PORTUGALIAE MATHEMATICA
Vol. 60 Fasc. 4 – 2003
Nova Série
DECAY OF SOLUTIONS OF
SOME NONLINEAR EQUATIONS
Mohammed Aassila
Abstract: For a class of scalar partial differential equations that incorporate convection, diffusion, and possibly dispersion in one space and one time dimension, the
stability of solutions is investigated.
1 – Introduction
The topic of this paper is the class of equations
(1.1)
ut − α uxx + β uxxx + γ uxxxxt + (g(u))x = 0 ,
x ∈ R, t > 0
2
where subscripts denote partial derivatives. The case g(u) = u2 and γ = 0 is
typical and has received much attention. If α > 0, β = 0, this is known as
Burgers equation. If α = 0, β > 0, this is essentially the Korteweg–de-Vries
equation. The case α > 0, β > 0 is thus refereed to as KdV-Burgers equation;
it also has been studied extensively as has been the case of general g. The
p+1
case g(u) = up+1 and γ = 1, β = 0 where p ≥ 1 is an integer is refereed to
as the Rosenau–Burgers equation. Indeed, if α = 0 then we have the Rosenau
equation proposed by Rosenau [8] for treating the dynamics of dense discrette
systems in order to overcome the shortenings by the KdV equation, since the
KdV equation describes unidimensional propagation of waves, but wave-wave
and wave-wall interactions cannot be treated by it. Such a model were studied
Received : April 15, 2001; Revised : April 15, 2002.
AMS Subject Classification: 35Q53, 35B40.
Keywords: stability; convergence rates; asymptotic profile; traveling wave.
390
MOHAMMED AASSILA
by Park [9] and Chung and Ha [3] for the global existence of the solution to
the IBVP. Equation (1.1) with α > 0, γ = 1, β = 0 is called the Rosenau–
Burgers equation and somehow corresponds to the KdV-Burgers equation and the
Benjamin–Bona–Mahoney–Burgers equation, but it is given neither by Rosenau
nor by Burgers. The Rosenau equation with the dissipative term −αuxx , or say,
the Rosenau–Burgers equation arises in some natural phenomena as for example,
in bore propagation and in water waves.
In section 2 of this paper, we study the problem (1.1) with β = 1, γ = 0:
(E1)
ut − α uxx + uxxx + (g(u))x = 0 ,
x ∈ R, t > 0
where α > 0 and g is a C 2 -class function. We give a general criterion for the
existence of traveling wave solutions of the form u(x, t) = φ(x − ct).
In section 3, we study the asymptotic behaviour of the solution for the Rosenau–
Burgers equation (problem (1.1) with β = 0, γ = 1):

µ
¶

up+1 = 0 ,
u − αu + u
+
t
xx
xxxxt
p+1
x


(E2)
u(x, 0) = u0 (x) → 0
x ∈ R, t > 0 ,
as x → ±∞ ,
where α > 0 and p ≥ 1 is an integer.
The problem (E2) was studied by Mei [7]. He proved that if
then
ku(t)kL2 ≤
And, if
R
R u0 (x) dx
c
(1 + t)4
and
Z
R
u0 (x) dx 6= 0
c
ku(t)k∞ ≤ √
,
1+t
∀t ≥ 0 .
c
,
1+t
∀t ≥ 0 .
= 0, then
ku(t)kL2 ≤
c
(1 + t)3/4
and
ku(t)k∞ ≤
Hence, it is proved in [7] that 0 is the asymptotic state of the solution u(x, t) for
the Rosenau–Burgers equation. In this ³paper,´ we prove that the solution of the
p+1
nonlinear parabolic equation ut −αuxx + up+1 = 0 is a better asymptotic profile
x
for the Rosenau–Burgers equation. Furthermore, we prove that the convergence
to this asymptotic profile is faster than the convergence to 0 proved in [7].
Before ending this section, we state and prove a general technical lemma which
will be needed later.
DECAY OF SOLUTIONS OF SOME NONLINEAR EQUATIONS
391
Lemma 1.1.
(i) If a > 0, b > 0, then we have for all t ≥ 0
Z
t
0
(1 + t − s)
−a
(1 + s)
−b

− min(a,b)

 c(1 + t)
ds ≤


t)− min(a,b) log(2
c(1 +
c(1 + t)1−a−b
if max(a, b) > 1,
+ t) if max(a, b) = 1,
if max(a, b) < 1 .
(ii) Let 0 < a < b with b > 1. Let f : (0, ∞) → R be bounded on [1, ∞) and
integrable on (0,1). Then we have for all t ≥ 0
Z
t
0
f (t − s) (1 + t − s)−a (1 + s)−b ds ≤ c t−a .
Proof:
(i) We give a brief outline of the proof. Let
I =
Z
³
∞
(1 + s)−a 1 + |t − s|
0
´−b
,
where a > 0, b > 0 and max(a, b) > 1.
Assuming t > 0, as is no essential loss of generality, since evidently
I(−t) ≤ I(t), we may write
I =
Z
εt
+
0
Z
t
+
εt
Z
∞
,
t
where 0 < ε < 1, and ε is held fixed. Now
Z
εt
0
≤
Z
εt
0
³
(1 + s)−a t(1 − ε)
´−b
ds = o(t−b )
Z
εt
(1 + s)−a ds .
0
In case a > 1, this expression is o(t−b ). In case a = 1, it is o(log t · t−b ), which is
o(t−a ) since b > 1. In case a < 1, it is
o(t−b )
Z
εt
s−a ds = o(t−b ) o(t−a+1 ) = o(t1−a−b ) ;
0
since 1 + min(a, b) ≤ a + b by virtue of the assumption that max(a, b) > 1, this
is in turn o(t− min(a,b) ).
Similarly,
Z
t
εt
≤ o(t−a )
Z t³
εt
1 + |t − s|
´−b
ds = o(t−a ) o(t−b+1 )
392
MOHAMMED AASSILA
if b 6= 1, and so is o(t− min(a,b) ). In case b = 1, the integral is o(t−a )o(log t) = o(t−b )
since in this case a > 1. If b > 1,
Z
∞
≤ o(t
t
−a
Z
)
t
∞³
1 + |t − s|
´−b
ds ,
which is o(t1−a−b ). Finally, if b ≤ 1,
Z
∞
t
−b
≤ o(t )
Z
t
∞³
1 + |t − s|
´−a
= o(t−b ) .
(ii) We have
Z
t
f (t − s) (1 + t − s)−a (1 + s)−b ds ≤
0
≤
≤
Z
Z
t
0
(1 + t − s)−a (1 + s)−b ds +
t
(1 + t − s)
0
−a
(1 + s)
−b
Z
1
0
f (s) (1 + s)−a (1 + t − s)−b ds
ds + c(1 + t)−b .
Now,
Z
t
0
(1 + t − s)−a (1 + s)−b ds =
=
Z
t
2
0
(1 + t − s)
−a
(1 + s)
−b
ds +
Z
t
t
2
(1 + t − s)−a (1 + s)−b ds .
If a, b > 1, we get
Z
t
0
(1 + t − s)
−a
(1 + s)
−b
µ
t
1
1+
ds ≤
b−1
2
¶−a
µ
t
+ (a − 1) 1 +
2
¶−b
≤ c(1 + t)−a .
If a < 1 < b, we get
Z
t
0
(1 + t − s)
−a
(1 + s)
−b
µ
1
t
1+
ds ≤
b−1
2
¶−a
+ (1 − a) (1 + t)
1−a
µ
t
1+
2
¶−b
≤ c (1 + t)−a .
If a = 1 < b, we get
Z
t
0
(1 + t − s)
−a
(1 + s)
−b
µ
1
t
ds ≤
1+
b−1
2
≤ c(1 + t)−1 .
¶−1
µ
t
+ log 1 +
2
¶µ
t
1+
2
¶−b
DECAY OF SOLUTIONS OF SOME NONLINEAR EQUATIONS
393
2 – Existence of traveling wave solutions
Consider the equation
(E1)
ut − α uxx + uxxx + (g(u))x = 0 ,
x ∈ R, t > 0
where α > 0 and g is a C 2 -class function.
Under certain conditions, the equation admits monotone traveling wave solutions u(x, t) = φ(x−ct) with speed c that connect the end states φ± = lim φ(r).
r→±∞
Such a wave profile must satisfy the third order ordinary differential equation
−c φ0 + g(φ)0 + φ000 − α φ00 = 0 .
An example is g(x) = 2 x(x − 1) (b − x) with b ≥ 2, which has the wave profile
1
φ(x) = 1+e
x for the parameter α = 2b − 1 and the speed c = 0. General profiles
(non necessarily monotone) have been constructed in [1, 5]. It is known that
p+1
√
monotone profiles exist for g(u) = up+1 and α ≥ 2 pc . More generally, we have
the following criterion:
Theorem 2.1. Let g ∈ C 2 be a strictly convex function. A monotone wave
profile φ for (E1) exists if and only if:
g(φ+ ) − g(φ− )
,
φ+ − φ −
(2.1)
c =
(2.2)
α ≥ 2
(2.3)
φ+ < φ − .
q
g 0 (φ− ) − c ,
The profile must therefore be monotonically decreasing.
Proof:
(=⇒) Clearly we have
−c φ + g(φ) + φ00 − α φ0 = constant
and hence −cφ− + g(φ− ) = −cφ+ + g(φ+ ) which implies (2.1).
Now, set ψ(z) = φ− − φ(−z) and
(2.4)
f (r) = g(φ− ) − c r − g(φ− − r) .
Then, f is concave, and −αψ 0 −ψ 00 = f (ψ) which is the equation for a wave profile
ψ of the Fisher–Kolmogorov–Petrovskii–Piskunov (F-KPP) equation v t − vxx =
f (v) that travels to the right with speed α and has limits ψ− = φ− − φ+ , ψ+ = 0.
394
MOHAMMED AASSILA
Thanks to [4], we know that such a monotone wave profile for concave f exists
p
p
if and only if α ≥ 2 f 0 (0) = 2 g 0 (φ− ) − c. In this case, ψ− > ψ+ and therefore
φ− > φ+ , since f is positive between ψ− and ψ+ . Thus (2.2)–(2.3) are true.
(⇐=) Define f as in (2.4), then by known results about the F-KPP equation,
+
that moves
there exists a unique decreasing wave profile ψ with ψ(0) = φ− −φ
2
to the right with speed α. Then, φ(z) = φ− − ψ(−z) is a monotone wave profile
for (E1) with φ(±∞) = φ± .
3 – Asymptotic profile of Rosenau–Burgers equation
Consider the Rosenau–Burgers equation in the form
(E2)

!
Ã
p+1

u

u − αu + u
= 0,
t
xx
xxxxt +
p+1 x



u(x, 0) = u0 (x) → 0
x ∈ R, t > 0,
as x → ±∞ ,
where α is any given constant, p ≥ 1 is integer.
Consider the following scalings to the variables
t→
t
,
ε2
x→
x
,
ε
u → ε1/p u ,
where ε ¿ 1, then we obtain from (E2)
(3.1)
ut − α uxx + ε4 uxxxxt + up ux = 0 .
For ε ¿ 1, neglecting the small term ε4 uxxxxt leads to the asymptotic state equation of the Rosenau–Burgers equation (E2) as follows
ut − α uxx + up ux = 0 .
The solution of this parabolic equation should be a better asymptotic profile of
equation (E2).
Concerning the parabolic equation
(3.2)
(
vt − α vxx + v p vx = 0 ,
v(x, 0) = v0 (x) → 0
we have the following result:
as x → ±∞
DECAY OF SOLUTIONS OF SOME NONLINEAR EQUATIONS
395
Theorem 3.1 ([10, 11]). Suppose that v0 (x) ∈ H 2 (R) ∩ L1 (R). Then, there
exists a positive constant δ0 such that if kv0 kL1 + kv0 kH 2 ≤ δ0 , then the problem
(3.2) has a unique global solution v(x, t) with
v ∈ C(R+ , H 2 (R) ∩ L1 (R)) ∩ L2 (R+ , H 1 (R))
and
(3.3)
³
Furthermore, if v0 ∈ L1 (R) ∩ H 6 (R), then
(3.4)
´
k∂xj v(t)kLq = O(1) kv0 kL1 + kv0 kH 2 (1 + t)
³
−
(j+1)q−1
2q
´
j
k∂xj vt (t)kL1 = O(1) kv0 kL1 + kv0 kH 6 (1 + t)−1− 2 ,
,
1≤q≤∞.
j = 0, 1, 2, 3, 4 .
The main result in this section is the following:
Theorem 3.2. Suppose that
w0 (x) =
and
Z
x
−∞
³
´
u0 (y) − v0 (y) dy ∈ W 3,1 (R)
v0 (x) ∈ L1 (R) ∩ H 6 (R) .
Let α : = kv0 kL1 + kv0 kH 6 , then there exists a positive constant δ0 such that if
kw0 kW 3,1 + α ≤ δ0 ,
then the Cauchy problem (E2) has a unique global solution u(x, t) with
u(x, t) − v(x, t) ∈ C(R+ , H 1 (R)) and satisfies
(i) if p = 1, for any η > 0 we have
3
(3.5)
k(u − v)(t)kL2 ≤ c(1 + t)− 4 +η ,
(3.6)
k(u − v)x (t)kL2 ≤ c(1 + t)−1+η ,
(3.7)
k(u − v)(t)kL∞ ≤ c(1 + t)− 8 +η .
7
(ii) If p ≥ 2, then we have
3
(3.8)
k(u − v)(t)kL2 ≤ c(1 + t)− 4 ,
(3.9)
k(u − v)x (t)kL2 ≤ c(1 + t)− 4 ,
(3.10)
k(u − v)(t)kL∞ ≤ c(1 + t)−1 .
5
396
MOHAMMED AASSILA
As a corollary, we obtain
Corollary 3.3. Under the hypotheses of theorem 3.2, we have for 2 ≤ q ≤ ∞
the decay rates
(3.11)

1
7
 c(1 + t)− 8 + 4q +η
k(u − v)(t)kLq ≤

c(1 + t)
1
−1+ 2q
if p = 1,
if p ≥ 2 .
The result in the corollary follows from the interpolation inequality
q−2
2
q
kf kLq 2 ,
kf kLq ≤ kf kL∞
2≤q≤∞.
The rest of the paper is devoted to the proof of theorem 3.2.
From (E2) and (3.2), we have
(3.12)
(u − v)t − α(u − v)xx − uxxxxt +
Ã
v p+1
up+1
−
p+1 p+1
!
= 0.
x
Since v(±∞, t) = 0, and we expect u(±∞, t) = 0, ux (±∞, t) = 0, then after
integrating (3.12) over R, we get
d
dt
(3.13)
Z ³
R
´
u(x, t) − v(x, t) dx = 0 .
Integration of (3.13) over [0, t] and thanks to the assumptions we get
Z ³
(3.14)
R
´
u(x, t) − v(x, t) dx =
Z ³
R
´
u0 (x) − v0 (x) dx = 0 .
Thus we have
wxt − α wxxx + wxxxxxt − vxxxxt +
Ã
v p+1
(v + wx )p+1
−
p+1
p+1
!
= 0,
x
where we set
(3.15)
w(x, t) =
Z
x
−∞
³
´
u(y, t) − v(y, t) dy
that is wx (x, t) = u(x, t) − v(x, t).
The integration over (−∞, x] with respect to x of the above equation yields
(3.16)
(
wt − α wxx + wxxxxt = Hp (w) ,
w(0, x) = w0 (x) ,
DECAY OF SOLUTIONS OF SOME NONLINEAR EQUATIONS
397
where
Hp (w) : = vxxxt −
´
1 ³
(v + wx )p+1 − v p+1
p+1
¶
p µ
1 X
j
= vxxxt −
v j wxp+1−j ,
p + 1 j=0 p+1
We have
p≥1.
¶
p µ
1 X
j
|v j wxp+1−j | ,
|Hp (w)| ≤ |vxxxt | +
p + 1 j=0 p+1
|∂x Hp (w)| ≤ |vxxxxt | + |wxp wxx |
p
(3.17)
+
µ
1 X j
p+1 j=1 p+1
¶½
j |v j−1 vx wxp+1−j | + (p+1−j) |v j wxp−j wxx |
¾
.
Taking the Fourier transform of (3.16), we get
bt +
w
which admits as solution
\
α ξ2
H
p (w)
b
w
=
1 + ξ4
1 + ξ4
b t) = e−a(ξ)t w
b0 (ξ) +
w(ξ,
Z
t
e−a(ξ)(t−s)
0
\
H
p (w)(ξ, s)
ds ,
1 + ξ4
α ξ2
where we set a(ξ) : = 1+ξ 4 .
The inverse Fourier transform of the above resultant equation yields
(3.18)
1
w(x, t) =
2π
Z
∞
−∞
1
+
2π
b0 (ξ) dξ
eiξx e−a(ξ)t w
Z tZ
∞
eiξx e−a(ξ)(t−s)
0 −∞
\
H
p (w)(ξ, s)
dξ ds .
1 + ξ4
The differentiation with respect to x of (3.18) gives
∂xj w(x, t) =
(3.19)
1
2π
Z
∞
−∞
1
−
2π
b0 (ξ) dξ
(iξ)j eiξx e−a(ξ)t w
Z tZ
∞
0 −∞
(iξ)j eiξx e−a(ξ)(t−s)
\
H
p (w)(ξ, s)
dξ ds .
1 + ξ4
Now, we define the solution spaces as follows, for any positive integer p ≥ 1 and
given δ > 0:
n
o
Spδ : = w ∈ C(R+ , H 2 (R)) | Ap (w) ≤ δ ,
398
MOHAMMED AASSILA
where

1
X
A1 (w) = sup
0≤t≤∞ 
2j+1
−η
4
j=0
2
X
Ap (w) = sup
(1+t)
(1+t)
2j+1
4
0≤t≤∞ j=0
k∂xj w(t)kL2 + (1+t)1−η kwxx (t)kL2
k∂xj w(t)kL2 ,
p≥2.



,
Rewriting (3.18) as the operational form w = Sw, we need to prove that S is a
contraction mapping from Spδ into Spδ where δ > 0 is a positive constant.
We have
Theorem 3.4. Under the hypotheses of theorem 3.2, there exists a positive
constant δ1 such that if
kw0 kW 3,1 + α < δ1
then Cauchy problem (3.16) has a unique global solution w(x, t) ∈ C(R+ , H 2 (R)).
Furthermore, we have the following estimates:
(i) If p = 1, then for any η > 0 we have
(3.20)
1
X
(1+t)
2j+1
−η
4
i=0
´
³
k∂xj w(t)kL2 + (1+t)1−η kwxx (t)kL2 ≤ c kw0 kW 3,1 +α .
(ii) If p ≥ 2, then we have
2
X
(3.21)
(1 + t)
j=0
2j+1
4
³
´
k∂xj w(t)kL2 ≤ c kw0 kW 3,1 + α .
Since u(x, t) − v(x, t) = wx (x, t), once we prove theorem 3.1, then theorem 3.2
can be easily proved. Hence, we prove theorem 3.1 in the rest of the paper.
We need to this end two lemmas and two well-known estimates quoted from [7]:
(3.22)
(3.23)
Z
∞
−∞
j+1
|ξ|j e−ca(ξ)t
dξ ≤ c(1 + t)− 2 ,
4
j
(1 + ξ ) (1 + |ξ|)
°
Z
° 1
°
° 2π
for j = 0, 1, 2.
∞
°
°
b0 (ξ) dξ °
(iξ)j eiξx e−a(ξ)t w
°
−∞
L2
j = 0, 1, 2, 3, 4 ,
≤ c kw0 kW j+1,1 (1 + t)−
2j+1
4
399
DECAY OF SOLUTIONS OF SOME NONLINEAR EQUATIONS
Lemma 3.5. Let w1 (x, t), w2 (x, t) ∈ Spδ , then we have
¯
¯
¯
¯
−1+η
\
,
sup¯H\
1 (w1 )(ξ, s) − H1 (w2 )(ξ, s)¯ ≤ c(α + δ) A1 (w1 − w2 ) (1 + s)
ξ∈R
¯
¯
¯
¯
p
−3
\
sup¯H\
p (w1 )(ξ, s) − Hp (w2 )(ξ, s)¯ ≤ c(α + δ) Ap (w1 − w2 ) (1 + s) 2 ,
ξ∈R
¯
p≥2,
¯
¯
¯
− 5 +η
\
sup |ξ| ¯H\
,
1 (w1 )(ξ, s) − H1 (w2 )(ξ, s)¯ ≤ c(α + δ) A1 (w1 − w2 ) (1 + s) 4
ξ∈R
¯
¯
¯
¯
p
−2
\
sup |ξ| ¯H\
,
p (w1 )(ξ, s) − Hp (w2 )(ξ, s)¯ ≤ c(α + δ) Ap (w1 −w2 ) (1+s)
ξ∈R
p≥2.
Lemma 3.6. Let w(x, t) ∈ Spδ , then
(i) if p = 1
°
°
°
Z t°
Z ∞
\
° 1
°
j iξx −a(ξ)(t−s) H1 (w)(ξ, s)
°
° ds ≤
(iξ) e e
dξ
° 2π
°
1 + ξ4
0 °
−∞
° 2
L
³
´
2j+1
≤ c α + (α + δ)2 (1 + t)−
4
+η
°
°
°
Z t°
Z ∞
\
° 1
°
H
(w)(ξ,
s)
1
2 iξx −a(ξ)(t−s)
°
(iξ) e e
dξ °
° 2π
° ds ≤
4
1+ξ
0 °
−∞
° 2
³ L
,
j = 0, 1 ,
´
≤ c α + (α + δ)2 (1 + t)−1+η ;
(ii) if p ≥ 2, then we have
°
°
°
Z ∞
Z t°
\
°
° 1
H
(w)(ξ,
s)
p
j iξx −a(ξ)(t−s)
° ds ≤
°
(iξ) e e
dξ
°
° 2π
1 + ξ4
−∞
0 °
° 2
³
´ L
2j+1
≤ c α + (α + δ)p+1 (1 + t)−
4
,
j = 0, 1, 2 .
Proof of Theorem 3.1: Rewriting (3.18) in the form w = Sw, we need
to prove that there exists a positive constant δ1 such that the operator S is a
contraction mapping from Spδ1 into Spδ1 .
We claim that S maps Spδ into itself. Indeed, for any w1 (x, t) ∈ Spδ and
denoting w = Sw1 we will prove that w ∈ Spδ for some small δ > 0. Thanks to
lemma 3.6, for any positive integer p there exists a constant c1 such that
³
Ap (w) ≤ c1 kw0 kW 3,1 + α + (α + δ)p+1
´
.
400
MOHAMMED AASSILA
o
n
Let n = max 2 + 2p+1 , c11 , and choose δ2 ≤
α≤
δ2
nc1
1
n c1 ,
and δ < δ2 we have
Ap (w) ≤ c1
≤ c1
Ã
δ2
δ2
+
+
n c1 n c1
µ
δ2
nc1 ,
then for kv0 kW 3,1 ≤
δ2
+ δ2
n c1
¶p+1 !
(2 + 2p+1 ) δ2
≤ δ2 .
n c1
Hence S : Spδ → Spδ for some small δ < δ2 .
Now, let us prove that S is a contraction in Spδ . Suppose that w1 (x, t), w2 (x, t) ∈
δ
Sp (δ < δ2 ), then we have by (3.18): for j = 0, 1, 2 and p ≥ 1
∂xj (Sw1 −Sw2 ) =
1
2π
Z tZ
∞
(iξ)j eiξx e−a(ξ)(t−s)
0 −∞
\
H\
p (w1 )(ξ, s) − Hp (w2 )(ξ, s)
dξ ds .
4
1+ξ
We estimate the term kSw1 − Sw2 kL2 : we have
°
Z t°
° 1 Z ∞
−a(ξ)(t−s) ³
´ °
°
°
iξx e
\
\
Hp (w1 )(ξ, s) − Hp (w2 )(ξ, s) dξ ° ds =
e
°
° 2
1 + ξ4
0 ° 2π −∞
L
!1
Ã
Z t Z ∞ −a(ξ)(t−s) ¯
¯2
2
e
¯ \
¯
\
ds
H
(w
)(ξ,
s)
−
H
(w
)(ξ,
s)
dξ
=
¯
¯
p
1
p
2
4 2
0
≤
Z
−∞
t
(1 + ξ )
Ã
¯
¯ Z
¯ \
¯
\
sup¯Hp (w1 )(ξ, s) − Hp (w2 )(ξ, s)¯
0 ξ∈R
∞
−∞
e−a(ξ)(t−s)
dξ
(1 + ξ 4 )2
!1
2
ds .
Using (3.22), lemma 1.1 and lemma 3.5, we obtain
°
Z t°
° 1 Z ∞
−a(ξ)(t−s) ³
´ °
°
°
iξx e
\
ds ≤
e
H\
°
1 (w1 )(ξ, s) − H1 (w2 )(ξ, s) dξ °
4
°
° 2
2π
1
+
ξ
0
−∞
L
Z t
≤ c(α + δ) A1 (w1 − w2 )
0
≤ c(α + δ) A1 (w1 − w2 ) (1 + t)−(1/4−η) ,
0 < η ≤ 1/2 ,
°
Z t°
° 1 Z ∞
−a(ξ)(t−s) ³
´ °
°
°
iξx e
\
e
H\
ds ≤
°
p (w1 )(ξ, s) − Hp (w2 )(ξ, s) dξ °
4
°
° 2
2π
1
+
ξ
−∞
0
L
Z t
≤ c(α + δ)p Ap (w1 − w2 )
p
1
(1 + s)−(1−η) (1 + t − s)− 4 ds
3
0
1
(1 + s)− 2 (1 + t − s)− 4 ds
1
≤ c(α + δ) Ap (w1 − w2 ) (1 + t)− 4 ,
p≥2.
401
DECAY OF SOLUTIONS OF SOME NONLINEAR EQUATIONS
That is, we obtain
kSw1 − Sw2 kL2 ≤
≤

 c(α + δ) A1 (w1 − w2 ) (1 + t)−(1/4−η) ,
 c(α + δ)p A (w − w ) (1 + t)
p
1
2
− 41
0 < η < 1/2, for p = 1 ,
,
for p ≥ 2 .
Similarly, by using (3.22), lemma 1.1 and lemma 3.5, we have the estimates for
k∂x (Sw1 − Sw2 )kL2 and k∂x2 (Sw1 − Sw2 )kL2 as follows:
k∂x (Sw1 − Sw2 )kL2 ≤
≤
and

 c(α + δ) A1 (w1 − w2 ) (1 + t)−(3/4−η) ,
0 < η < 1/2, for p = 1 ,
 c(α + δ)p A (w − w ) (1 + t)− 34 ,
p
1
2
for p ≥ 2 ,
k∂x2 (Sw1 − Sw2 )kL2 ≤
≤

 c(α + δ) A1 (w1 − w2 ) (1 + t)−(1−η) ,
0 < η < 1/2, for p = 1 ,
 c(α + δ)p A (w − w ) (1 + t)− 54 ,
p
1
2
for p ≥ 2 .
Hence, we deduce that, for some constant c1 :
n
Ap (Sw1 − Sw2 ) ≤ c1 (α + δ)p Ap (w1 − w2 ) .
o
Let n = max c21 , 2 and choose δ ≤ δ3 < nc11 , then for α < δ3 and Ap (w2 ) < δ3
we deduce that
Ap (Sw1 − Sw2 ) < Ap (w1 − w2 ) ,
that is S : Spδ → Spδ is a contraction for small δ < δ3 .
Finally, let δ1 < min {δ2 , δ3 }, then we have proved that S is a contraction from
δ
1
Sp to Spδ1 , and consequently by Banach’s fixed point theorem, S has a unique
fixed point in Spδ1 , and then we have the existence of a unique global solution.
Proof of Lemma 3.5: For p = 1, we have
¯
¯
¯
¯
\
sup¯H\
1 (w1 )(ξ, s) − H1 (w2 )(ξ, s)¯ ≤
ξ∈R
(3.24)
≤ kv(s)kL2 k(w1x − w2x )(s)kL2
1
+ k(w1x + w2x )(s)kL2 k(w1x − w2x )(s)kL2
2
³
1
3
´
3
≤ c α(1 + s)− 4 + δ(1 + s)− 4 +η A1 (w1 − w2 ) (1 + s)− 4 +η
≤ c(α + δ) A1 (w1 − w2 ) (1 + s)−1+η ,
402
MOHAMMED AASSILA
and
¯
¯
¯
¯
\
sup |ξ| ¯H\
1 (w1 )(ξ, s) − H1 (w2 )(ξ, s)¯ ≤
ξ∈R
≤
Z
∞
−∞
¯
¯
¯
¯
¯H1x (w1 )(x, s) − H1x (w2 )(x, s)¯ dx
≤ kvx (s)kL2 k(w1x − w2x )(s)kL2 + kv(s)kL2 k(w1xx − w2xx )(s)kL2
+ kw1x (s)kL2 k(w1xx − w2xx )(s)kL2 + kw2xx (s)kL2 k(w1x − w2x )(s)kL2
(3.25)
≤ c
½³
³
´
3
3
α(1 + s)− 4 + δ(1 + s)−1+η A1 (w1 − w2 ) (1 + s)− 4 +η
+ α(1 + s)
− 41
+ δ(1 + s)
− 34 +η
´
A1 (w1 − w2 ) (1 + s)
−1+η
¾
5
≤ c(α + δ) A1 (w1 − w2 ) (1 + s)− 4 +η .
For p ≥ 2, we have
¯
¯
¯
¯
\
sup¯H\
p (w1 )(ξ, s) − Hp (w2 )(ξ, s)¯ ≤
ξ∈R
≤
≤
(3.26)
Z
Z
∞
−∞
∞
¯
¯
¯
¯
¯Hp (w1 )(x, s) − Hp (w2 )(x, s)¯ dx
−∞
1
≤
p+1
+
¶¯
p µ
¯
1 X
j
¯ j p+1−j
p+1−j ¯
− v j w2x
¯v w1x
¯ dx
p+1 j=0 p+1
(µ
p−1
Xµ
j=0
c
≤
p+1
p
p+1
¶
¶
p−1
kv(s)kL
∞ kv(s)kL2 k(w1x − w2x )(s)kL2
p−j
X
j
p−j−i
kw1x (s)kiL2 kw2x (s)kL
kv(s)kjL∞ k(w1x−w2x )(s)kL2
2
p+1
i=0
(µ
¶
p−1µ
¶
)
X
p+1
p
j
αp +
(p−j) αj δ p−j Ap (w1−w2 ) (1+s)− 2
p+1
p+1
j=0
3
≤ c(α + δ)p Ap (w1 − w2 ) (1 + s)− 2
)
403
DECAY OF SOLUTIONS OF SOME NONLINEAR EQUATIONS
and
¯
¯
¯
¯
\
sup |ξ|¯H\
p (w1 )(ξ, s) − Hp (w2 )(ξ, s)¯ ≤
ξ∈R
≤
Z
∞
−∞
¯
¯
¯
¯
¯∂x Hp (w1 )(x, s) − ∂x Hp (w2 )(x, s)¯ dx
°
°
°
°
p
p
≤ °(w1xx w1x
− w2xx w2x
)(s)° 1
L
µ
¶½ °
°
1
p
°
°
p °v p−1 (s) vx (s) (w1x − w2x )(s)°
+
p+1
p+1
°
°
°
°
+ °v p (s)(w1xx − w2xx )(s)°
+
p−1 µ
1 X
p+1 j=1
j
p+1
°
°
L1
L1
¾
¶½ °
°
°
°
p+1−j
p+1−j
−w2x
)(s)°
j °v j−1 (s) vx (s) (w1x
°
°
p−j
p−j
+ (p+1−j) °v j (s) (w1x
w1xx − w2x
w2xx )(s)°
L1
¾
L1
.
Since
°
°
°
°
p
p
°(w1xx w1x − w2xx w2x )(s)°
L1
≤
≤ kw1xx (s)kL2 k(w1x − w2x (s)kL2
p−1
X
i=0
kw1x (s)kiL2 kw2x (s)kp−1−i
L2
+ kw2x (s)kpL2 k(w1xx − w2xx )(s)kL2
≤ c δ p Ap (w1 − w2 ) (p + 1) (1 + s)−2
and
1
p+1
µ
p
p+1
¶½ °
°
°
°
p °v p−1 (s) vx (s) (w1x −w2x )(s)°
½
p
≤ c p α Ap (w1 − w2 ) (1 + s)
L
°
°
° p
°
+
v
(s)
(w
−w
)(s)
°
°
1xx
2xx
1
− p+2
2
p
+ α Ap (w1 − w2 ) (1 + s)
≤ c αp (p + 1) Ap (w1 − w2 ) (1 + s)−2
L1
¾
− p+2
2
≤
¾
404
MOHAMMED AASSILA
and
¶ °
p−1 µ
°
1 X
j
°
°
p+1−j
p+1−j
j °v j−1 (s) vx (s) (w1x
− w2x
)(s)° 1 ≤
L
p+1 j=1 p+1
≤
¶
p−1 µ
j−1
c X
j
j αj (1+s)− 2 −1 Ap (w1 − w2 )
p+1 j=1 p+1
× (1+s)
− 34
p−j
X
δ p−j (1 + s)−
3(p−j)
4
i=0
≤ c Ap (w1 − w2 ) (1+s)
−2
p−1 µ
1 X
p+1 j=1
j
p+1
¶
j(p−j +1) αj δ p−j
and
¶
p−1 µ
°
°
1 X
j
°
°
p−j
p−j
w1xx − w2x
w2xx )(s)° 1 ≤
(p+1−j) °v j (s)(w1x
L
p+1 j=1 p+1
¶
p−1 µ
j
c X
j
≤
(p+1−j) αj (1+s)− 2
p+1 j=1 p+1
×
½
δ p−j Ap (w1 − w2 ) (1 + s)−
+δ
p−j
Ap (w1 − w2 ) (1 + s)
≤ c Ap (w1 − w2 )(1 + s)−2
3p+5−3j
4
−2
(1 + s)
−
3(p−j−1)
4
(p − j)
¾
¶
p−1 µ
1 X
j
(p+1−j)2 αj δ p−j ,
p+1 j=1 p+1
we obtain that
¯
¯
¯
¯
\
sup |ξ| ¯H\
p (w1 )(ξ, s) − Hp (w2 )(ξ, s)¯ ≤
ξ∈R
(3.27)
(
≤ c Ap (w1 − w2 ) (1 + s)−2 δ p (p+1) + αp (p+1)
!)
¶
µ
¶
p−1õ
1 X
j
j
j p−j
2 j p−j
+
j(p+1−j) α δ
+
(p+1−j) α δ
p+1
p+1
p+1 j=1
≤ c(α + δ)p Ap (w1 − w2 ) (1 + s)−2 .
Thanks to (3.24)–(3.27), we deduce the result of lemma 3.5.
405
DECAY OF SOLUTIONS OF SOME NONLINEAR EQUATIONS
Proof of Lemma 3.6: Since for f ∈ H 1 we have
1
1
√
kf kL∞ ≤ 2 kf kL2 2 · kfx kL2 2
we easily deduce that
(i) for p = 1,
1
1
√
√
1
2 kw(t)kL2 2 kwx (t)kL2 2 ≤ 2 δ(1 + t)− 2 +η ,
1
1
√
√
7
≤ 2 kwx (t)kL2 2 kwxx (t)kL2 2 ≤ 2 δ(1 + t)− 8 +η ;
kw(t)kL∞ ≤
(3.28)
kwx (t)kL∞
(ii) for p ≥ 2:
1
1
√
√
1
2 kw(t)kL2 2 kwx (t)kL2 2 ≤ 2 δ(1 + t)− 2 ,
kw(t)kL∞ ≤
(3.29)
kwx (t)kL∞ ≤
1
1
√
√
2 kwx (t)kL2 2 kwxx (t)kL2 2 ≤ 2 δ(1 + t)−1 .
Thanks to (3.17) we have
\
sup |H
p (w)(ξ, s)| ≤
ξ∈R
≤
Z
Z
∞
−∞
∞
−∞
|Hp (w)(x, s)| dx
(
)
¶
p µ
1 X
j
|v j wxp+1−j | dx
|vxxxt | +
p+1 j=0 p+1
≤ kvxxxt (s)kL1 +
1
p+1
µ
¶
p
p−1
kv(s)kL
∞ kv(s)kL2 kwx (s)kL2
p+1
¶
p−1 µ
1 X
j
+
kv(s)kjL∞ kwx (s)kp−1−j
kwx (s)k2L2 .
L∞
p+1 j=1 p+1
Now, because of (3.28)–(3.29) we have
½
−5
2
−(3/2−2η)
\
sup |H
+ α δ(1+s)−(1−η)
1 (w)(ξ, s)| ≤ c α(1+s) 2 + δ (1+s)
ξ∈R
n
o
≤ c α + (α + δ)2 (1+s)−(1−η) ,
(3.30)
by choosing η such that 0 < η ≤ 1/2, and
(
−
\
sup |H
p (w)(ξ, s)| ≤ c α(1+s) 2 +
ξ∈R
(3.31)
+
p−1
Xµ
j=0
n
5
j
p+1
¶
j
µ
α δ
¶
p+1
p
αp δ(1+s)− 2
p+1
p+1−j
o
(1+s)
3
≤ c α + (α+δ)p+1 (1+s)− 2 ,
− 2p−j+1
2
)
p≥2,
¾
406
MOHAMMED AASSILA
where for p ≥ 2, we used (1 + s)−
we have
2p−j+1
2
< (1 + s)−
p+1
2
3
≤ (1 + s)− 2 . Similarly,
\
sup |ξ| |H
p (w)(ξ, s)| ≤
ξ∈R
p−1
≤ kvxxxxt (s)kL1 + kwx (s)kL
2 kwx (s)kL2 kwxx (s)kL2
+
+
¶½
p µ
1 X
j
p−j
j kv(s)kj−1
L∞ kwx (s)kL∞ kvx (s)kL2 kwx (s)kL2
p+1 j=1 p+1
p−j
(p+1−j) kv(s)kj−1
L∞ kwx (s)kL∞
kv(s)kL2 kwxx (s)kL2
¾
.
Now, since for p ≥ 2 we have (1 + s)−(p+1) < (1 + s)−2 and (1 + s)−
p+2
(1 + s)− 2 ≤ (1 + s)−2 , we have
½
2p−j+2
2
−3
\
sup |ξ| |H
+ δ 2 (1+s)−2(1−η) + α δ(1+s)−(3/2−η)
1 (w)(ξ, s)| ≤ c α(1+s)
ξ∈R
(3.32)
n
o
≤ c α + (α+δ)2 (1+s)−(3/2−η) ,
(
0<η≤
−3
\
sup |ξ| |H
+ δ p+1 (1+s)−(p+1)
p (w)(ξ, s)| ≤ c α(1+s)
ξ∈R
(3.33)
+
¶
p µ
X
j
p+1
j=1
n
αj δ p−j+1 (1+s)
− p+2
2
o
≤ c α + (α+δ)p+1 (1+s)−2 ,
1
,
2
¾
)
p≥2.
By the Parseval equality we have
°
Z t°
° 1 Z ∞
°
−a(ξ)(t−s)
e
°
°
\
eiξx
H
(w)(ξ,
s)
dξ
°
° ds =
p
4
°
° 2
1+ξ
0 2π −∞
L
°
Z t ° −a(ξ)(t−s)
°e
°
\
°
°
=
° 1 + ξ 4 Hp (w)(ξ, s)°
0
≤
Z
t
\
sup |H
p (w)(ξ, s)|
0 ξ∈R
ÃZ
∞
−∞
ds
L2
e−2a(ξ)(t−s)
dξ
(1+ξ 4 )2
!1
2
ds .
≤
407
DECAY OF SOLUTIONS OF SOME NONLINEAR EQUATIONS
Thanks to (3.30)–(3.31), lemma 1.1 and (3.22) we obtain
°
Z t°
°
° 1 Z ∞
−a(ξ)(t−s)
e
°
°
\
H
(w)(ξ,
s)
dξ
eiξx
° ds ≤
°
1
4
° 2
°
1+ξ
0 2π −∞
L
n
oZ t
≤ c α + (α+δ)2
(3.34)
n
1
(1+s)−(1−η) (1+t−s)− 4 ds
0
o
≤ c α + (α+δ)2 (1+t)−(1/4−η) ,
°
Z t°
° 1 Z ∞
°
−a(ξ)(t−s)
°
°
iξx e
\
e
H
(w)(ξ,
s)
dξ
°
° ds ≤
p
° 2
1 + ξ4
0 ° 2π −∞
L
n
oZ
≤ c α + (α+δ)p+1
(3.35)
n
o
t
0<η≤
1
,
2
1
3
(1+s)− 2 (1+t−s)− 4 ds
0
1
≤ c α + (α+δ)p+1 (1+t)− 4 ,
p≥2.
Similarly, we have
°
Z t°
° 1 Z ∞
°
−a(ξ)(t−s)
e
°
°
\
iξ eiξx
H
(w)(ξ,
s)
dξ
°
° ds ≤
p
4
°
° 2
1+ξ
0 2π −∞
L
ÃZ
Z
t
≤
\
sup |H
p (w)(ξ, s)|
0 ξ∈R
∞
−∞
|ξ|2 e−2a(ξ)(t−s)
dξ
(1 + ξ 4 )2
!1
2
ds .
Using (3.30)–(3.31), lemma 1.1 and (3.22), we obtain
°
Z t°
°
° 1 Z ∞
−a(ξ)(t−s)
°
°
iξx e
\
iξ e
H
(w)(ξ,
s)
dξ
° ds ≤
°
1
° 2
1 + ξ4
0 ° 2π −∞
L
n
oZ t
(3.36)
≤ c α + (α+δ)2
n
3
(1+s)−(1−η) (1+t−s)− 4 ds
0
o
≤ c α + (α+δ)2 (1+t)−(3/4−η) ,
(3.37)
°
Z t°
° 1 Z ∞
°
−a(ξ)(t−s)
°
°
iξx e
\
iξ e
H
(w)(ξ,
s)
dξ
°
° ds ≤
p
4
°
° 2
2π
1
+
ξ
0
−∞
L
n
o
0<η≤
3
≤ c α + (α+δ)p+1 (1+t)− 4 ,
1
,
2
p≥2.
408
MOHAMMED AASSILA
Furthermore, we have
°
Z t°
° 1 Z ∞
°
−a(ξ)(t−s)
°
°
2 iξx e
\
(iξ) e
H
(w)(ξ,
s)
dξ
°
°
p
4
°
°
2π
1
+
ξ
0
−∞
ds =
L2
°
Z t°
−a(ξ)(t−s)
°
°
2 e
\
°
=
Hp (w)(ξ, s)°
°(iξ)
° 2 ds
4
1+ξ
0
L
!1
Ã
Z t
¯
¯´ Z ∞ |ξ|4 e−2a(ξ)(t−s)
2
³
¯\
¯
ds .
dξ
sup (1 + |ξ|) ¯Hp (w)(ξ, s)¯
≤
4
4
0 ξ∈R
−∞
(1 + ξ ) (1 + |ξ|)
Using (3.30)–(3.33), lemma 1.1 and (3.22), we have
°
Z t°
° 1 Z ∞
°
−a(ξ)(t−s)
°
°
2 iξx e
\
(iξ) e
H
(w)(ξ,
s)
dξ
°
° ds ≤
1
4
°
° 2
2π
1
+
ξ
−∞
0
L
Z
n
o tn
(3.38)
≤ c α + (α+δ)2
n
0
o
≤ c α + (α+δ)2 (1+t)−(1−η) ,
0<η≤
1
,
2
°
Z t°
° 1 Z ∞
°
−a(ξ)(t−s)
°
°
2 iξx e
\
(iξ) e
H
(w)(ξ,
s)
dξ
°
° ds ≤
p
4
°
° 2
2π
1
+
ξ
−∞
0
L
Z
n
o tn
3
(3.39)
≤ c α + (α+δ)p+1
n
o
5
(1+s)−(1−η) + (1+s)−(3/2−η) (1+t−s)− 4 ds
≤ c α + (α+δ)
p+1
o
5
(1+s)− 2 + (1+s)−2 (1+t−s)− 4 ds
o
0
5
(1+t)− 4 ,
p≥2.
Now, thanks to (3.34)–(3.39), the proof of lemma 3.6 is complete.
ACKNOWLEDGEMENT – The author thanks the referee for the useful remarks and
suggestions.
REFERENCES
[1] Bona, J.L. and Shonbek, M.E. – Traveling-wave solutions to the Korteweg–
de-Vries–Burgers equation, Proc. Roy. Soc. Edinburgh, Sect A. 101 (1985), 207–226.
[2] Chern, I.L. and Liu, T.P. – Convergence to diffusion waves of solutions for
viscous conservation laws, Comm. Math. Phys., 110 (1987), 503–517.
DECAY OF SOLUTIONS OF SOME NONLINEAR EQUATIONS
409
[3] Chung, S.K. and Ha, S.H. – Finite element Galerkin solutions for the Rosenau
equation, Applicable Analysis, 54 (1994), 39–56.
[4] Henry, D. – Geometric Theory of Semilinear Parabolic Equations, LNM 840,
Springer, Berlin, 1981.
[5] Jacobs, D.; McKinney, B. and Shearer, M. – Traveling-wave solutions of
the modified Korteweg–de-Vries–Burgers equation, J. Differential Equations, 116
(1995), 448–467.
[6] Jeffrey, A. and Zhao, H. – Global existence and optimal temporal decay
estimates for system parabolic conservation laws I. The one-dimensional case,
Applicable Analysis, 70 (1998), 175–193.
[7] Mei, M. – Long-time behavior of solution for Rosenau–Burgers equation,
Applicable Analysis, 68 (1998), 333–356.
[8] Nishihara, K. and Rajopadhye, S. – Asymptotic behaviour of solutions to
the Korteweg–de-Vries–Burgers equations, Differential and Integral Equations, 11
(1998), 85–93.
[9] Park, M.A. – On nonlinear dispersive equations, Differential and Integral Equations, 9 (1996), 1331–1335.
[10] Pego, R.L. – Remarks on the stability of shock profiles for conservation laws with
dissipation, Trans. Amer. Math. Soc., 291 (1985), 353–361.
[11] Rosenau, P. – Dynamics of dense discrete systems, Prog. Theoretical Phys., 79
(1988), 1028–1042.
Mohammed Aassila,
Mathematisches Institut,
Weyertal 86-90, D-50931 Koln – GERMANY
E-mail: aassila@mi.uni-koeln.de