). Both points are in [−R,R].
Furthermore, we have from (23):
√ √
∂ u¯ (ξ∗(η),η) = 1(−∂ u −∂ u )(x+t−η 2,0) = 1(−u −∂ u )(x+t−η 2),
η n 2 t n x n √ 2 1,n x 0,n √
∂ u¯ (ξ,η∗(ξ)) = 1(−∂ u +∂ u )(x−t+η 2,0) = 1(−u +∂ u )(x−t+η 2).
ξ n 2 t n x n 2 1,n x 0,n
(27)
Using (27), the Cauchy-Schwarz inequality and the fact that u and ∂ u are uni-
1,n x 0,n
formly bounded in L2(−R,R) since they are convergent, we have:
(cid:82)0∂ u¯ (ξ∗(η),η)dη ≤ C(R),
η¯ η n (28)
(cid:82)0∂ u¯ (ξ,η∗(ξ))dξ ≤ C(R).
ξ¯ ξ n
Using (25), (26) and (28), we reach to conclusion of Lemma 2.3.
Let us show the following:
Lemma 2.4. (Convergence of u as n → ∞) Consider (x,t) ∈ R×[0,+∞). We have
n
the following:
− If t > T(x), then u (x,t) → +∞,
n
− If t < T(x), then u (x,t) → u(x,t).
n
Proof. We claim that it is enough to show the convergence for a subsequence. Indeed, this
is clear from the fact that the limit is explicit and doesn’t depend on the subsequence.
Consider (x,t) ∈ R×[0,+∞), up to extracting a subsequence, there is l(x,t) ∈ R such
that u (x,t) → l(x,t) as n → ∞.
n
Let us show that l (cid:54)= −∞. Since F (u) ≥ 0, it follows that u (x,t) ≥ u (x,t) where
n n n
(cid:26) ∂2u = ∂2u ,
t n x n (29)
u (0) = u and ∂ u (0) = u .
n 0,n t n 1,n
Since u ∈ L∞(R+,H1(−R,R)) ⊂ L∞(R+,L∞(−R,R)), for any R > 0, from the fact
n loc loc
that (u ,u ) is convergent in H1 ×L2 , it follows that l(x,t) ≥ limsup u (x,t) >
0,n 1,n loc loc n→+∞ n
−∞.
Note from the fact that F (u) ≤ eu that we have
n
∀x ∈ R,t < T(x), u (x,t) ≤ u(x,t). (30)
n
10