Local existence of solutions to dissipative nonlinear evolution equations with mixed types

Nonlinear Analysis 71 (2009) 5897–5905

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Local existence of solutions to dissipative nonlinear evolution equations with mixed types Hu Wei, Mina Jiang ∗ Laboratory of Nonlinear Analysis, Department of Mathematics, Huazhong Normal University, Wuhan 430079, PR China

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Article history: Received 5 June 2007 Accepted 11 March 2009

abstract In this paper, we consider the local existence of solutions to the Cauchy problems for the following nonlinear evolution equations with mixed types

 MSC: 35B40 35F25 35K45

ψt = −(1 − α)ψ − θx + αψxx , θt = −(1 − α)θ + γ ψx + 2ψθx + αθxx ,

with initial data

(ψ, θ )(x, 0) = (ψ0 (x), θ0 (x)) → (ψ± , θ± ),

Keywords: Evolution equation Local existence Contraction mapping principle

as x → ±∞,

where α and γ are positive constants satisfying α < 1, γ < α(1 −α). Through constructing an approximation solution sequence, we obtain the local existence by using the contraction mapping principle. © 2009 Elsevier Ltd. All rights reserved.

1. Introduction In physical and mechanical fields, many phenomena can be modeled by systems of nonlinear interaction between ellipticity and dissipation. Due to their complexity, these systems are far from well understood. A set of simplified equations was thus proposed by Hsieh in [1]:



ψt = −(σ − α)ψ − σ θx + αψxx , θt = −(1 − β)θ + γ ψx + 2ψθx + βθxx ,

(1.1)

where α , β , σ and γ are positive constants satisfying α < σ , β < 1. The complexity of system (1.1) can be explained by a rough argument. If we ignore the damping and diffusion terms temporarily, the system (1.1) becomes:



ψt = −σ θx , θt = γ ψx + 2ψθx .

(1.2)

√

We can find that the system (1.2) is elliptic for |ψ| < σ γ and hyperbolic, otherwise. Around the zero equilibrium, the system (1.2) subject to initial small disturbances is unstable due to ellipticity, and the inherent instability will cause growth √ of |ψ| if it overcomes the effect of damping and diffusion terms. But when |ψ| > σ γ , the system is hyperbolic and ψ ceases to grow. Moreover, the dissipative terms would tend to draw the system back to the elliptic regime; then a ‘‘switching back and forth’’ phenomenon is expected due to the interplay between ellipticity, hyperbolicity and dissipation for suitable coefficients, which makes the system (1.2) quite complicated. There are only a few rigorous results available so far regarding

∗

Corresponding author. E-mail address: [email protected] (M. Jiang).

0362-546X/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2009.03.011

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H. Wei, M. Jiang / Nonlinear Analysis 71 (2009) 5897–5905

this phenomenon, cf. [2–5]. Jian and Chen in [3] first obtained the existence result for the following modified form of the system (1.1)



ψt = −(σ − α)ψ − σ θx + αψxx , θt = −(1 − β)θ + γ ψx + (ψθ )x + βθxx .

(1.3)

Tang and Zhao in [4] discussed the Cauchy problem of the system (1.1) with α = β and σ = 1



ψt = −(1 − α)ψ − θx + αψxx , θt = −(1 − α)θ + γ ψx + 2ψθx + αθxx ,

(1.4)

with initial data

(ψ, θ )(x, 0) = (ψ0 (x), θ0 (x)) → (0, 0),

as x → ±∞.

(1.5)

They obtained the global existence and decay rates of the solutions to the system (1.4)–(1.5) under the assumption γ < 4α(1 − α) and the initial data

(ψ0 (x), θ0 (x)) ∈ L2 ∩ W 1,∞ (R, R2 ). Zhu and Wang in [5] considered the system (1.4) with more generalized initial data

(ψ, θ )(x, 0) = (ψ0 (x), θ0 (x)) → (ψ± , θ± ),

as x → ±∞,

(1.6)

and they also obtained the global existence and decay rates under the assumption of small initial data. Observing the results obtained in [4,5], we can find out that the assumptions imposed on the initial data mean that the system (1.4) is elliptic. But for the system (1.4) with mixed types, i.e., ellipticity and hyperbolicity, the global existence of the solution is an open problem. In order to get the global existence of the solutions to (1.4) with mixed types, obtaining the local existence is the first and essential step, which is the main purpose of this paper. Speaking roughly, we get the local existence of solutions to (1.4) with mixed types by constructing an approximation solution sequence and by the contraction mapping principle in this paper. Before stating our results precisely, we introduce the following notations. Notations: Hereafter, we denote several generic positive constants depending on a, b, . . . by Ca,b,... or only by C or O(1) without any confusion and δ := |ψ+ − ψ− | + |θ+ − θ− |. Lp = Lp (R)(1 ≤ p ≤ ∞) denotes the usual Lebesgue space with the norm

Z kf kLp =

|f (x)|p dx

 1p

,

1 ≤ p < ∞,

R

kf kL∞ = sup |f (x)|, R

and the integral region R will be omitted without any confusion. H l (l ≥ 0) denotes the usual lth-order Sobolev space with the norm

kf kH l =

l X

! 21 j xf

k∂ k

2

.

j =0

For simplicity, kf (·, t )kLp and kf (·, t )kH l are denoted by kf (t )kLp and kf (t )kH l respectively. First, we reformulate the system (1.4)–(1.6). As in [4,5], we introduce the following system



ψ¯t = −(1 − α)ψ¯ − θ¯x + α ψ¯ xx , −(1 − α)θ¯ + γ ψ¯ x = 0,

(1.7)

or

   γ  ψ¯t = −(1 − α)ψ¯ + α − ψ¯ xx , 1−α γ  θ¯ = ψ¯ x , 1−α

(1.8)

where the diffusion equation is obtained by Darcy’s law, cf. [6,7]. By direct calculation, the solutions of (1.8) can be written explicitly as

 Z ¯ x, t ) = e−(1−α)t (ψ+ − ψ− ) ψ(

x

G(y, t + 1)dy + ψ− −∞



,

(1.9)

H. Wei, M. Jiang / Nonlinear Analysis 71 (2009) 5897–5905

5899

and

γ γ ψ¯ x (x, t ) = (ψ+ − ψ− )e−(1−α)t G(x, t + 1), 1−α 1−α  2 γ x where G(x, t ) = √ 1 exp − 4at , a = α − 1−α > 0; the details can be found in [5]. 4π at To eliminate the values of θ (x, t ) at x = ±∞, the following correct function is introduced as in [6,5],   Z x ˆθ (x, t ) = e−(1−α)t θ− + (θ+ − θ− ) m0 (y)dy , θ¯ (x, t ) =

(1.10)

(1.11)

−∞

where m0 (x) is a smooth function with compact support satisfying +∞

Z

m0 (x)dx = 1. −∞

Under the above notations, we let



¯ x, t ), u(x, t ) = ψ(x, t ) − ψ( ¯ v(x, t ) = θ (x, t ) − θ (x, t ) − θˆ (x, t ).

(1.12)

Then from (1.4), (1.6) and (1.9)–(1.12), we can deduce that (u, v)(x, t ) satisfies the following Cauchy problem



ut = −(1 − α)u − vx + α uxx − θˆx , ¯ x + 2(θ¯x + θˆx )u + F (x, t ), vt = −(1 − α)v + γ ux + 2uvx + αvxx + 2ψv

(1.13)

with initial data



¯ x, 0) → 0, x → ±∞, u(x, 0) := u0 (x) = ψ0 (x) − ψ( v(x, 0) := v0 (x) = θ0 (x) − θ¯ (x, 0) − θˆ (x, 0) → 0, x → ±∞,

(1.14)

where

¯ θ¯x + θˆx ) + α(θ¯xx + θˆxx ). F (x, t ) = −θ¯t + 2ψ(

(1.15)

Define

 X (0, T ) = (u(x, t ), v(x, t ))|u(x, t ), v(x, t ) ∈ L∞ ([0, T ], H 2 ) ∩ L2 ([0, T ], H 3 ) . Our main results can be stated as follows: Theorem 1.1. If (u0 (x), v0 (x)) ∈ H 2 (R, R2 ), then there exists a constant t0 which depends only on k(u0 (x), v0 (x))kH 2 (R,R2 ) , such that the Cauchy problem (1.13)–(1.14) admits a unique smooth solution (u(x, t ), v(x, t )) ∈ X (0, t0 ) satisfying

k(u(x, t ), v(x, t ))kH 2 (R,R2 ) ≤ 2k(u0 (x), v0 (x))kH 2 (R,R2 ) . 2. Preliminaries In this section, we give some preliminary lemmas for our later use. ¯ θ¯ ) constructed in (1.9)–(1.10), cf. [6,5,8]. First we list several properties of (ψ,

¯ x, t ) and θ¯ (x, t ) to (1.8) satisfy the following properties: Lemma 2.1. The solutions ψ( ¯ t )kL∞ = O(1)e−(1−α)t , l = 0, 1, 2, . . . ; (i) k∂tl ψ( (ii) for any p with 1 ≤ p ≤ +∞, it holds that 1

¯ t )kLp = O(1)|ψ+ − ψ− |e−(1−α)t (1 + t )− 2 k∂ ∂ ψ( l k t x





1 1− 1p − k− 2

  − 21 1− 1p − 2k

k∂tl ∂xk θ¯ (t )kLp = O(1)|ψ+ − ψ− |e−(1−α)t (1 + t )

,

,

l = 0, 1, 2, . . . , k = 1, 2, . . . ; l, k = 0, 1, 2, . . . .

Lemma 2.2. The function θˆ (x, t ) defined by (1.11) satisfies the following properties: (i) k∂tl θˆ (t )kL∞ = O(1)e−(1−α)t , l = 0, 1, 2, . . . ; (ii) for any p with 1 ≤ p ≤ +∞, then

k∂tl ∂xk θˆ (t )kLp = O(1)|θ+ − θ− |e−(1−α)t ,

l = 0, 1, 2, . . . , k = 1, 2, . . . .

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H. Wei, M. Jiang / Nonlinear Analysis 71 (2009) 5897–5905

For the Green function G(x, t ), it has the following properties. Lemma 2.3. When 1 ≤ p ≤ +∞, 0 ≤ l, k < +∞, we have

k∂ ∂

(x, t )kLp = O(1)t

l k t xG

  − 12 1− 1p −l− 2k

.

Now we cite some fundamental inequalities for our later use. Lemma 2.4 (The Hausdorff–Young Inequality). If f ∈ L1 (R), g ∈ Lp (R)(1 ≤ p ≤ ∞), then

kf ∗ g kLp ≤ kf kL1 kg kLp . Lemma 2.5 (Minkowski’s Inequality for Integrals). Let 1 ≤ p < ∞. Suppose that f is measurable on Rm × Rn , that R f (·, y) ∈ Lp (Rm ) for almost all y ∈ Rn , and that the function y →k f (·, y) kp,Rm belongs to Lp (Rn ). Then the function x → Rn f (x, y)dy belongs to Lp (Rm ) and

Z





Rn

f (x, y)dy

Z ≤ p

Lx (Rm )

Rn

kf (x, y)kLpx (Rm ) dy.

3. The proof of the main result In this section, we prove our main result: Theorem 1.1. First we rewrite the problem (1.13)–(1.14) in the following integral forms

Z t Z t    u ( x , t ) = G ( x , t ) ∗ u ( x ) − ( 1 − α) G ( x , t − s ) ∗ u ( x , s ) ds + Gx (x, t − s) ∗ v(x, s)ds 0    0 0 Z t      − G(x, t − s) ∗ θˆx (x, s)ds,   0  Z t Z t  v(x, t ) = G(x, t ) ∗ v0 (x) − (1 − α) G(x, t − s) ∗ v(x, s)ds − γ Gx (x, t − s) ∗ u(x, s)ds  0 0  Z Z  t t    ¯ x, s)vx (x, s))ds +2 G(x, t − s) ∗ (u(x, s)vx (x, s))ds + 2 G(x, t − s) ∗ (ψ(    0 0  Z Z  t t    +2 G(x, t − s) ∗ ((θ¯x + θˆx )u(x, s))ds + G(x, t − s) ∗ F (x, s)ds, 0

(3.1)

0

where the convolutions are taken with respect to the space variable x. Then for n = 1, 2, 3, . . ., we can construct the approximation solution sequence as follows:

Z t Z t  (n) (n−1)   u ( x , t ) = G ( x , t ) ∗ u ( x ) − ( 1 − α) G ( x , t − s ) ∗ u ( x , s ) ds + Gx (x, t − s) ∗ v (n−1) (x, s)ds 0    0 0 Z  t     − G(x, t − s) ∗ θˆx (x, s)ds,   0  Z t Z t  v (n) (x, t ) = G(x, t ) ∗ v0 (x) − (1 − α) G(x, t − s) ∗ v (n−1) (x, s)ds − γ Gx (x, t − s) ∗ u(n−1) (x, s)ds  0 0  Z t Z t    (n−1) (n−1)  ¯ + 2 G ( x , t − s ) ∗ ( u ( x , s )v ( x , s )) ds + 2 G ( x , t − s ) ∗ ( ψ( x, s)vx(n−1) (x, s))ds  x   0 0  Z t Z t     +2 G(x, t − s) ∗ ((θ¯x + θˆx )u(n−1) (x, s))ds + G(x, t − s) ∗ F (x, s)ds, 0

(0)

(3.2)

0

(0)

where (u (x, t ), v (x, t )) = (G(x, t ) ∗ u0 (x), G(x, t ) ∗ v0 (x)). To prove Theorem 1.1, we only need to prove that there exists a constant t0 which depends only on k(u0 (x), v0 (x))kH 2 (R,R2 ) such that

 (n) ku (x, t ), v (n) (x, t )kH 2 (R,R2 ) ≤ 2ku0 (x), v0 (x)kH 2 (R,R2 ) , n ≥ 1,    sup ku(n) (x, t ) − u(n−1) (x, t ), v (n) (x, t ) − v (n−1) (x, t )k 2 2 H (R,R )

0 ≤t ≤t

0    ≤ ξ sup ku(n−1) (x, t ) − u(n−2) (x, t ), v (n−1) (x, t ) − v (n−2) (x, t )kH 2 (R,R2 ) ,

0≤t ≤t0

where ξ ∈ (0, 1) is a constant independent of n.

(3.3) n ≥ 2,

H. Wei, M. Jiang / Nonlinear Analysis 71 (2009) 5897–5905

5901

3.1. The first step: The proof of (3.3)1 We will prove (3.3)1 by the method of induction. For simplicity, let

ku0 (x)kH 2 = M ,

kv0 (x)kH 2 = N .

When n = 0, employing Lemmas 2.3 and 2.4, we have

ku(0) kH 2 = kG(x, t ) ∗ u0 (x)kH 2 ≤ kG(x, t )kL1 ku0 kH 2 = M .

(3.4)

kv (0) kH 2 = kG(x, t ) ∗ v0 (x)kH 2 ≤ kG(x, t )kL1 kv0 kH 2 = N .

(3.5)

So when n = 0, (3.3)1 is true. Suppose (3.3)1 is true when n ≤ m − 1. Then by Lemmas 2.2–2.5, the Sobolev inequality, the induction assumption and (3.2), we have

ku

(m)

Z

t

Z

t

kGx (t − s) ∗ v (m−1) kH 2 ds kG(t − s) ∗ u kH 2 ds + C kH 2 ≤ kG ∗ u0 (x)kH 2 + C (1 − α) 0 0 Z t +C kG(t − s) ∗ θˆx kH 2 ds 0 Z t Z t ≤ kG(x, t )kL1 ku0 kH 2 + C (1 − α) kG(t − s)kL1 ku(m−1) kH 2 ds + C kGx (t − s)kL1 kv (m−1) kH 2 ds 0 0 Z t +C kG(t − s)kL1 kθˆx kH 2 ds (m−1)

0 1

≤ M + C (1 − α)(M + N )t + C (M + N )t 2 + C δ(1 − e−(1−α)t ).

(3.6)

Similarly,

Z t Z t kGx (t − s) ∗ u(m−1) kH 2 ds kv (m) kH 2 ≤ kG ∗ v0 (x)kH 2 + C (1 − α) kG(t − s) ∗ v (m−1) kH 2 ds + C γ 0 0 Z t Z t (m−1) (m−1) ¯ x(m−1) )kH 2 ds +C kG(t − s) ∗ (u vx )kH 2 ds + C kG(t − s) ∗ (ψv 0 0 Z t Z t +C kG(t − s) ∗ ((θ¯x + θˆx )u(m−1) )kH 2 ds + C kG(t − s) ∗ F kH 2 ds 0 0 Z t Z t ≤ kG(x, t )kL1 kv0 kH 2 + C (1 − α) kG(t − s)kL1 kv (m−1) kH 2 ds + C kGx (t − s)kL1 ku(m−1) kH 2 ds 0 0 Z t Z t ¯ x(m−1) )kH 2 ds +C kG(t − s) ∗ (u(m−1) vx(m−1) )kH 2 ds + C kG(t − s) ∗ (ψv 0 0 Z t Z t (m−1) ¯ ˆ +C kG(t − s) ∗ ((θx + θx )u )kH 2 ds + C kG(t − s) ∗ F kH 2 ds 0

0 1

≤ N + C (1 − α)(M + N )t + C (M + N )t 2 +

4 X

Ii .

(3.7)

i=1

Now we estimate Ii (i = 1, 2, 3, 4) in the right-hand side of (3.7) respectively. Using Lemma 2.3, the Sobolev inequality and the induction assumption, we have t

Z

kG(t − s)kL1 ku(m−1) vx(m−1) kL2 + kGx (t − s)kL1 ku(m−1) vx(m−1) kL2
(m−1) + kGx (t − s)kL1 ku(xm−1) vx(m−1) + u(m−1) vxx kL2 ds Z t 1 ≤C ku(m−1) kL∞ kvx(m−1) kL2 + (t − s)− 2 ku(m−1) kL∞ kvx(m−1) kL2 0
 1 (m−1) + (t − s)− 2 ku(xm−1) kL∞ kvx(m−1) kL2 + ku(m−1) kL∞ kvxx kL2 ds Z t  1 (m−1) ≤C M kvx(m−1) kL2 + (t − s)− 2 (2M kvx(m−1) kL2 + M kvxx kL2 ) ds

I1 ≤ C

0

0 1

≤ C (M + N )2 (t + t 2 ).

(3.8)

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H. Wei, M. Jiang / Nonlinear Analysis 71 (2009) 5897–5905

Using Lemmas 2.1, 2.3 and the induction assumption, we have t

Z

¯ x(m−1) kL2 + kGx (t − s)kL1 kψv ¯ x(m−1) kL2 kG(t − s)kL1 kψv 0

(m−1) ¯ xx + kGx (t − s)kL1 kψ¯ x vx(m−1) kL2 + kψv kL2 ds Z t
 (m−1) ¯ L∞ kvxx ¯ L∞ kvx(m−1) kL2 + (t − s)− 21 kψk ¯ L∞ kvx(m−1) kL2 + kψ¯ x kL∞ kvx(m−1) kL2 + kψk kψk ≤C kL2 ds

I2 ≤ C

0 1

≤ C δ(M + N )(t + t 2 ).

(3.9)

By using Lemmas 2.2–2.4 and the induction assumption, we have

Z t

kG(t − s)kL1 k(θ¯x + θˆx )u(m−1) kL2 + kGx (t − s)kL1 k(θ¯x + θˆx )u(m−1) kL2   + kGx (t − s)kL1 k(θ¯xx + θˆxx )u(m−1) kL2 + k(θ¯x + θˆx )ux(m−1) kL2 ds Z t kG(t − s)kL1 kθ¯x + θˆx kL∞ ku(m−1) kL2 + kGx (t − s)kL1 kθ¯x + θˆx kL∞ ku(m−1) kL2 ≤C 0   + kGx (t − s)kL1 kθ¯xx + θˆxx kL∞ ku(m−1) kL2 + kθ¯x + θˆx kL∞ kux(m−1) kL2 ds

I3 ≤ C

0

1

≤ C δ(M + N )(t + t 2 ).

(3.10)

Using Lemmas 2.2–2.4, the induction assumption and (1.15), we have t

Z

kG(t − s)kL1 kF kH 2 ds

I4 = C 0 t

Z ≤C

  ¯ θ¯x + θˆx )kH 2 + αkθ¯xx kH 2 + αkθˆxx kH 2 ds kG(t − s)kL1 kθ¯t kH 2 + 2kψ(

0

≤ C δ(1 − e−(1−α)t ).

(3.11)

Substituting (3.8)–(3.11) into (3.7), we have 1

1

1

kv (m) kH 2 ≤ N + C (1 − α)(M + N )t + C (M + N )t 2 + C (M + N )2 (t + t 2 ) + C δ(M + N )(t + t 2 ) + C δ(1 − e−(1−α)t ).

(3.12)

From (3.6) and (3.12), we can easily deduce that if we choose t0 = t0 (M , N ) sufficiently small such that 1

M + C (1 − α)(M + N )t0 + C (M + N )t02 + C δ(1 − e−(1−α)t0 ) ≤ 2 max(M , N ), 1

1

N + C (1 − α)(M + N )t0 + C (M + N )t02 + C (M + N )2 (t0 + t02 )

(3.13)

1

+ C δ(M + N )(t0 + t02 ) + C δ(1 − e−(1−α)t0 ) ≤ 2 max(M , N ), then (3.3)1 is proved. 3.2. The second step: The proof of (3.3)2 Firstly,

ku

(n)

(n−1)

−u

kH 2

Z t


(n−1) (n−2)

≤ −(1 − α) G(x, t − s) ∗ u (x, s) − u (x, s) ds

2 0 H

Z t


(n−1) + (x, s) − v (n−2) (x, s) ds

Gx (x, t − s) ∗ v H2

0

Z

t

≤ (1 − α) kG(x, t − s) ∗ u(n−1) (x, s) − u(n−2) (x, s) kH 2 ds 0 Z t
+ kGx (x, t − s) ∗ v (n−1) (x, s) − v (n−2) (x, s) kH 2 ds 0 Z t ≤ (1 − α) kG(x, t − s)kL1 ku(n−1) (x, s) − u(n−2) (x, s)kH 2 ds 0




H. Wei, M. Jiang / Nonlinear Analysis 71 (2009) 5897–5905 t

Z +

5903

kGx (x, t − s)kL1 kv (n−1) (x, s) − v (n−2) (x, s)kH 2 ds

0


1 ≤ ((1 − α)t + Ct 2 ) sup ku(n−1) − u(n−2) kH 2 + kv (n−1) − v (n−2) kH 2 .

(3.14)

t

Similarly,

Z t
kG(x, t − s) ∗ v (n−1) (x, s) − v (n−2) (x, s) kH 2 ds kv (n) − v (n−1) kH 2 ≤ (1 − α) 0 Z t
+γ kGx (x, t − s) ∗ u(n−1) (x, s) − u(n−2) (x, s) kH 2 ds 0 Z t
kG(x, t − s) ∗ u(n−1) (x, s)vx(n−1) (x, s) − u(n−2) (x, s)vx(n−2) (x, s) kH 2 ds +2 0 Z t

¯ x, s) vx(n−1) (x, s) − vx(n−2) (x, s) kH 2 ds kG(x, t − s) ∗ ψ( +2 0 Z t 
 kG(x, t − s) ∗ (θˆx + θ¯x )(x, s) u(n−1) (x, s) − u(n−2) (x, s) kH 2 ds +2 0 9

=

X

Ii .

(3.15)

i =5

Now we estimate Ii (i = 5, . . . , 9) in the right-hand side of (3.15) respectively as follows: I5 = (1 − α)

t

Z


kG(t − s) ∗ v (n−1) − v (n−2) kH 2 ds

0

≤ (1 − α)

t

Z

kG(t − s)kL1 kv (n−1) − v (n−2) kH 2 ds

0

≤ (1 − α)t sup kv (n−1) − v (n−2) kH 2 ,

(3.16)

t

and t

Z


kGx (t − s) ∗ u(n−1) − u(n−2) kH 2 ds 0 Z t ≤ C (γ ) sup ku(n−1) − u(n−2) kH 2 kGx (t − s)kL1 ds

I6 = γ

t

0

1 2

≤ C (γ )t sup ku

(n−1)

−u

t

(n−2)

kH 2 .

(3.17)

By the Sobolev inequality, we have

Z

t

I7 = 2


kG(t − s) ∗ u(n−1) vx(n−1) − u(n−2) vx(n−2) kH 2 ds

Z0 t

kG(t − s)kL1 ku(n−1) vx(n−1) − u(n−2) vx(n−2) kL2 + kGx (t − s)kL1 ku(n−1) vx(n−1) − u(n−2) vx(n−2) kL2
(n−1) (n−2) + kGx (t − s)kL1 ku(xn−1) vx(n−1) + u(n−1) vxx − u(xn−2) vx(n−2) − u(n−2) vxx kL2 ds Z t
≤ C ku(n−1) − u(n−2) kL2 kvx(n−1) kL∞ + ku(n−2) kL∞ kvx(n−1) − vx(n−2) kL2 ds 0Z t
1 +C (t − s)− 2 ku(n−1) − u(n−2) kL2 kvx(n−1) kL∞ + ku(n−2) kL∞ kvx(n−1) − vx(n−2) kL2 ds Z0 t
1 (n−1) (n−2) +C (t − s)− 2 kvx(n−1) − vx(n−2) kL2 ku(xn−1) kL∞ + ku(n−1) kL∞ kvxx − vxx kL2 ds Z0 t
1 (n−2) +C (t − s)− 2 ku(xn−1) − u(xn−2) kL2 kvx(n−2) kL∞ + kvxx kL2 ku(n−1) − u(n−2) kL∞ ds Z t0
1 ≤ C (1 + (t − s)− 2 ) ku(n−1) − u(n−2) kH 2 + kv (n−1) − v (n−2) kH 2 ds 0
1 ≤ C (M + N )(t + t 2 ) sup ku(n−1) − u(n−2) kH 2 + kv (n−1) − v (n−2) kH 2 . ≤ 2

0

t

(3.18)

5904

H. Wei, M. Jiang / Nonlinear Analysis 71 (2009) 5897–5905 t

Z


¯ x, s)(vx(n−1) − vx(n−2) ) kH 2 ds kG(t − s) ∗ ψ(

I8 = 2

Z0 t

¯ x(n−1) − vx(n−2) )kL2 + kGx (t − s)kL1 kψ(v ¯ x(n−1) − vx(n−2) )kL2 kG(t − s)kL1 kψ(v
(n−1) (n−2) ¯ xx + kGx (t − s)kL1 kψ¯ x (vx(n−1) − vx(n−2) ) + ψ(v − vxx )kL2 ds Z t  ¯ L∞ + (t − s)− 21 kvx(n−1) − vx(n−2) kL2 kψk ¯ L∞ ds kvx(n−1) − vx(n−2) kL2 kψk ≤ C Z0 t
1 (n−1) (n−2) ¯ L∞ kvxx (t − s)− 2 kψ¯ x kL∞ kvx(n−1) − vx(n−2) kL2 + kψk − vxx kL2 ds + ≤ 2

0

0

1

≤ C δ(t 2 + t ) sup kv (n−1) − v (n−2) kH 2

(3.19)

t

and t

Z I9 = 2

  kG(t − s) ∗ (θ¯x + θˆx )(u(n−1) − u(n−2) ) kH 2 ds

0

Z t ≤2 kG(t − s)kL1 k(θ¯x + θˆx )(u(n−1) − u(n−2) )kL2 + kGx (t − s)kL1 k(θ¯x + θˆx )(u(n−1) − u(n−2) )kL2 0  + kGx (t − s)kL1 k(θ¯xx + θˆxx )(u(n−1) − u(n−2) ) + (θ¯x + θˆx )(ux(n−1) − ux(n−2) )kL2 ds Z t  1 ≤C ku(n−1) − u(n−2) kL2 kθ¯x + θˆx kL∞ + (t − s)− 2 ku(n−1) − u(n−2) kL2 kθ¯x + θˆx kL∞ ds 0 Z t   1 + (t − s)− 2 kθ¯xx + θˆxx kL∞ ku(n−1) − u(n−2) kL2 + kθ¯x + θˆx kL∞ ku(xn−1) − u(xn−2) kL2 ds 0 1

≤ C δ(t + t 2 ) sup ku(n−1) − u(n−2) kH 2 .

(3.20)

t

Substituting (3.16)–(3.20) into (3.15), we get

  1 1 kv (n) − v (n−1) kH 2 ≤ (1 − α)t + C (M + N )(t + t 2 ) + C δ(t + t 2 ) sup kv (n−1) − v (n−2) kH 2 t   1 1 1 + C (M + N )(t 2 + t ) + C δ(t + t 2 ) + C (γ )t 2 sup ku(n−1) − u(n−2) kH 2 t

1 2

≤ C (M + N )(1 + δ)(t + t ) sup ku t

(n−1)

−u

(n−2)


kH 2 + kv (n−1) − v (n−2) kH 2 .

(3.21)

From (3.14) and (3.21), we have

ku(n) − u(n−1) kH 2 + kv (n) − v (n−1) kH 2





1 ≤ C (M + N )(1 + δ)(t + t 2 ) sup ku(n−1) − u(n−2) kH 2 + kv (n−1) − v (n−2) kH 2 .

(3.22)

t

If we choose t0 = t0 (M , N ) sufficiently small such that 1

C (M + N )(1 + δ)(t0 + t02 ) < 1,

(3.23)

then (3.3)2 is proved. If we let t0 satisfy (3.13) and (3.23) simultaneously, by the combination of the two steps and by the contraction mapping principle, we can prove Theorem 1.1. Acknowledgements The research was supported by the National Natural Science Foundation of China, #10625105 and #10431060 and the Program for New Century Excellent Talents in University, #NCET-04-0745. References [1] [2] [3] [4]

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