Nova S´erie

DECAY OF SOLUTIONS OF SOME NONLINEAR EQUATIONS

Mohammed Aassila

Abstract: For a class of scalar partial differential equations that incorporate con- vection, 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) u_{t}−α u_{xx}+β u_{xxx}+γ u_{xxxxt}+ (g(u))_{x} = 0, x∈R, t >0
where subscripts denote partial derivatives. The case g(u) = ^{u}_{2}^{2} 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
case g(u) = ^{u}_{p+1}^{p+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.

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−αu_{xx}, 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) u_{t}−α u_{xx}+u_{xxx}+ (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 formu(x, t) =φ(x−ct).

In section 3, we study the asymptotic behaviour of the solution for the Rosenau–

Burgers equation (problem (1.1) withβ = 0, γ = 1):

(E2)

u_{t}−α u_{xx}+u_{xxxxt}+
µu^{p+1}

p+1

¶

x

= 0, x∈R, t >0, u(x,0) =u0(x)→0 as x→ ±∞ ,

whereα >0 and p≥1 is an integer.

The problem (E2) was studied by Mei [7]. He proved that if Z

Ru_{0}(x)dx6= 0
then

ku(t)kL^{2} ≤ c

(1 +t)^{4} and ku(t)k^{∞}≤ c

√1 +t, ∀t≥0.
And, if^{R}_{R}u_{0}(x)dx= 0, then

ku(t)kL^{2} ≤ c

(1 +t)^{3/4} and ku(t)k^{∞}≤ c

1 +t, ∀t≥0 .

Hence, it is proved in [7] that 0 is the asymptotic state of the solutionu(x, t) for
the Rosenau–Burgers equation. In this paper, we prove that the solution of the
nonlinear parabolic equationu_{t}−αu_{xx}+^{³}^{u}_{p+1}^{p+1}^{´}

x= 0 is a better asymptotic profile 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.

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}ds ≤

c(1 +t)^{−}^{min(a,b)} if max(a, b)>1,
c(1 +t)^{−}^{min(a,b)}log(2 +t) if max(a, b) = 1,
c(1 +t)^{1−a−b} if max(a, b)<1 .
(ii) Let 0< a < bwithb >1. Let f: (0,∞)→R be bounded on [1,∞) and

integrable on (0,1). Then we have for allt≥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 ∞ 0

(1 +s)^{−a}^{³}1 +|t−s|^{´}^{−b} ,
wherea >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 εis held fixed. Now

Z _{εt}

0 ≤

Z _{εt}

0 (1 +s)^{−a}^{³}t(1−ε)^{´}^{−b}ds = o(t^{−b})
Z _{εt}

0 (1 +s)^{−a}ds .

In casea >1, this expression iso(t^{−b}). In casea= 1, it iso(logt·t^{−b}), which is
o(t^{−a}) since b >1. In casea <1, it is

o(t^{−}^{b})
Z εt

0

s^{−}^{a}ds = o(t^{−}^{b})o(t^{−}^{a+1}) = o(t^{1}^{−}^{a}^{−}^{b}) ;

since 1 + min(a, b) ≤a+b by virtue of the assumption that max(a, b) >1, this
is in turno(t^{−}^{min(a,b)}).

Similarly,
Z _{t}

εt ≤ o(t^{−a})
Z _{t}

εt

³1 +|t−s|^{´}^{−b}ds = o(t^{−a})o(t^{−b+1})

ifb6= 1, and so iso(t^{−}^{min(a,b)}). In caseb= 1, the integral is o(t^{−}^{a})o(logt) =o(t^{−}^{b})
since in this casea >1. Ifb >1,

Z ∞

t ≤ o(t^{−}^{a})
Z ∞

t

³1 +|t−s|^{´}^{−}^{b}ds ,

which iso(t^{1−a−b}). Finally, ifb≤1,
Z ∞

t ≤ o(t^{−b})
Z ∞

t

³1 +|t−s|^{´}^{−a}= o(t^{−b}) .

(ii) We have
Z _{t}

0 f(t−s) (1 +t−s)^{−a}(1 +s)^{−b}ds ≤

≤
Z _{t}

0 (1 +t−s)^{−a}(1 +s)^{−b}ds +
Z _{1}

0 f(s) (1 +s)^{−a}(1 +t−s)^{−b}ds

≤
Z _{t}

0

(1 +t−s)^{−}^{a}(1 +s)^{−}^{b}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 .
Ifa, b >1, we get

Z _{t}

0 (1 +t−s)^{−a}(1 +s)^{−b}ds ≤ 1
b−1

µ 1+t

2

¶−a

+ (a−1) µ

1 + t 2

¶−b

≤ c(1 +t)^{−a} .
Ifa <1< b, we get

Z _{t}

0

(1 +t−s)^{−}^{a}(1 +s)^{−}^{b}ds ≤ 1
b−1

µ 1 + t

2

¶−a

+ (1−a) (1 +t)^{1}^{−}^{a}
µ

1 + t 2

¶−b

≤ c(1 +t)^{−a} .
Ifa= 1< b, we get

Z _{t}

0 (1 +t−s)^{−a}(1 +s)^{−b}ds ≤ 1
b−1

µ 1 + t

2

¶−1

+ log µ

1 + t 2

¶ µ 1 + t

2

¶−b

≤ c(1 +t)^{−1} .

2 – Existence of traveling wave solutions

Consider the equation

(E1) ut−α uxx+uxxx+ (g(u))_{x} = 0, x∈R, t >0
whereα >0 and gis aC^{2}-class function.

Under certain conditions, the equation admits monotone traveling wave solu- tionsu(x, t) =φ(x−ct) with speedcthat connect the end states φ±= lim

r→±∞φ(r).

Such a wave profile must satisfy the third order ordinary differential equation

−c φ^{0}+g(φ)^{0}+φ^{000}−α φ^{00} = 0.

An example is g(x) = 2x(x−1) (b−x) with b≥ 2, which has the wave profile
φ(x) = _{1+e}^{1}x for the parameter α= 2b−1 and the speedc= 0. General profiles
(non necessarily monotone) have been constructed in [1, 5]. It is known that
monotone profiles exist forg(u) = ^{u}_{p+1}^{p+1} and α≥2√pc. More generally, we have
the following criterion:

Theorem 2.1. Letg∈C^{2} be a strictly convex function. A monotone wave
profileφfor (E1) exists if and only if:

c = g(φ_{+})−g(φ−)
φ+−φ−

, (2.1)

α ≥ 2^{q}g^{0}(φ−)−c ,
(2.2)

φ_{+} < φ− .
(2.3)

The profile must therefore be monotonically decreasing.

Proof:

(=⇒) Clearly we have

−c φ+g(φ) +φ^{00}−α φ^{0} = 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) equationv_{t}−v_{xx} =
f(v) that travels to the right with speedα and has limitsψ−=φ−−φ_{+}, ψ_{+}= 0.

Thanks to [4], we know that such a monotone wave profile for concavef exists
if and only ifα≥2^{p}f^{0}(0) = 2^{p}g^{0}(φ−)−c. In this case,ψ−> ψ_{+} and therefore
φ−> φ_{+}, sincef is positive betweenψ− and ψ_{+}. Thus (2.2)–(2.3) are true.

(⇐=) Definef as in (2.4), then by known results about the F-KPP equation,
there exists a unique decreasing wave profile ψ with ψ(0) = ^{φ}^{−}^{−φ}_{2} ^{+} that moves
to the right with speedα. Then,φ(z) =φ−−ψ(−z) is a monotone wave profile
for (E1) withφ(±∞) =φ±.

3 – Asymptotic profile of Rosenau–Burgers equation Consider the Rosenau–Burgers equation in the form

(E2)

u_{t}−α u_{xx}+u_{xxxxt}+
Ãu^{p+1}

p+ 1

!

x

= 0, x∈R, t >0,
u(x,0) =u_{0}(x)→0 as x→ ±∞ ,

whereα is any given constant,p≥1 is integer.

Consider the following scalings to the variables t→ t

ε^{2}, x→ x

ε, u→ε^{1/p}u ,
whereε¿1, then we obtain from (E2)

(3.1) u_{t}−α u_{xx}+ε^{4}u_{xxxxt}+u^{p}u_{x} = 0.

Forε¿1, neglecting the small termε^{4}u_{xxxxt} leads to the asymptotic state equa-
tion of the Rosenau–Burgers equation (E2) as follows

ut−α uxx+u^{p}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 as x→ ±∞

we have the following result:

Theorem 3.1 ([10, 11]). Suppose thatv_{0}(x)∈H^{2}(R)∩L^{1}(R). Then, there
exists a positive constantδ_{0} such that ifkv_{0}kL^{1}+kv_{0}kH^{2} ≤δ_{0}, then the problem
(3.2) has a unique global solutionv(x, t) with

v ∈ C(R^{+}, H^{2}(R)∩L^{1}(R)) ∩ L^{2}(R^{+}, H^{1}(R))
and

(3.3) k∂_{x}^{j}v(t)kL^{q} = O(1)^{³}kv_{0}kL^{1} +kv_{0}kH^{2}

´(1 +t)^{−}^{(j+1)q−1}^{2}^{q} , 1≤q ≤ ∞.
Furthermore, ifv_{0} ∈L^{1}(R)∩H^{6}(R), then

(3.4) k∂_{x}^{j}vt(t)kL^{1} = O(1)^{³}kv0kL^{1}+kv0kH^{6}

´(1 +t)^{−1−}^{j}^{2}, j= 0,1,2,3,4.

The main result in this section is the following:

Theorem 3.2. Suppose that
w_{0}(x) =

Z _{x}

−∞

³u_{0}(y)−v_{0}(y)^{´}dy ∈ W^{3,1}(R)
and

v0(x) ∈ L^{1}(R)∩H^{6}(R) .

Letα: =kv_{0}kL^{1}+kv_{0}kH^{6}, then there exists a positive constant δ_{0} such that if
kw_{0}kW^{3,1}+α ≤ δ_{0} ,

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) ifp= 1, for any η >0 we have

k(u−v)(t)kL^{2} ≤ c(1 +t)^{−}^{3}^{4}^{+η} ,
(3.5)

k(u−v)_{x}(t)kL^{2} ≤ c(1 +t)^{−}^{1+η} ,
(3.6)

k(u−v)(t)kL^{∞} ≤ c(1 +t)^{−}^{7}^{8}^{+η} .
(3.7)

(ii) If p≥2, then we have

k(u−v)(t)kL^{2} ≤ c(1 +t)^{−}^{3}^{4} ,
(3.8)

k(u−v)_{x}(t)kL^{2} ≤ c(1 +t)^{−}^{5}^{4} ,
(3.9)

k(u−v)(t)kL^{∞} ≤ c(1 +t)^{−1} .
(3.10)

As a corollary, we obtain

Corollary 3.3. Under the hypotheses of theorem 3.2, we have for2≤q≤ ∞ the decay rates

(3.11) k(u−v)(t)kL^{q} ≤

c(1 +t)^{−}^{7}^{8}^{+}^{4q}^{1}^{+η} if p= 1,
c(1 +t)^{−1+}^{2q}^{1} if p≥2 .
The result in the corollary follows from the interpolation inequality

kfkL^{q} ≤ kfk

q−2 q

L^{∞} kfk

2 q

L^{2}, 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}−u_{xxxxt}+

Ãu^{p+1}

p+ 1− v^{p+1}
p+ 1

!

x

= 0.

Since v(±∞, t) = 0, and we expect u(±∞, t) = 0, u_{x}(±∞, t) = 0, then after
integrating (3.12) overR, we get

(3.13) d

dt Z

R

³u(x, t)−v(x, t)^{´}dx = 0 .

Integration of (3.13) over [0, t] and thanks to the assumptions we get (3.14)

Z

R

³u(x, t)−v(x, t)^{´}dx =
Z

R

³u_{0}(x)−v_{0}(x)^{´}dx = 0 .

Thus we have

wxt−α wxxx+wxxxxxt−vxxxxt+

Ã(v+wx)^{p+1}

p+ 1 − v^{p+1}
p+ 1

!

x

= 0 , where we set

(3.15) w(x, t) =

Z x

−∞

³u(y, t)−v(y, t)^{´}dy

that isw_{x}(x, t) =u(x, t)−v(x, t).

The integration over (−∞, x] with respect tox of the above equation yields (3.16)

(w_{t}−α w_{xx}+w_{xxxxt} =H_{p}(w) ,
w(0, x) =w_{0}(x),

where

Hp(w) : = vxxxt− 1 p+ 1

³(v+wx)^{p+1}−v^{p+1}^{´}

= vxxxt− 1 p+ 1

Xp

j=0

µ j p+1

¶

v^{j}w^{p+1−j}_{x} , p≥1 .
We have

|H_{p}(w)| ≤ |v_{xxxt}|+ 1
p+ 1

Xp

j=0

µ j p+1

¶

|v^{j}w_{x}^{p+1−j}|,

|∂_{x}H_{p}(w)| ≤ |v_{xxxxt}|+|w^{p}_{x}w_{xx}|

+ 1

p+1 Xp

j=1

µ j p+1

¶ ½

j|v^{j−1}v_{x}w^{p+1−j}_{x} |+ (p+1−j)|v j w_{x}^{p−j}w_{xx}|

¾ . (3.17)

Taking the Fourier transform of (3.16), we get b

wt+ α ξ^{2}

1 +ξ^{4}w^{b} = H\_{p}(w)
1 +ξ^{4}
which admits as solution

w(ξ, t) =b e^{−}^{a(ξ)t}w_{b}_{0}(ξ) +
Z _{t}

0

e^{−}^{a(ξ)(t}^{−}^{s)} H\_{p}(w)(ξ, s)
1 +ξ^{4} ds ,
where we set a(ξ) : = _{1+ξ}^{α ξ}^{2}_{4}.

The inverse Fourier transform of the above resultant equation yields

(3.18)

w(x, t) = 1 2π

Z ∞

−∞

e^{iξx}e^{−a(ξ)t}wb_{0}(ξ)dξ
+ 1

2π
Z _{t}

0

Z ∞

−∞

e^{iξx}e^{−a(ξ)(t−s)}H\_{p}(w)(ξ, s)

1 +ξ^{4} dξ ds .
The differentiation with respect tox of (3.18) gives

(3.19)

∂_{x}^{j}w(x, t) = 1
2π

Z ∞

−∞

(iξ)^{j}e^{iξx}e^{−a(ξ)t}w_{b}_{0}(ξ)dξ

− 1 2π

Z _{t}

0

Z ∞

−∞

(iξ)^{j}e^{iξx}e^{−}^{a(ξ)(t}^{−}^{s)}H\_{p}(w)(ξ, s)

1 +ξ^{4} dξ ds .
Now, we define the solution spaces as follows, for any positive integerp≥1 and
givenδ >0:

Sp^{δ} : = ^{n}w∈C(R^{+}, H^{2}(R))|Ap(w)≤δ^{o} ,

where

A_{1}(w) = sup

0≤t≤∞

X1

j=0

(1+t)^{2j+1}^{4} ^{−}^{η}k∂_{x}^{j}w(t)kL^{2} + (1+t)^{1}^{−}^{η}kw_{xx}(t)kL^{2}

,
A_{p}(w) = sup

0≤t≤∞

X2

j=0

(1+t)^{2j+1}^{4} k∂_{x}^{j}w(t)kL^{2}, p≥2 .

Rewriting (3.18) as the operational form w=Sw, we need to prove that S is a
contraction mapping fromSp^{δ} intoSp^{δ} where δ >0 is a positive constant.

We have

Theorem 3.4. Under the hypotheses of theorem 3.2, there exists a positive
constantδ_{1} such that if

kw_{0}kW^{3,1}+α < δ_{1}

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 anyη >0we have (3.20)

X1

i=0

(1+t)^{2j+1}^{4} ^{−η}k∂_{x}^{j}w(t)kL^{2} + (1+t)^{1−η}kw_{xx}(t)kL^{2} ≤ c^{³}kw_{0}kW^{3,1}+α^{´}.

(ii) If p≥2, then we have (3.21)

X2

j=0

(1 +t)^{2j+1}^{4} k∂_{x}^{j}w(t)kL^{2} ≤ c^{³}kw0kW^{3,1}+α^{´} .

Sinceu(x, t)−v(x, t) =w_{x}(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]:

Z ∞

−∞

|ξ|^{j}e^{−ca(ξ)t}

(1 +ξ^{4}) (1 +|ξ|)^{j} dξ ≤ c(1 +t)^{−}^{j+1}^{2} , j= 0,1,2,3,4 ,
(3.22)

°°

°° 1 2π

Z ∞

−∞

(iξ)^{j}e^{iξx}e^{−a(ξ)t}w_{b}_{0}(ξ)dξ

°°

°°

L^{2} ≤ ckw_{0}kW^{j+1,1}(1 +t)^{−}^{2j+1}^{4}
(3.23)

for j= 0,1,2.

Lemma 3.5. Let w_{1}(x, t), w_{2}(x, t)∈ Sp^{δ}, then we have
sup

ξ∈R

¯¯

¯H\1(w1)(ξ, s)−H\1(w2)(ξ, s)^{¯}^{¯}_{¯} ≤ c(α+δ)A1(w1−w2) (1 +s)^{−1+η} ,
sup

ξ∈R

¯¯

¯H\_{p}(w_{1})(ξ, s)−H\_{p}(w_{2})(ξ, s)^{¯}^{¯}_{¯} ≤ c(α+δ)^{p}A_{p}(w_{1}−w_{2}) (1 +s)^{−}^{3}^{2}, p≥2,
sup

ξ∈R|ξ|^{¯}^{¯}¯H\_{1}(w_{1})(ξ, s)−H\_{1}(w_{2})(ξ, s)^{¯}^{¯}_{¯} ≤ c(α+δ)A_{1}(w_{1}−w_{2}) (1 +s)^{−}^{5}^{4}^{+η} ,
sup

ξ∈R|ξ|^{¯}^{¯}¯H\_{p}(w_{1})(ξ, s)−H\_{p}(w_{2})(ξ, s)^{¯}^{¯}_{¯} ≤ c(α+δ)^{p}A_{p}(w_{1}−w_{2}) (1+s)^{−2}, p≥2 .
Lemma 3.6. Let w(x, t)∈ Sp^{δ}, then

(i) if p= 1
Z _{t}

0

°°

°°

°° 1 2π

Z ∞

−∞

(iξ)^{j}e^{iξx}e^{−a(ξ)(t−s)}H\_{1}(w)(ξ, s)
1 +ξ^{4} dξ

°°

°°

°°_{L}_{2}ds ≤

≤ c^{³}α+ (α+δ)^{2}^{´}(1 +t)^{−}^{2j+1}^{4} ^{+η}, j = 0,1 ,
Z _{t}

0

°°

°°

°° 1 2π

Z ∞

−∞

(iξ)^{2}e^{iξx}e^{−}^{a(ξ)(t}^{−}^{s)}H\_{1}(w)(ξ, s)
1 +ξ^{4} dξ

°°

°°

°°

L^{2}

ds ≤

≤ c^{³}α+ (α+δ)^{2}^{´}(1 +t)^{−}^{1+η} ;
(ii) if p≥2, then we have

Z _{t}

0

°°

°°

°° 1 2π

Z ∞

−∞

(iξ)^{j}e^{iξx}e^{−a(ξ)(t−s)}H\p(w)(ξ, s)
1 +ξ^{4} dξ

°°

°°

°°

L^{2}

ds ≤

≤ c^{³}α+ (α+δ)^{p+1}^{´}(1 +t)^{−}^{2j+1}^{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 fromSp^{δ}^{1} intoSp^{δ}^{1}.

We claim that S maps Sp^{δ} into itself. Indeed, for any w_{1}(x, t) ∈ Sp^{δ} and
denotingw =Sw_{1} we will prove that w ∈ Sp^{δ} for some small δ > 0. Thanks to
lemma 3.6, for any positive integerpthere exists a constant c_{1} such that

Ap(w) ≤ c1

³kw0kW^{3,1} +α+ (α+δ)^{p+1}^{´} .

Let n = max^{n}2 + 2^{p+1},_{c}^{1}

1

o, and choose δ_{2} ≤ _{n c}^{1}_{1}, then for kv_{0}kW^{3,1} ≤ _{nc}^{δ}^{2}_{1},
α≤ _{nc}^{δ}^{2}_{1} and δ < δ_{2} we have

A_{p}(w) ≤ c_{1}
Ã δ_{2}

n c1

+ δ_{2}
n c1

+
µ δ_{2}

n c1

+δ_{2}

¶p+1!

≤ c_{1}(2 + 2^{p+1})δ_{2}

n c1 ≤ δ_{2} .
HenceS: Sp^{δ}→ Sp^{δ} for some smallδ < δ_{2}.

Now, let us prove thatSis a contraction inSp^{δ}. Suppose thatw_{1}(x, t), w_{2}(x, t)∈
Sp^{δ} (δ < δ_{2}), then we have by (3.18): for j= 0,1,2 and p≥1

∂_{x}^{j}(Sw_{1}−Sw_{2}) = 1
2π

Z _{t}

0

Z ∞

−∞

(iξ)^{j}e^{iξx}e^{−a(ξ)(t−s)}H\_{p}(w_{1})(ξ, s)−H\_{p}(w_{2})(ξ, s)
1 +ξ^{4} dξ ds .
We estimate the termkSw_{1}−Sw_{2}kL^{2}: we have

Z _{t}

0

°°

°°

° 1 2π

Z ∞

−∞

e^{iξx}e^{−}^{a(ξ)(t}^{−}^{s)}
1 +ξ^{4}

³H\_{p}(w_{1})(ξ, s)−H\_{p}(w_{2})(ξ, s)^{´}dξ

°°

°°

°L^{2}

ds =

=
Z _{t}

0

ÃZ ∞

−∞

e^{−a(ξ)(t−s)}
(1 +ξ^{4})^{2}

¯¯

¯H\_{p}(w_{1})(ξ, s)−H\_{p}(w_{2})(ξ, s)^{¯}^{¯}_{¯}^{2}dξ

!^{1}_{2}
ds

≤
Z _{t}

0 sup

ξ∈R

¯¯

¯H\p(w1)(ξ, s)−H\p(w2)(ξ, s)^{¯}^{¯}_{¯}
ÃZ ∞

−∞

e^{−a(ξ)(t−s)}
(1 +ξ^{4})^{2} dξ

!^{1}_{2}
ds .

Using (3.22), lemma 1.1 and lemma 3.5, we obtain
Z _{t}

0

°°

°°

° 1 2π

Z ∞

−∞

e^{iξx}e^{−a(ξ)(t−s)}
1 +ξ^{4}

³H\_{1}(w_{1})(ξ, s)−H\_{1}(w_{2})(ξ, s)^{´}dξ

°°

°°

°L^{2}

ds ≤

≤ c(α+δ)A_{1}(w_{1}−w_{2})
Z _{t}

0

(1 +s)^{−}^{(1}^{−}^{η)}(1 +t−s)^{−}^{1}^{4} ds

≤ c(α+δ)A1(w1−w2) (1 +t)^{−(1/4−η)}, 0< η ≤1/2,
Z _{t}

0

°°

°°

° 1 2π

Z ∞

−∞

e^{iξx}e^{−}^{a(ξ)(t}^{−}^{s)}
1 +ξ^{4}

³H\_{p}(w_{1})(ξ, s)−H\_{p}(w_{2})(ξ, s)^{´}dξ

°°

°°

°L^{2}

ds ≤

≤ c(α+δ)^{p}Ap(w1−w2)
Z _{t}

0 (1 +s)^{−}^{3}^{2} (1 +t−s)^{−}^{1}^{4} ds

≤ c(α+δ)^{p}A_{p}(w_{1}−w_{2}) (1 +t)^{−}^{1}^{4} , p≥2.

That is, we obtain
kSw_{1}−Sw_{2}kL^{2} ≤

≤

c(α+δ)A_{1}(w_{1}−w_{2}) (1 +t)^{−(1/4−η)}, 0< η <1/2, for p= 1 ,
c(α+δ)^{p}A_{p}(w_{1}−w_{2}) (1 +t)^{−}^{1}^{4}, for p≥2 .

Similarly, by using (3.22), lemma 1.1 and lemma 3.5, we have the estimates for
k∂x(Sw1−Sw2)kL^{2} and k∂_{x}^{2}(Sw1−Sw2)kL^{2} as follows:

k∂_{x}(Sw_{1}−Sw_{2})kL^{2} ≤

≤

c(α+δ)A_{1}(w_{1}−w_{2}) (1 +t)^{−}^{(3/4}^{−}^{η)}, 0< η <1/2, for p= 1 ,
c(α+δ)^{p}Ap(w1−w2) (1 +t)^{−}^{3}^{4}, for p≥2 ,

and

k∂_{x}^{2}(Sw_{1}−Sw_{2})kL^{2} ≤

≤

c(α+δ)A_{1}(w_{1}−w_{2}) (1 +t)^{−(1−η)}, 0< η <1/2, for p= 1 ,
c(α+δ)^{p}A_{p}(w_{1}−w_{2}) (1 +t)^{−}^{5}^{4} , for p≥2.

Hence, we deduce that, for some constantc_{1}:

A_{p}(Sw_{1}−Sw_{2}) ≤ c_{1}(α+δ)^{p}A_{p}(w_{1}−w_{2}) .
Letn= max^{n}_{c}^{2}

1,2^{o}and choose δ ≤δ_{3} < _{nc}^{1}

1, then for α < δ_{3} and A_{p}(w_{2})< δ_{3}
we deduce that

A_{p}(Sw_{1}−Sw_{2})< A_{p}(w_{1}−w_{2}) ,
that isS: Sp^{δ} → Sp^{δ} is a contraction for small δ < δ_{3}.

Finally, letδ_{1} <min{δ_{2}, δ_{3}}, then we have proved thatSis a contraction from
Sp^{δ}^{1} to Sp^{δ}^{1}, and consequently by Banach’s fixed point theorem, S has a unique
fixed point inSp^{δ}^{1}, and then we have the existence of a unique global solution.

Proof of Lemma 3.5: For p= 1, we have sup

ξ∈R

¯¯

¯H\_{1}(w_{1})(ξ, s)−H\_{1}(w_{2})(ξ, s)^{¯}^{¯}_{¯} ≤

(3.24)

≤ kv(s)kL^{2}k(w_{1x}−w_{2x})(s)kL^{2}

+ 1

2k(w_{1x}+w_{2x})(s)kL^{2}k(w_{1x}−w_{2x})(s)kL^{2}

≤ c^{³}α(1 +s)^{−}^{1}^{4} +δ(1 +s)^{−}^{3}^{4}^{+η}^{´}A1(w1−w2) (1 +s)^{−}^{3}^{4}^{+η}

≤ c(α+δ)A_{1}(w_{1}−w_{2}) (1 +s)^{−1+η} ,

and

sup

ξ∈R|ξ|^{¯}^{¯}¯H\_{1}(w_{1})(ξ, s)−H\_{1}(w_{2})(ξ, s)^{¯}^{¯}_{¯} ≤

(3.25)

≤ Z ∞

−∞

¯¯

¯H_{1x}(w_{1})(x, s)−H_{1x}(w_{2})(x, s)^{¯}^{¯}_{¯}dx

≤ kv_{x}(s)kL^{2}k(w_{1x}−w_{2x})(s)kL^{2} + kv(s)kL^{2}k(w_{1xx}−w_{2xx})(s)kL^{2}

+kw1x(s)kL^{2}k(w1xx−w2xx)(s)kL^{2} + kw2xx(s)kL^{2}k(w1x−w2x)(s)kL^{2}

≤ c^{½³}α(1 +s)^{−}^{3}^{4} +δ(1 +s)^{−1+η}^{´}A_{1}(w_{1}−w_{2}) (1 +s)^{−}^{3}^{4}^{+η}
+^{³}α(1 +s)^{−}^{1}^{4} +δ(1 +s)^{−}^{3}^{4}^{+η}^{´}A_{1}(w_{1}−w_{2}) (1 +s)^{−1+η}

¾

≤ c(α+δ)A_{1}(w_{1}−w_{2}) (1 +s)^{−}^{5}^{4}^{+η} .

For p≥2, we have

sup

ξ∈R

¯¯

¯H\_{p}(w_{1})(ξ, s)−H\_{p}(w_{2})(ξ, s)^{¯}^{¯}_{¯} ≤

(3.26)

≤ Z ∞

−∞

¯¯

¯H_{p}(w_{1})(x, s)−H_{p}(w_{2})(x, s)^{¯}^{¯}_{¯}dx

≤ Z ∞

−∞

1 p+1

Xp

j=0

µ j p+1

¶ ¯¯¯v^{j}w_{1x}^{p+1−j}−v^{j}w^{p+1−j}_{2x} ^{¯}^{¯}_{¯}dx

≤ 1 p+1

(µ p p+1

¶

kv(s)k^{p−1}L^{∞} kv(s)kL^{2}k(w_{1x}−w_{2x})(s)kL^{2}

+

p−1X

j=0

µ j p+1

¶

kv(s)k^{j}L^{∞}k(w_{1x}−w_{2x})(s)kL^{2}
p−jX

i=0

kw_{1x}(s)k^{i}L^{2}kw_{2x}(s)k^{p−j−i}_{L}^{2}
)

≤ c p+1

(µ p p+1

¶
α^{p}+

p−1X

j=0

µ j p+1

¶

(p−j)α^{j}δ^{p−j}
)

Ap(w1−w2) (1+s)^{−}^{p+1}^{2}

≤ c(α+δ)^{p}A_{p}(w_{1}−w_{2}) (1 +s)^{−}^{3}^{2}

and

sup

ξ∈R|ξ|^{¯}^{¯}¯H\_{p}(w_{1})(ξ, s)−H\_{p}(w_{2})(ξ, s)^{¯}^{¯}_{¯} ≤

≤ Z ∞

−∞

¯¯

¯∂_{x}H_{p}(w_{1})(x, s)−∂_{x}H_{p}(w_{2})(x, s)^{¯}^{¯}_{¯}dx

≤ ^{°}^{°}°(w_{1xx}w^{p}_{1x}−w_{2xx}w_{2x}^{p} )(s)^{°}^{°}_{°}

L^{1}

+ 1

p+ 1 µ p

p+1

¶ ½

p^{°}^{°}_{°}v^{p}^{−}^{1}(s)v_{x}(s) (w_{1x}−w_{2x})(s)^{°}^{°}_{°}

L^{1}

+^{°}^{°}_{°}v^{p}(s)(w_{1xx}−w_{2xx})(s)^{°}^{°}_{°}

L^{1}

¾

+ 1

p+1

p−1

X

j=1

µ j p+1

¶ ½

j^{°}^{°}_{°}v^{j−1}(s)vx(s) (w^{p+1−j}_{1x} −w_{2x}^{p+1−j})(s)^{°}^{°}_{°}

L^{1}

+ (p+1−j)^{°}^{°}_{°}v^{j}(s) (w^{p−j}_{1x} w_{1xx}−w^{p−j}_{2x} w_{2xx})(s)^{°}^{°}_{°}

L^{1}

¾ .

Since

°°

°(w1xxw^{p}_{1x}−w2xxw_{2x}^{p} )(s)^{°}^{°}_{°}

L^{1} ≤

≤ kw_{1xx}(s)kL^{2}k(w_{1x}−w_{2x}(s)kL^{2}
p−1X

i=0

kw_{1x}(s)k^{i}L^{2}kw_{2x}(s)k^{p−1−i}_{L}^{2}
+kw_{2x}(s)k^{p}_{L}^{2}k(w_{1xx}−w_{2xx})(s)kL^{2}

≤ 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)^{°}^{°}_{°}

L^{1}+^{°}^{°}_{°}v^{p}(s) (w1xx−w2xx)(s)^{°}^{°}_{°}

L^{1}

¾

≤

≤ c

½

p α^{p}A_{p}(w_{1}−w_{2}) (1 +s)^{−}^{p+2}^{2} +α^{p}A_{p}(w_{1}−w_{2}) (1 +s)^{−}^{p+2}^{2}

¾

≤ c α^{p}(p+ 1)A_{p}(w_{1}−w_{2}) (1 +s)^{−2}

and 1 p+1

p−1X

j=1

µ j p+1

¶

j^{°}^{°}_{°}v^{j}^{−}^{1}(s)v_{x}(s) (w_{1x}^{p+1−j}−w^{p+1−j}_{2x} )(s)^{°}^{°}_{°}

L^{1} ≤

≤ c p+1

p−1X

j=1

µ j p+1

¶

j α^{j}(1+s)^{−}^{j−1}^{2} ^{−}^{1}A_{p}(w_{1}−w_{2})

× (1+s)^{−}^{3}^{4}

pX−j

i=0

δ^{p−j}(1 +s)^{−}^{3(p−j)}^{4}

≤ c Ap(w1−w2) (1+s)^{−2} 1
p+1

p−1X

j=1

µ j p+1

¶

j(p−j+1)α^{j}δ^{p−j}

and 1 p+1

p−1X

j=1

µ j p+1

¶

(p+1−j)^{°}^{°}_{°}v^{j}(s)(w_{1x}^{p−j}w1xx−w^{p−j}_{2x} w2xx)(s)^{°}^{°}_{°}

L^{1} ≤

≤ c p+1

p−1X

j=1

µ j p+1

¶

(p+1−j)α^{j}(1+s)^{−}^{j}^{2}

×

½

δ^{p}^{−}^{j}A_{p}(w_{1}−w_{2}) (1 +s)^{−}^{3p+5−3j}^{4}

+ δ^{p−j}Ap(w1−w2) (1 +s)^{−2}(1 +s)^{−}^{3(p−j−1)}^{4} (p−j)

¾

≤ c A_{p}(w_{1}−w_{2})(1 +s)^{−}^{2} 1
p+1

p−1X

j=1

µ j p+1

¶

(p+1−j)^{2}α^{j}δ^{p}^{−}^{j} ,

we obtain that sup

ξ∈R|ξ|^{¯}^{¯}¯H\_{p}(w_{1})(ξ, s)−H\_{p}(w_{2})(ξ, s)^{¯}^{¯}_{¯} ≤

(3.27)

≤ c A_{p}(w_{1}−w_{2}) (1 +s)^{−2}
(

δ^{p}(p+1) +α^{p}(p+1)

+ 1

p+1

p−1X

j=1

Ãµ j p+1

¶

j(p+1−j)α^{j}δ^{p−j} +
µ j

p+1

¶

(p+1−j)^{2}α^{j}δ^{p−j}

!)

≤ c(α+δ)^{p}A_{p}(w_{1}−w_{2}) (1 +s)^{−}^{2} .

Thanks to (3.24)–(3.27), we deduce the result of lemma 3.5.

Proof of Lemma 3.6: Since for f ∈H^{1} we have
kfkL^{∞} ≤√

2kfk

1 2

L^{2} · kfxk

1 2

L^{2}

we easily deduce that (i) for p= 1,

(3.28) kw(t)kL^{∞} ≤ √

2kw(t)k_{L}^{1}^{2}^{2}kw_{x}(t)k_{L}^{1}^{2}^{2} ≤ √

2δ(1 +t)^{−}^{1}^{2}^{+η} ,
kwx(t)k^{L}^{∞} ≤ √

2kwx(t)k

1 2

L^{2}kwxx(t)k

1 2

L^{2} ≤ √

2δ(1 +t)^{−}^{7}^{8}^{+η} ;
(ii) for p≥2:

(3.29) kw(t)kL^{∞} ≤ √

2kw(t)k

1 2

L^{2}kw_{x}(t)k

1 2

L^{2} ≤ √

2δ(1 +t)^{−}^{1}^{2} ,
kwx(t)k^{L}^{∞} ≤ √

2kwx(t)k

1 2

L^{2}kwxx(t)k

1 2

L^{2} ≤ √

2δ(1 +t)^{−1} .
Thanks to (3.17) we have

sup

ξ∈R|H\_{p}(w)(ξ, s)| ≤
Z ∞

−∞|H_{p}(w)(x, s)|dx

≤ Z ∞

−∞

(

|vxxxt|+ 1 p+1

Xp

j=0

µ j p+1

¶

|v^{j}w_{x}^{p+1−j}|
)

dx

≤ kv_{xxxt}(s)kL^{1} + 1
p+1

µ p p+1

¶

kv(s)k^{p−1}L^{∞} kv(s)kL^{2}kw_{x}(s)kL^{2}

+ 1

p+1

pX−1

j=1

µ j p+1

¶

kv(s)k^{j}_{L}^{∞}kw_{x}(s)k^{p}_{L}^{−}^{∞}^{1}^{−}^{j}kw_{x}(s)k^{2}L^{2} .
Now, because of (3.28)–(3.29) we have

sup

ξ∈R|H\_{1}(w)(ξ, s)| ≤ c

½

α(1+s)^{−}^{5}^{2} +δ^{2}(1+s)^{−(3/2−2η)}+α δ(1+s)^{−(1−η)}

¾

≤ c^{n}α+ (α+δ)^{2}^{o}(1+s)^{−(1−η)} ,
(3.30)

by choosingη such that 0< η≤1/2, and

(3.31)

sup

ξ∈R|H\_{p}(w)(ξ, s)| ≤ c
(

α(1+s)^{−}^{5}^{2} +
µ p

p+1

¶

α^{p}δ(1+s)^{−}^{p+1}^{2}

+

p−1X

j=0

µ j p+1

¶

α^{j}δ^{p+1−j}(1+s)^{−}^{2p−j+1}^{2}
)

≤ c^{n}α+ (α+δ)^{p+1}^{o}(1+s)^{−}^{3}^{2} , p≥2,