Electronic Journal of Differential Equations, Vol. 2012 (2012), No. 96, pp. 1–11.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
EXISTENCE AND ASYMPTOTIC BEHAVIOR OF SOLUTIONS
TO THE GENERALIZED DAMPED BOUSSINESQ EQUATION
YINXIA WANG
Abstract. We consider the Cauchy problem for the n-dimensional generalized damped Boussinesq equation. Based on decay estimates of solutions
to the corresponding linear equation, we define a solution space with time
weighted norms. Under small condition on the initial value, the existence and
asymptotic behavior of global solutions in the corresponding Sobolev spaces
are established by the contraction mapping principle.
1. Introduction
We study the Cauchy problem of the generalized damped Boussinesq equation
in n space dimensions
utt − a∆utt − 2b∆ut − α∆3 u + β∆2 u − ∆u = ∆f (u)
(1.1)
with the initial value
t=0:
u = u0 (x),
ut = u1 (x).
(1.2)
Here u = u(x, t) is the unknown function of x = (x1 , · · · , xn ) ∈ Rn and t > 0,
a, b, α, β are positive constants. The nonlinear term f (u) = O(u1+θ ) and θ is a
positive integer.
The first initial boundary value problem for
utt − a∆utt − 2b∆ut − α∆3 u + β∆2 u − ∆u = γ∆(u2 )
(1.3)
in a unit circle was investigated in [16], where a, b, α, β are positive constants and γ
is a constant. The existence and the uniqueness of strong solution was established
and the solutions were constructed in the form of series in the small parameter
present in the initial conditions. The long-time asymptotics was also obtained in
the explicit form. In [1], the authors considered the initial-boundary value problem
for (1.3) in the unit ball B ⊂ R3 , similar results were established. It is well-known
that the equation (1.3) is closely contacted with many wave equations. For example,
the equation (which we call the Bq equation)
utt − uxx + uxxxx = (u2 )xx ,
2000 Mathematics Subject Classification. 35L30, 35L75.
Key words and phrases. Generalized damped equation; global solution; asymptotic behavior.
c
2012
Texas State University - San Marcos.
Submitted May 31, 2012. Published June 10, 2012.
1
2
Y. WANG
EJDE-2012/96
which was derived by Boussinesq in 1872 to describe shallow water waves. The
improved Bq equation(which we call IBq equation) is
utt − uxx − uxxtt = (u2 )xx .
A modification of the IBq equation analogous of the MKdV equation yields
utt − uxx − uxxtt = (u3 )xx ,
which we call the IMBq equation (see [5]). (1.1) is a higher order wave equation. In
[8], we considered the Cauchy problem for the Cahn-Hilliard equation with inertial
term. Combining high frequency, low frequency technique and energy methods,
we obtained global existence and asymptotic behavior of solutions. Wang, Liu
and Zhang [13] investigated a fourth wave equation that is of the regularity-loss
type. Based on the decay property of the solution operators, global existence and
asymptotic behavior of solutions are obtained. For global existence and asymptotic
behavior of solutions to higher order wave equations, we refer to [2]-[3] and [6]-[15]
and references therein.
The main purpose of this paper is to establish global existence and asymptotic
behavior of solutions to (1.1), (1.2) by using the contraction mapping principle.
Firstly, we consider the decay property of the following linear equation
utt − a∆utt − 2b∆ut − α∆3 u + β∆2 u − ∆u = 0.
(1.4)
We obtain the following decay estimate of solutions to (1.4) associated with initial
condition (1.2),
n
k
1
k∂xk u(t)kL2 ≤ C(1 + t)− 4 − 2 − 2 (ku0 kḢ −1 + ku1 kḢ −2 + ku0 kH s+2 + ku1 kH s ) (1.5)
1
1
(k ≤ s + 2),
n
h
k∂xh ut (t)kL2 ≤ C(1 + t)− 4 − 2 −1 (ku0 kḢ −1 + ku1 kḢ −2 + ku0 kH s+2 + ku1 kH s ) (1.6)
1
1
(h ≤ s) Based on the estimates (1.5) and (1.6), we define a solution space with
time weighted norms. Then global existence and asymptotic behavior of classical
solutions to (1.1), (1.2) are obtained by using the contraction mapping principle.
We give notation which is used in this paper. Let F [u] denote the Fourier
transform of u defined by
Z
û(ξ) = F [u] =
e−iξ·x u(x)dx,
Rn
and we denote its inverse transform by F −1 .
For 1 ≤ p ≤ ∞, Lp = Lp (Rn ) denotes the usual Lebesgue space with the
norm k · kLp . The usual Sobolev space of s is defined by Hps = (I − ∆)−s/2 Lp
with the norm kf kHps = k(I − ∆)s/2 f kLp ; the homogeneous Sobolev space of s is
defined by Ḣps = (−∆)−s/2 Lp with the norm kf kHps = k(−∆)s/2 f kLp ; especially
H s = H2s , Ḣ s = Ḣ2s . Moreover, we know that Hps = Lp ∩ Ḣps for s ≥ 0.
Finally, in this paper, we denote every positive constant by the same symbol C
or c without confusion. [·] is the Gauss symbol.
The article is organized as follows. In Section 2 we derive the solution formula
of our semi-linear problem. We study the decay property of the solution operators
appearing in the solution formula in section 3. Then, in Section 4, we discuss the
linear problem and show the decay estimates. Finally, we prove global existence
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EXISTENCE AND ASYMPTOTIC BEHAVIOR
3
and asymptotic behavior of solutions for the Cauchy problem (1.1), (1.2) in Section
5.
2. Solution formula
The aim of this section is to derive the solution formula for problem (1.1), (1.2).
We first investigate the equation (1.4). Taking the Fourier transform, we have
(1 + a|ξ|2 )ûtt + 2b|ξ|2 ût + (α|ξ|6 + β|ξ|4 + |ξ|2 )û = 0.
(2.1)
The corresponding initial value are
t=0:
û = û0 (ξ),
ût = û1 (ξ).
(2.2)
The characteristic equation of (2.1) is
(1 + a|ξ|2 )λ2 + 2b|ξ|2 λ + α|ξ|6 + β|ξ|4 + |ξ|2 = 0.
Let λ = λ± (ξ) be the corresponding eigenvalues of (2.3), we obtain
p
−b|ξ|2 ± |ξ| −1 − (a + β − b2 )|ξ|2 − (α + aβ)|ξ|4 − aα|ξ|6
λ± (ξ) =
.
1 + a|ξ|2
(2.3)
(2.4)
The solution to the problem (2.1)-(2.2) is given in the form
û(ξ, t) = Ĝ(ξ, t)û1 (ξ) + Ĥ(ξ, t)û0 (ξ),
(2.5)
where
Ĝ(ξ, t) =
1
(eλ+ (ξ)t − eλ− (ξ)t )
λ+ (ξ) − λ− (ξ)
(2.6)
and
1
(λ+ (ξ)eλ− (ξ)t − λ− (ξ)eλ+ (ξ)t ).
λ+ (ξ) − λ− (ξ)
We define G(x, t) and H(x, t) by
Ĥ(ξ, t) =
G(x, t) = F −1 [Ĝ(ξ, t)](x),
(2.7)
H(x, t) = F −1 [Ĥ(ξ, t)](x),
respectively, where F −1 denotes the inverse Fourier transform. Then, applying
F −1 to (2.5), we obtain
u(t) = G(t) ∗ u1 + H(t) ∗ u0 .
By the Duhamel principle, we obtain the solution formula to (1.1), (1.2),
Z t
u(t) = G(t) ∗ u1 + H(t) ∗ u0 +
G(t − τ ) ∗ (I − a∆)−1 ∆f (u)(τ )dτ.
(2.8)
(2.9)
0
3. Decay Property
The aim of this section is to establish decay estimates of the solution operators
G(t) and H(t) appearing in the solution formula (2.8).
Lemma 3.1. The solution of problem (2.1), (2.2) satisfies
|ξ|2 (1+|ξ|2 )|û(ξ, t)|2 +|ût (ξ, t)|2 ≤ Ce−cω(ξ)t (|ξ|2 (1+|ξ|2 )|û0 (ξ)|2 +|û1 (ξ)|2 ), (3.1)
for ξ ∈ Rn and t ≥ 0, where ω(ξ) =
|ξ|2
1+|ξ|2 .
4
Y. WANG
EJDE-2012/96
¯t and taking the real part yields
Proof. Multiplying (2.1) by û
1 d
{(1 + a|ξ|2 )|ût |2 + (α|ξ|6 + β|ξ|4 + |ξ|2 )|û|2 } + 2b|ξ|2 |ût |2 = 0.
2 dt
¯ and taking the real part, we obtain
Multiplying (2.1) by û
(3.2)
1 d
¯ + (α|ξ|6 + β|ξ|4 + |ξ|2 )|û|2 − (1 + a|ξ|2 )|ût |2 = 0.
{b|ξ|2 |û|2 + 2(1 + a|ξ|2 )Re(ût û)}
2 dt
(3.3)
Multiplying both sides of (3.2) and (3.3) by (1 + a|ξ|2 ) and b|ξ|2 respectively, summing up the products yields
d
E + F = 0,
(3.4)
dt
where
1
E = (1 + a|ξ|2 )2 |ût |2 + (1 + a|ξ|2 )(α|ξ|6 + β|ξ|4 + |ξ|2 )|û|2 + b2 |ξ|4 |û|2
2
¯
+ b|ξ|2 (1 + a|ξ|2 ) Re(ût û)
and
F = b|ξ|2 (α|ξ|6 + β|ξ|4 + |ξ|2 )|û|2 + b|ξ|2 (1 + a|ξ|2 )|ût |2 .
A simple computation implies that
C(1 + |ξ|2 )2 E0 ≤ E ≤ C(1 + |ξ|2 )2 E0 ,
(3.5)
where
E0 = |ξ|2 (1 + |ξ|2 )|û|2 + |ût |2 .
Note that F ≥ c|ξ|2 E0 . It follows from (3.5) that
F ≥ cω(ξ)E,
(3.6)
where
ω(ξ) =
|ξ|2
.
1 + |ξ|2
Using (3.4) and (3.6), we obtain
d
E + cω(ξ)E ≤ 0.
dt
Thus E(ξ, t) ≤ e−cw(ξ)t E(ξ, 0), which together with (3.5) proves the desired estimates (3.1). Then the proof is complete.
Lemma 3.2. Let Ĝ(ξ, t) and Ĥ(ξ, t) be the fundamental solution of (1.4) in the
Fourier space, which are given in (2.6) and (2.7), respectively. Then we have the
estimates
|ξ|2 (1 + |ξ|2 )|Ĝ(ξ, t)|2 + |Ĝt (ξ, t)|2 ≤ Ce−cω(ξ)t
(3.7)
|ξ|2 (1 + |ξ|2 )|Ĥ(ξ, t)|2 + |Ĥt (ξ, t)|2 ≤ C|ξ|2 (1 + |ξ|2 )e−cω(ξ)t
(3.8)
and
for ξ ∈ Rn and t ≥ 0, where ω(ξ) =
|ξ|2
1+|ξ|2 .
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EXISTENCE AND ASYMPTOTIC BEHAVIOR
5
Proof. If û0 (ξ) = 0, from (2.5), we obtain
û(ξ, t) = Ĝ(ξ, t)û1 (ξ),
ût (ξ, t) = Ĝt (ξ, t)û1 (ξ).
Substituting the equalities into (3.1) with û0 (ξ) = 0, we obtain (3.7). In what
follows, we consider û1 (ξ) = 0, it follows from (2.5) that
û(ξ, t) = Ĥ(ξ, t)û0 (ξ), ût (ξ, t) = Ĥt (ξ, t)û0 (ξ).
Substituting the equalities into (3.1) with û1 (ξ) = 0, we obtain the desired estimate
(3.8). The Lemma is proved.
Lemma 3.3. Let k ≥ 0 and 1 ≤ p ≤ 2. Then we have
n
1
1
k
l
1
k∂xk G(t) ∗ φkL2 ≤ C(1 + t)−( 2 ( p − 2 )+ 2 + 2 − 2 ) kφkḢp−l + Ce−ct k∂x(k−2)+ φkL2 , (3.9)
n
1
1
k
l
k∂xk H(t) ∗ φkL2 ≤ C(1 + t)−( 2 ( p − 2 )+ 2 + 2 ) kφkḢp−l + Ce−ct k∂xk φkL2
k∂xk Gt (t)
1
1
k
l
−( n
2 ( p − 2 )+ 2 + 2 )
∗ φkL2 ≤ C(1 + t)
n
1
1
k
l
−ct
kφkḢp−l + Ce
k∂xk φkL2 ,
1
k∂xk Ht (t) ∗ φkL2 ≤ C(1 + t)−( 2 ( p − 2 )+ 2 + 2 + 2 ) kφkḢp−l + Ce−ct k∂xk+2 φkL2
n
k
(3.10)
(3.11)
(3.12)
1
k∂xk G(t) ∗ (I − a∆)−1 ∆gkL2 ≤ C(1 + t)−( 4 + 2 + 2 ) kgkL1 + Ce−ct k∂xk gkL2 , (3.13)
n
k
k∂xk Gt (t) ∗ (I − a∆)−1 ∆gkL2 ≤ C(1 + t)−( 4 + 2 +1) kgkL1 + Ce−ct k∂xk gkL2 , (3.14)
where (k − 2)+ = max{0, k − 2}.
Proof. Firstly, we prove (3.9). By the Plancherel theorem and (3.7), we obtain
k∂xk G(t) ∗ φk2L2
Z
Z
=
|ξ|2k |Ĝ(ξ, t)|2 |φ̂(ξ)|2 dξ +
|ξ|2k |Ĝ(ξ, t)|2 |φ̂(ξ)|2 dξ
|ξ|≤R0
|ξ|≥R0
Z
2k−2 −c|ξ|2 t
2
≤C
|ξ|
e
|φ̂(ξ)| dξ
|ξ|≤R0
Z
−ct
+ Ce
|ξ|2k (|ξ|2 (1 + |ξ|2 ))−1 |φ̂(ξ)|2 dξ
(3.15)
|ξ|≥R0
−l
≤ Ck|ξ|
φ̂(ξ)k2Lp0
−ct
+ Ce
Z
2
|ξ|(2k−2+2l)q e−cq|ξ| t dξ
1/q
|ξ|≤R0
k∂x(k−2)+ φk2L2 ,
where R0 is a small positive constant and
Hausdorff-Young inequality that
1
p
+
1
p0
= 1,
2
p0
+
1
q
= 1. It follows from
l
k |ξ|−l φ̂(ξ)kLp0 ≤ Ck(−∆)− 2 φkLp .
(3.16)
By a straight computation, we obtain
Z
1/q
2
n
|ξ|(2k−2+2l)q e−cq|ξ| t dξ
≤ C(1 + t)−( 2q +k−1+l)
(3.17)
|ξ|≤R0
1
1
≤ C(1 + t)−(n( p − 2 )+k−1+l) .
Combining (3.15), (3.16) and (3.17) yields (3.9).
Similarly, using (3.7) and (3.8), respectively, we can prove (3.10)-(3.12).
6
Y. WANG
EJDE-2012/96
In what follows, we prove (3.13). By the Plancherel theorem, (3.7), and
Hausdorff-Young inequality, we have
k∂xk G(t) ∗ (I − a∆)−1 ∆gk2L2
Z
=
|ξ|2k |Ĝ(ξ, t)|2 |ξ|4 (1 + a|ξ|2 )−2 |ĝ(ξ)|2 dξ
|ξ|≤R0
Z
+
|ξ|2k |Ĝ(ξ, t)|2 |ξ|4 (1 + |ξ|2 )−2 |ĝ(ξ)|2 dξ
|ξ|≥R0
Z
Z
2
≤C
|ξ|2k+2 e−c|ξ| t |ĝ(ξ)|2 dξ + Ce−ct
|ξ|2k |ĝ(ξ)|2 dξ
|ξ|≤R0
|ξ|≥R0
Z
2
≤ Ckĝ(ξ)k2L∞
|ξ|2k+2 e−c|ξ| t dξ + Ce−ct k∂xk gk2L2
|ξ|≤R0
−( n
2 +k+1)
≤ C(1 + t)
kgk2L1 + Ce−ct k∂xk gk2L2 .
where R0 is a small positive constant. Thus (3.13) follows. Similarly, we can prove
(3.14). Thus we have completed the proof of lemma.
4. Decay estimate for solutions to the linear equation
Theorem 4.1. Assume that u0 ∈ H s+2 (Rn ) ∩ Ḣ1−1 (Rn ), u1 ∈ H s (Rn ) ∩ Ḣ1−2 (Rn )
(s ≥ [ n2 ] + 5). Then the classical solution u(x, t) to (1.4) associated with initial
condition (1.2), which is given by the formula (2.8), satisfies the decay estimates
n
k
1
k∂xk u(t)kL2 ≤ C(1 + t)− 4 − 2 − 2 (ku0 kḢ −1 + ku1 kḢ −2 + ku0 kH s+2 + ku1 kH s ) (4.1)
1
1
for k ≤ s + 2,
n
h
k∂xh ut (t)kL2 ≤ C(1 + t)− 4 − 2 −1 (ku0 kḢ −1 + ku1 kḢ −2 + ku0 kH s+2 + ku1 kH s ) (4.2)
1
1
for h ≤ s,
n
m
1
k∂xm u(t)kL∞ ≤ C(1 + t)− 2 − 2 − 2 (ku0 kḢ −1 + ku1 kḢ −2 + ku0 kH s+2 + ku1 kH s ) (4.3)
1
for m ≤ s + 1 −
1
[ n2 ].
Proof. Firstly, we prove (4.1). Using (3.9) and (3.10), we obtain
k∂xk u(t)kL2
≤ k∂xk G(t) ∗ u1 kL2 + Ck∂xh H(t) ∗ u0 kL2
n
k
1
≤ C(1 + t)− 4 − 2 − 2 (ku0 kḢ −1 + ku1 kḢ −2 ) + Ce−ct (ku0 kH s+2 + ku1 kH s )
1
k
1
−n
4 −2−2
≤ C(1 + t)
1
(ku0 kḢ −1 + ku1 kḢ −2 + ku0 kH s+2 + ku1 kH s ).
1
1
Similar to the proof of (4.1), using (3.11) and (3.12), we can prove (4.2). In what
follows, we prove (4.3). Using (4.1) and Gagliardo-Nirenberg inequality, it is not
difficult to get (4.3). The Lemma is proved.
5. Existence of global solution and asymptotic behavior
The purpose of this section is to prove the existence and asymptotic behavior of
global solutions to the Cauchy problem (1.1), (1.2). We need the following Lemma,
which come from [4] (see also [17]).
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EXISTENCE AND ASYMPTOTIC BEHAVIOR
7
Lemma 5.1. Let s and θ be positive integers, δ > 0, p, q, r ∈ [1, ∞] satisfy 1r =
1
1
s
p + r , and let k ∈ {0, 1, 2, · · · , s}. Assume that F (v) is a class of C and satisfies
|∂vl F (v)| ≤ Cl,δ |v|θ+1−l ,
|v| ≤ δ,
0 ≤ l ≤ s, l < θ + 1
and
|∂vl F (v)| ≤ Cl,δ , |v| ≤ δ, l ≤ s, θ + 1 ≤ l.
If v ∈ Lp ∩ W k,q ∩ L∞ and kvkL∞ ≤ δ, then
kF (v)kW k,r ≤ Ck,δ kvkW k,q kvkLp kvkθ−1
L∞ ,
k∂xα F (v)kLr ≤ Ck,δ k∂xα vkLq kvkLp kvkθ−1
L∞ ,
|α| ≤ k.
Lemma 5.2. Let s and θ be positive integers, δ > 0, p, q, r ∈ [1, ∞] satisfy 1r = p1 +
1
r , and let k ∈ {0, 1, 2, · · · , s}. Let F (v) be a function that satisfies the assumptions
of Lemma 5.1. Moreover, assume that
|∂vs F (v1 ) − ∂vs F (v2 )| ≤ Cδ (|v1 | + |v2 |)max{θ−s,θ} |v1 − v2 |,
p
If v1 , v2 ∈ L ∩ W
k,q
∞
∩L
|v1 | ≤ δ,
|v2 | ≤ δ.
and kv1 kL∞ ≤ δ, kv2 kL∞ ≤ δ, then for |α| ≤ k, we have
k∂xα (F (v1 ) − F (v2 ))kLr
≤ Ck,δ {(k∂xα v1 kLq + k∂xα v2 kLq )kv1 − v2 kLp
+ (kv1 kLp + kv2 kLp )k∂xα (v1 − v2 )kLq }(kv1 kL∞ + kv2 kL∞ )θ−1 .
Based on the estimates (4.1)-(4.3) of solutions to (1.4) associated with initial
condition (1.2), we define the following solution space
X = {u ∈ C([0, ∞); H s+2 (Rn )) ∩ C 1 ([0, ∞); H s (Rn )) : kukX < ∞},
where
kukX = sup
t≥0
n
X
k
1
(1 + t) 4 + 2 + 2 k∂xk u(t)kL2 +
X
n
h
(1 + t) 4 + 2 +1 k∂xh ut (t)kL2 },
h≤s
k≤s+2
For R > 0, we define XR = {u ∈ X : kukX ≤ R}. For m ≤ s + 1 − [ n2 ], using
Gagliardo-Nirenberg inequality, we obtain
m
n
1
k∂xm u(t)kL∞ ≤ C(1 + t)−( 2 + 2 + 2 ) kukX .
(5.1)
Theorem 5.3. Assume that u0 ∈ H s+2 (Rn ) ∩ Ḣ1−1 (Rn ), u1 ∈ H s (Rn ) ∩ Ḣ1−2 (Rn )
(s ≥ [ n2 ] + 5) and integer θ ≥ 2. Let f (u) be a function of class C s+2 and satisfy
Lemmas 5.1 and 5.2. Put
E0 = ku0 kḢ −1 + ku1 kḢ −2 + ku0 kH s+2 + ku1 kH s .
1
1
If E0 is suitably small, the Cauchy problem (1.1)-(1.2) has a unique global classical
solution u(x, t) satisfying u ∈ C([0, ∞); H s+2 (Rn )), ut ∈ C([0, ∞); H s (Rn )), utt ∈
L∞ ([0, ∞); H s−2 (Rn )). Moreover, the solution satisfies the decay estimate
n
k
1
k∂xk u(t)kL2 ≤ CE0 (1 + t)− 4 − 2 − 2 ,
h
−n
4 − 2 −1
k∂xh ut (t)kL2 ≤ CE0 (1 + t)
for k ≤ s + 2 and h ≤ s.
(5.2)
(5.3)
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Y. WANG
EJDE-2012/96
Proof. Define the mapping
t
Z
G(t − τ ) ∗ (I − a∆)−1 ∆f (u(τ ))dτ.
Ψ(u) = G(t) ∗ u1 + H(t) ∗ u0 +
(5.4)
0
Using (3.9)-(3.10), (3.13), Lemma 5.1 and (5.1), for k ≤ s + 2 we obtain
k∂xk Ψ(u)kL2
≤ Ck∂xk G(t) ∗ u1 kL2 + Ck∂xk H(t) ∗ u0 kL2
Z t
+C
k∂xk G(t − τ ) ∗ (I − a∆)−1 ∆f (u(τ ))kL2 dτ
0
k
n
1
≤ C(1 + t)− 4 − 2 − 2 (ku0 kḢ −1 + ku1 kḢ −2 ) + Ce−ct (ku0 kH s+2 +ku1 kH s )
1
1
Z t/2
n
k
1
+C
(1 + t − τ )− 4 − 2 − 2 kf (u)kL1 dτ
0
Z t
n
1
+C
(1 + t − τ )− 4 − 2 k∂xk f (u)kL1 dτ
t/2
t
Z
+C
e−c(t−τ ) k∂xk f (u)kL2 dτ
0
k
n
1
≤ C(1 + t)− 4 − 2 − 2 (ku0 kḢ −1 + ku1 kḢ −2 ) + Ce−ct (ku0 kH s+2 +ku1 kH s )
1
1
Z t/2
n
k
1
+C
(1 + t − τ )− 4 − 2 − 2 kuk2L2 kukθ−1
L∞ dτ
0
Z t
Z t
θ−1
− 21
−n
k
2
4
+C
(1 + t − τ )
k∂x ukL2 kukL∞ dτ +C
e−c(t−τ ) k∂xk ukL2 kukθL∞ dτ
t/2
0
k
1
−n
4 −2−2
−ct
≤ C(1 + t)
(ku0 kḢ −1 + ku1 kḢ −2 ) + Ce (ku0 kH s+2 +ku1 kH s )
1
1
Z t/2
k
1
n
1
n
n
+CRθ+1
(1 + t − τ )− 4 − 2 − 2 (1 + τ )−( 2 +1) (1 + τ )−( 2 + 2 )(θ−1) dτ
0
Z t
n
1
n
1
n
+CRθ+1
(1 + t − τ )− 4 − 2 (1 + τ )− 2 −k−1 (1 + τ )−( 2 + 2 )(θ−1) dτ
t/2
t
+CRθ+1
Z
0
k
1
−n
4 −2−2
≤ C(1 + t)
n
k
1
n
1
e−c(t−τ ) (1 + τ )− 4 − 2 − 2 (1 + τ )−( 2 + 2 )θ dτ
{(ku0 kḢ −1 + ku1 kḢ −2 + ku0 kH s+2 + ku1 kH s ) + Rθ+1 }.
1
1
Thus
n
k
1
(1 + t) 4 + 2 + 2 k∂xk Ψ(u)kL2 ≤ CE0 + CRθ+1 .
It follows from (5.4) that
Z t
Ψ(u)t = Gt (t) ∗ u1 + Ht (t) ∗ u0 +
Gt (t − τ ) ∗ (I − a∆)−1 ∆f (u(τ ))dτ.
0
Using (3.11)-(3.12), (3.14) Lemma 5.1 and (5.1), for h ≤ s we have
k∂xh Ψ(u)t kL2
≤ Ck∂xh Gt (t) ∗ u1 kL2 + Ck∂xh Ht (t) ∗ u0 kL2
Z t
+C
k∂xh Gt (t − τ ) ∗ (I − a∆)−1 ∆f (u(τ ))kL2 dτ
0
(5.5)
(5.6)
EJDE-2012/96
EXISTENCE AND ASYMPTOTIC BEHAVIOR
n
9
h
≤ C(1 + t)− 4 − 2 −1 (ku0 kḢ −1 + ku1 kḢ −2 ) + Ce−ct (ku0 kH s+2 + ku1 kH s )
1
1
Z t/2
h
n
+C
(1 + t − τ )− 4 − 2 −1 kf (u)kL1 dτ
0
Z t
Z t
−1
h
−n
4
+C
k∂x f (u)kL1 dτ + C
(1 + t − τ )
e−c(t−τ ) k∂xh f (u)kL2 dτ
t/2
0
h
−n
4 − 2 −1
(ku0 kḢ −1 + ku1 kḢ −2 ) + Ce−ct (ku0 kH s+2 + ku1 kH s )
≤ C(1 + t)
1
1
Z t/2
h
n
+C
(1 + t − τ )− 4 − 2 −1 kuk2L2 kukθ−1
L∞ dτ
0
Z t
Z t
n
+C
dτ
+
C
(1 + t − τ )− 4 −1 k∂xh uk2L2 kukθ−1
e−c(t−τ ) k∂xh ukL2 kukθL∞ dτ
L∞
t/2
0
h
−n
4 − 2 −1
−ct
(ku0 kḢ −1 + ku1 kḢ −2 ) + Ce (ku0 kH s+2 + ku1 kH s )
≤ C(1 + t)
1
1
Z t/2
n
h
n
n
1
+CRθ+1
(1 + t − τ )− 4 − 2 −1 (1 + τ )−( 2 +1) (1 + τ )−( 2 + 2 )(θ−1) dτ
0
Z t
n
n
1
n
+CRθ+1
(1 + t − τ )− 4 −1 (1 + τ )− 2 −h−1 (1 + τ )−( 2 + 2 )(θ−1) dτ
t/2
t
+CRθ+1
Z
n
h
n
1
e−c(t−τ ) (1 + τ )− 4 − 2 −1 (1 + τ )−( 2 + 2 )θ dτ
0
h
−n
4 − 2 −1
≤ C(1 + t)
{(ku0 kḢ −1 + ku1 kḢ −2 + ku0 kH s+2 + ku1 kH s ) + Rθ+1 }.
1
1
Thus
n
h
(1 + t) 4 + 2 +1 k∂xh Ψ(u)t kL2 ≤ CE0 + CRθ+1 .
Combining (5.5), (5.7) and taking E0 and R suitably small yields
kΨ(u)kX ≤ R.
For ũ, ū ∈ XR , by using (5.4), we have
Z t
Ψ(ũ) − Ψ(ū) =
G(t − τ ) ∗ (I − a∆)−1 ∆[f (ũ) − f (ū)]dτ.
0
Using (5.9), (3.13) and Lemma 5.2, (5.1), for k ≤ s + 2 we obtain
k∂xk Ψ(ũ) − Ψ(ū))kL2
Z t
≤
k∂xk G(t − τ ) ∗ (I − a∆)−1 ∆[f (ũ) − f (ū)]kL2 dτ
0
Z
t/2
n
k
n
1
1
(1 + t − τ )− 4 − 2 − 2 k(f (ũ) − f (ū))kL1 dτ
≤C
0
Z
t
(1 + t − τ )− 4 − 2 k∂xk (f (ũ) − f (ū))kL1 dτ
+C
t/2
t
Z
+C
Z
≤C
e−c(t−τ ) k∂xk (f (ũ) − f (ū))kL2 dτ
0
t/2
n
k
1
(1 + t − τ )− 4 − 2 − 2 (kũkL2 + kūkL2 )kũ − ūkL2
0
×(kũkL∞ + kūkL∞ )θ−1 dτ
(5.7)
(5.8)
(5.9)
10
Y. WANG
Z
t
+C
EJDE-2012/96
1
n
(1 + t − τ )− 4 − 2 {(k∂xk ũkL2 + k∂xk ũkL2 )kũ − ūkL2
t/2
+(kũkL2 + kūkL2 )k∂xk (ũ − ū)kL2 }(kũkL∞ + kūkL∞ )θ−1 dτ
Z t
e−c(t−τ ) {(k∂xk ũkL2 + k∂xk ũkL2 )kũ − ūkL∞
+C
0
+(kũkL∞ + kūkL∞ )k∂xk (ũ − ū)kL2 }(kũkL∞ + kūkL∞ )θ−1 dτ
Z t/2
n
k
1
1
n
θ
≤ CR kũ − ūkX
(1 + t − τ )− 4 − 2 − 2 (1 + τ )−( 2 + 2 )θ dτ
0
Z t
k+1
n
θ
1
+CRθ kũ − ūkX
(1 + t − τ )− 4 − 2 (1 + τ )−( 2 (n+1)+ 2 ) dτ
t/2
t
Z
+CCRθ kũ − ūkX
n
n
k
1
e−c(t−τ ) (1 + τ )−( 4 + 2 θ+ 2 + 2 ) dτ
0
k
n
1
≤ CRθ (1 + t)− 4 − 2 − 2 kũ − ūkX ,
which implies
n
k
1
(1 + t) 4 + 2 + 2 k∂xk (Ψ(ũ) − Ψ(ū))kL2 ≤ CRθ kũ − ūkX .
(5.10)
Similarly for h ≤ s, from (5.6), (3.14) and (5.1), we have
Z t
k∂xh (Ψ(ũ) − Ψ(ū))t kL2 ≤
k∂xh Gt (t − τ ) ∗ (I − a∆)−1 ∆[f (ũ) − f (ū)]kL2 dτ
0
Z
t/2
n
h
(1 + t − τ )− 4 − 2 −1 k(f (ũ) − f (ū))kL1 dτ
≤C
0
Z
t
n
(1 + t − τ )− 4 −1 k∂xh (f (ũ) − f (ū))kL1 dτ
+C
t/2
t
Z
+C
e−c(t−τ ) k∂xh (f (ũ) − f (ū))kL2 dτ
0
n
h
≤ CRθ (1 + t)− 4 − 2 −1 kũ − ūkX ,
which implies
n
h
(1 + t) 4 + 2 +1 k∂xh (Ψ(ũ) − Ψ(ū))t kL2 ≤ CRθ kũ − ūkX .
(5.11)
Using (5.10), (5.11) and taking R suitably small yields
kΨ(ũ) − Ψ(ū)kX ≤
1
kũ − ūkX .
2
(5.12)
From (5.8) and (5.12), we know that Ψ is strictly contracting mapping. Consequently, we conclude that there exists a fixed point u ∈ XR of the mapping Ψ,
which is a classical solution to (1.1), (1.2). This completes the proof.
Acknowledgements. The author would like to thank the anonymous referee for
his/her comments and suggestions. This work was supported in part by grants
11101144 from the NNSF of China, and 201031 from the Research Initiation Project
for High-level Talents of North China University of Water Resources and Electric
Power.
EJDE-2012/96
EXISTENCE AND ASYMPTOTIC BEHAVIOR
11
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Yinxia Wang
School of Mathematics and Information Sciences, North China University of Water
Resources and Electric Power, Zhengzhou 450011, China
E-mail address: yinxia117@126.com