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Optimal decay rates to diffusion wave for nonlinear evolution equations with ellipticity
J. Math. Anal. Appl. 319 (2006) 740–763 www.elsevier.com/locate/jmaa
Optimal decay rates to diffusion wave for nonlinear evolution equations with ellipticity ✩ Zhian Wang Department of Mathematical and Statistical Sciences, 632 CAB, University of Alberta, Edmonton, Alberta, Canada T6G 2G1 Received 27 January 2005 Available online 2 August 2005 Submitted by D. O’Regan
Abstract We derive the optimal convergence rates to diffusion wave for the Cauchy problem of a set of nonlinear evolution equations with ellipticity and dissipative effects ψt = −(1 − α)ψ − θx + αψxx , θt = −(1 − α)θ + νψx + (ψθ)x + αθxx , subject to the initial data with end states (ψ, θ )(x, 0) = ψ0 (x), θ0 (x) → (ψ± , θ± )
as x → ±∞,
where α and ν are positive constants such that α < 1, ν < 4α(1 − α). Introducing the auxiliary function to avoid the difference of the end states, we show that the solutions to the reformulated problem decay as t → ∞ with the optimal decay order. The decay properties of the solution in the L2 -sense, which are not optimal, were already established in paper [C.J. Zhu, Z.Y. Zhang, H. Yin, Convergence to diffusion waves for nonlinear evolution equations with ellipticity and damping, and with different end states, Acta Math. Sinica (English ed.), in press]. The main element of this paper is to obtain the optimal decay order in the sense of Lp space for 1 p ∞, which is based on the application of Fourier analysis and interpolation inequality under some suitable restrictions on coefficients α and ν. Moreover, we discuss the asymptotic behavior of the solution to general system (1.1) at the end. However, the optimal decay rates of the solution to general system (1.1) remains unknown. ✩
The research was supported by the F.S. Chia Scholarship. E-mail addresses: [email protected], [email protected].
0022-247X/$ – see front matter © 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2005.06.046
Z. Wang / J. Math. Anal. Appl. 319 (2006) 740–763
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© 2005 Elsevier Inc. All rights reserved. Keywords: Evolution equation; Fourier transform; Optimal decay rate; Interpolation inequality
1. Introduction It is well known that Lorenz derived his famous equations in [9] from Rayleigh–Benard equations [2]. The rich contents of the Lorenz equations has been explored extensively [11]. Thus the Lorenz equations can actually stand alone without their historical association with the Rayleigh–Benard equations. However, it is still legitimate to ask the questions in what sense and to what degree the Lorenz equations represent a valid description of the original Rayleigh–Benard problem. Since most physical problems are formulated in terms of partial differential equations in a larger context, we may also ask in what sense the approximation with a finite number of modes by truncation or other means can reveal the behavior of the original problem. Numerical solutions of the Rayleigh–Benard problem [2] have shown that some of the salient features of the Lorenz system disappear when more modes are retained than the number Lorenz retained. It also seems that the occurrence of certain phenomenon depends on the number and which modes are retained. Since the Rayleigh–Benard problem is a system of two highly nonlinear partial differential equations with three independent variables, except for the linearized system, it is difficult to analyze those equations in any way besides the numerical computations. Therefore it will be useful to construct a manageable partial differential equation in order to make a comparative study of the original partial differential equation and its reduced system. Toward this end, Hsieh [7] proposed the following alternative system to try to yield more and new insight to Rayleigh–Benard equations ψt = −(σ − α)ψ − σ θx + αψxx , (1.1) θt = −(1 − β)θ + νψx + (ψθ )x + βθxx , where (t, x) ∈ [0, +∞) × R, α, β, σ and ν are all positive constants satisfying the relation α < σ and β < 1. With the change of variables √ ψ =√ − 2X(t) sin x, θ = 2Y (t) cos x + 2Z(t) cos 2x, the Lorenz equation can be derived from (1.1) by only retaining the coefficients of sin x, cos x and cos 2x. Then the rich contents of the Lorenz equations are presumably also contained in the solutions of system (1.1). More detailed physical background can be seen in [7,12,13]. The exploration of (1.1) is supplemented with the initial data (1.2) ψ(0, x), θ (0, x) = ψ0 (x), θ0 (x) . As a preliminary work, Jian and Chen [8] first established the global existence of solutions to system (1.1) when (ψ0 , θ0 ) ∈ H 1 (R, R 2 ) ∩ L1 (R, R 2 ). Hsiao and Jian [5] obtained the global existence of classical solutions for the initial boundary value problem of system
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(1.1) with initial condition (ψ0 , θ0 ) ∈ C 2,δ ([0, 1]) ∗ C 2,δ ([0, 1]) (0 < δ < 1) and periodic boundary condition (ψ0 )x , (θ0 )x (0) = (ψ0 )x , (θ0 )x (1), 0 t T . (ψ0 , θ0 )(0) = (ψ0 , θ0 )(1), Moreover, Tang and Zhao [13] studied the following slightly modified system ψt = −(σ − α)ψ − σ θx + αψxx , θt = −(1 − β)θ + νψx + 2ψθx + βθxx .
(1.3)
When (ψ0 , θ0 ) ∈ L2 (R, R 2 ), they established the global existence, nonlinear stability and optimal decay rate of the solution to (1.3) with suitable restrictions on coefficients α, β, σ and ν. Furthermore, if (ψ0 , θ0 ) ∈ L1 (R, R 2 ), they obtained the optimal decay rates of solutions for system (1.3). Zhu and Wang [15] recently extended the above results to more general case in which initial data (1.2) satisfy (1.4) ψ0 (x), θ0 (x) → (ψ± , θ± ) as x → ±∞, where (ψ+ − ψ− , θ+ − θ− ) = (0, 0). They derived correction functions for system (1.3) by using the idea of asymptotical analysis [6,10] and reformulated the original system (1.3) with (1.4) into a new system with initial data which go to zero as |x| → ∞. Based on the technique of a priori assumption and energy methods, they established the global existence, nonlinear stability and decay rates of the solution under suitable restrictions on coefficients. Moreover, Duan and Zhu [3] investigated the asymptotics of diffusion wave toward the solution of system (1.3). Using the same idea and method, Zhu et al. [16] recently obtained the global existence, nonlinear stability and decay rates of the solution to the diffusion wave for system (1.1). But the optimal decay estimates of the solution to diffusion wave were not derived in that paper. In the relevant exploration of PDEs, it is more essential to find the asymptotic profile than to find the decay rates of solutions. Here the decay rates of asymptotic profile is consistent with that of the solutions to linearized PDEs. Such decay rates are generally called optimal decay rates. However, the difference of the solution and asymptotic profile could satisfy higher decay rates. The elaborate statement about the optimal decay rates is presented in [14]. The aim of this paper is to show that the optimal decay rates of the diffusion wave may attain by using the Fourier analysis and interpolation inequality as did in [13]. Notation Most of the notations we adopt in this paper are standard. For the sake of reading, we specify some notations here. Throughout the paper, we use C to denote generic constants which can change from line to line. When the dependence of the constant on some index or a function is important, we highlight it in the notation. Lp (R) (1 p ∞) denotes usual Lebesgue space with the norm f Lp (R) = ( R |f (x)|p dx)1/p , 1 p < ∞, as well as f L∞ (R) = supR |f (x)|. Lp (R, R 2 ), for 1 p ∞, denotes the usual Lebesgue space of R 2 -valued functions equipped with the norm (u, v) Lp (R,R 2 ) = u Lp (R) + v Lp (R) . Moreover, we denote by H l (R) the usual lth order Sobolev space with its norm
Z. Wang / J. Math. Anal. Appl. 319 (2006) 740–763
f H l (R) =
l i 2 ∂ f x
743
1/2 .
i=0
Z+ denotes the class of all positive integers and N = Z+ ∪ {0}.
2. Asymptotic profile and preliminaries To make the analysis easier, as did in [13,15], we discuss the system (1.1) for the case α = β and σ = 1. That is, we consider the following system: ψt = −(1 − α)ψ − θx + αψxx , (2.1) θt = −(1 − α)θ + νψx + (ψθ )x + αθxx , with initial data (ψ0 (x), θ0 (x)) which satisfies initial condition (1.4). To study the asymptotic behavior of the solution to system (2.1), we first need to find the appropriate asymptotic profile. As in [16], the solution of the system is expected to behave as those of the following linear system: ψ¯ t = −(1 − α)ψ¯ + α ψ¯ xx , (2.2) θ¯t = −(1 − α)θ¯ + α θ¯xx . Note that both equations in (2.2) are independent and have the same form. It then suf¯ x) = φ(t, ¯ x)e−(1−α)t , fices to solve one of them, say (2.2)1 . With the transformation ψ(t, Eq. (2.2)1 becomes a heat equation φ¯ t = α φ¯ xx .
(2.3)
Note that the linear diffusion equation (2.3) is invariant under the transformation (t, x) → (c2 t, cx) for any c > 0 and the function f (t + 1, x) is a solution to (2.3) too if f (t, x) is a solution of (2.3). Then we easily observe that (2.3) possesses the solution of the form
x ¯ := p(ξ ), −∞ < ξ < +∞, (2.4) φ(t, x) = p √ 1+t √ subject to the boundary condition p(±∞) = ψ± , where ξ = x/ 1 + t. Substituting (2.4) into (2.3) and solving the resulting equation yields that ψ − ψ− ¯ t) = p(ξ ) = √ + φ(x, 4πα(1 + t) Therefore
x −∞
¯ x) = e−(1−α)t (ψ+ − ψ− ) ψ(t,
x
−∞
where
x2 exp − G(x, t) = √ 4αt 4παt 1
exp −
y2 dy + ψ− . 4α(1 + t)
G(t + 1, y) dy + ψ− ,
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is the heat kernel function. Performing the same procedure as above, one has that x θ¯ (t, x) = e−(1−α)t (θ+ − θ− )
G(t + 1, y) dy + θ− .
−∞
Then it is straightforward to get ¯ x) → ψ± e−(1−α)t ψ(t,
as x → ±∞,
and θ¯ (t, x) → θ± e−(1−α)t
as x → ±∞.
¯ ¯ t) and their derivatives in We now consider the asymptotic behavior of ψ(x, t), θ(x, Lp (R). First for the heat kernel function G(t, x), it has the following properties. Lemma 2.1. When 1 p +∞, 0 l, k < +∞, we have l k 1 1 k ∂ ∂ G(t, ·) p Ct − 2 (1− p )−l− 2 . t x L (R) Employing the above lemma, we easily get the following results by simple calculations. ¯ Lemma 2.2. The solutions ψ(x, t) and θ¯ (x, t) to (2.2) satisfy the following properties: ¯ ·) L∞ (R) Ce−(1−α)t , ∂tl θ¯ (t, ·) L∞ (R) Ce−(1−α)t , l = 0, 1, 2, . . . ; (i) ∂tl ψ(t, (ii) for any p with 1 p +∞, it holds that l k 1 k ∂ ∂ ψ(t, ¯ ·) p C|ψ+ − ψ− |e−(1−α)t (1 + t) 2p − 2 , t x L (R) k = 1, 2, . . . , l = 0, 1, 2, . . . , l k 1 k ∂ ∂ θ¯ (t, ·) p C|θ+ − θ− |e−(1−α)t (1 + t) 2p − 2 , t x L (R) k = 1, 2, . . . , l = 0, 1, 2, . . . . Now let ¯ x), u(t, x) = ψ(t, x) − ψ(t, ¯ v(t, x) = θ (t, x) − θ (t, x), then system (2.1) can be recasted to the following reformulated equations: ut = −(1 − α)u + αuxx − vx − θ¯x , ¯ x + H (ψ, ¯ θ¯ ) vt = −(1 − α)v + αvxx + νux + (uv)x + (θ¯ u)x + (ψv) with initial data ¯ 0) → 0, u(x, 0) = u0 (x) = ψ0 (x) − ψ(x, ¯ v(x, 0) = v0 (x) = θ0 (x) − θ (x, 0) → 0, where
x → ±∞, x → ±∞,
(2.5)
(2.6)
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¯ θ¯ ) = ν ψ¯ x + ψ¯ θ¯x + θ¯ ψ¯ x . H (ψ, Zhu et al. [16] established the global existence, nonlinear stability and decay rate of a solution to (2.2) and (2.3) with some suitable restrictions on coefficients. The aim of this paper is to derive the optimal decay estimates of solutions to (2.5), (2.6). First, we cite the main results obtained in [16] as follows for later presentation. Lemma 2.3. [16, Theorem 3.1] Suppose that both δ = |ψ+ − ψ− | + |θ+ − θ− | and δ0 = u0 2H 2 (R) + v0 2H 2 (R) are sufficiently small. If (u0 , v0 ) ∈ H 2 (R, R 2 ), then for any 0 < α < 1, ν < 4α(1 − α), problem (2.5), (2.6) admits a unique global solution (u(t, x), v(t, x)) := (u, v) ∈ X(0, T ) satisfying u(t, ·)2 2 + v(t, ·)2 2 + H (R) H (R)
t
u(τ, ·)2
H 3 (R)
2 + v(τ, ·)H 3 (R) dτ
0
C(δ + δ0 ), and
(2.7)
sup (u, v)(t, x) + (ux , vx )(t, x) → 0,
as t → ∞.
(2.8)
x∈R
That is, there exists a unique global solution (ψ(t, x), θ (t, x)) of (2.1), (1.4) satisfying ¯ θ − θ¯ ) ∈ X(0, T ) (ψ − ψ, and
sup (ψ, θ )(t, x) + (ψx , θx )(t, x) → 0,
as t → ∞,
x∈R
where the solution space X(0, T ) is defined by X(0, T ) = (u, v) | u, v ∈ L∞ 0, T ; H 2 (R, R 2 ) ∩ L2 0, T ; H 3 (R, R 2 ) . Lemma 2.4. [16, Theorem 4.2] Suppose that (u(t, x), v(t, x)) is a solution to problem (2.5), (2.6) under the assumptions imposed in Lemma 2.3, then for any ν < 4α(1 − α), we have k ∂ 2 1 u(t, ·), v(t, ·) C(δ + δ0 ) 4 e−lt , k = 0, 1, 2, (2.9) 2 ∂x k 2 L (R,R ) where l is determined by c0 ν 2 , l = min (2 − )(1 − α), 2(1 − α) − α and ∈ (0, 2), c0 > 0 satisfy the relation 1 > 0, 2c0 α − (1−α) 2(1 − α) −
c0 ν 2 α
> 0.
(2.10)
(2.11)
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What follows are some well-known results. We present them here for later use. Lemma 2.5 (Convolution inequality [4]). By ∗ denotes the convolution. If u ∈ Lp (R) and v ∈ Lq (R), then u ∗ v ∈ Lr (R), where 1/r = 1/p + 1/q − 1, 1 p, q ∞, and u ∗ v Lr (R) u Lp (R) v Lq (R) . In particular, it follows that u ∗ v L∞ (R) u L∞ (R) v L1 (R)
and u ∗ v L1 (R) u L1 (R) v L1 (R) .
Lemma 2.6 (Interpolation inequality in Lp -space [1]). For any f ∈ Lq (R) ∩ Lr (R), if 1 q p r ∞ and 1/p = λ/q + (1 − λ)/r, then f ∈ Lp (R) and satisfies the following inequality: 1−λ f Lp (R) f λLq (R) f L r (R) .
Lemma 2.7 (Ehrling–Browder’s inequality [1]). Let v ∈ Lq (R N , R n ) and D m v ∈ Lr (R N , R n ) with 1 q, r ∞. Then for any integer j with 0 j m, there exists a constant C = C(m, p, N ), such that 1−α D j v Lp (R N ,R n ) C D m v αLr (R N ,R n ) v L q (R N ,R n ) ,
where p is determined by
j 1 m 1−α 1 = +α − + , p N r N q
j α 1. m
3. Linear analysis To investigate the optimal decay rates of solution to system (2.6), we first study the homogeneous linearized system corresponding to (2.5) ut = −(1 − α)u − vx + αuxx , (3.1) vt = −(1 − α)v + νux + αvxx , with initial data u(0, x), v(0, x) = u0 (x), v0 (x) .
(3.2)
Taking Fourier transform to (3.1), it becomes the following ordinary differential equation iξ uˆ uˆ 1 − α + αξ 2 , (3.3) =− vˆ vˆ t −iνξ 1 − α + αξ 2 with initial data u(0, ˆ x), v(0, ˆ x) = uˆ 0 (x), vˆ0 (x) ,
(3.4)
where uˆ and vˆ denotes the Fourier transform with the frequency ξ . Using the standard approaches to solve ODEs, we get the fundamental matrix to (3.3) which reads
Z. Wang / J. Math. Anal. Appl. 319 (2006) 740–763
e−(1−α+αξ M= 2
2 )t
e
√ νξ t
√
+ e−
1 √ i ν
νξ t
√ √ √ i ν e νξ t − e− νξ t
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√ √νξ t e − e− νξ t
e
√
νξ t
√
+ e−
νξ t
.
Hence, the solution of (3.3), (3.4) takes the form ⎧ 2 √ √ √ ⎪ e−(1−α+αξ )t √νξ t ⎪ ⎪ (vˆ0 + i ν uˆ 0 ) − e− νξ t (vˆ0 − i ν uˆ 0 ) , e √ ⎨ uˆ = 2i ν 2 )t ⎪ −(1−α+αξ √ ⎪ −√νξ t √ √ ⎪ ⎩ vˆ = e e (vˆ0 − i ν uˆ 0 ) + e νξ t (vˆ0 + i ν uˆ 0 ) . 2 Therefore, the inverse Fourier transform gives the solution of (3.1), (3.2) in the form √ √ 1 K− (t, x) ∗ (v0 + i νu0 ) − K+ (t, x) ∗ (v0 − i νu0 ) , u(t, x) = 2i √ ν (3.5) √ √ v(t, x) = 12 K+ (t, x) ∗ (v0 − i νu0 ) + K− (t, x) ∗ (v0 + i νu0 ) , where the kernel functions K± (t, x) are given by √ 2 √ 2 K± (t, x) = F −1 e−(1−α+αξ ± νξ )t = e−(1−α)t F −1 e−αξ t∓ νξ t
√
i ν x2 1 ν t exp ∓ x exp − , =√ exp − 1 − α − 4α 2α 4αt 4παt here F −1 denotes the inverse Fourier transform. From the explicit expression of the solution (3.5) and kernel function K± (t, x), we observe that the system (3.1) is stable if and only if ν < 4α(1 − α). Moreover, when p ∈ [1, +∞] and t → +∞, it follows that
x2 ν K± (t, ·) p = √ 1 t exp − exp − 1 − α − 4παt L (R) 4α 4αt Lp (R)
ν x2 − 12 −(1−α− 4α )t = O(1)t e exp − 4αt p L (R) = O(1)t
1 ν − 12 + 2p −(1−α− 4α )t
e
.
Furthermore, from Lemma 2.1, it is easy to derive that
∂ 1 ν K± (t, ·) ∂x p = √4παt exp − 1 − α − 4α t L (R)
√ x2 ∂ x2 i ν exp − + exp − × ∓ p 2α 4αt ∂x 4αt L (R) Ct
1 ν − 12 + 2p −(1−α− 4α )t
Ct
1 − 12 + 2p
e e
ν −(1−α− 4α )t
+ Ct
1 ν −1+ 2p −(1−α− 4α )t
e
.
Actually, using mathematical induction, we may easily prove that the above optimal decay rate holds for (∂ k /∂x k )K± (t, x) for any k ∈ N and p ∈ [1, ∞]. That is k ∂ 1 ν − 12 + 2p K (t, ·) e−(1−α− 4α )t . (3.6) p Ct ∂x k ± L (R)
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From the solution expression (3.5), we find that the decay rate of the solution of (3.1), (3.2) is indicated by the decay rates of K± (t, x). Namely, (3.6) holds for (u(t, x), v(t, x)) as well. We summarize in the following theorem. Theorem 3.1. Suppose (u0 , v0 ) ∈ L1 (R, R 2 ) ∩ Lp (R, R 2 ), then there exists a unique smooth solution (u(t, x), v(t, x)) to system (3.1), (3.2), which decays if and only if ν < 4α(1 − α). Moreover, for any k ∈ N and p ∈ [1, ∞], the following decay rate is optimal: k ∂ ν −1+ 1 C(1 + t) 2 2p e−(1−α− 4α )t . ∂x k u(t, ·), v(t, ·) p L (R,R 2 ) 4. Main results and proof In this section, we are devoted to proving that the solution of nonlinear system (2.5) has the same decay rate as that of the corresponding homogeneous linearized system (3.1). That is, the solution of system (2.5), (2.6) may attain the optimal decay rates. The main result is given in the following theorem. Theorem 4.1. Assume that ν < 4α(1 − α) and (ψ0 , θ0 ) ∈ H 2 (R, R 2 ) ∩ L1 (R, R 2 ). If both δ = |ψ+ − ψ− | + |θ+ − θ− |
and δ0 = u0 2H 2 (R) + v0 2H 2 (R)
are small enough, then for any k ∈ N and p with 1 p ∞, the solution of (2.5), (2.6) obtained in Lemma 2.3 may decay with the following optimal decay rate: k ∂ ν −1+ 1 u(t, ·), v(t, ·) C(1 + t) 2 2p e−(1−α− 4α )t . (4.1) ∂x k p L (R,R 2 ) To prove Theorem 4.1, we need the following lemmas. Lemma 4.2. For any m ∈ N, there exists constants Cm dependent of m, such that the solution (ψ(t, x), θ (t, x)) of system (2.5), (2.6) satisfies
m m ∂ u(t, x) 2 ∂ v(t, x) 2 dx + ∂x m ∂x m R
t
+
∂ m u(τ, x) ∂x m
2
+
∂ m v(τ, x) ∂x m
2 dx dτ
0 R
t
+
∂ m+1 u(τ, x) ∂x m+1
2
∂ m+1 v(τ, x) + ∂x m+1
2 dx dτ Cm .
(4.2)
0 R
Proof. We shall prove Lemma 4.2 by performing the mathematical induction. Indeed, the inequality in Lemma 2.3 has given (4.2) for m = 0, 1, 2. So we only need to prove (4.2)
Z. Wang / J. Math. Anal. Appl. 319 (2006) 740–763
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holds for m 3. To this end, we assume that (4.2) is true for m k − 1 with k 3. Under this assumption, it follows that
i i ∂ u(t, x) 2 ∂ v(t, x) 2 dx Ci , i = 0, 1, . . . , k − 1; + (4.3) ∂x i ∂x i R
and t
∂ i u(t, x) ∂x i
2
+
2
∂ i v(t, x) ∂x i
dx dt Ci ,
i = 0, 1, . . . , k.
(4.4)
0 R 1/2
1/2
By Sobolev inequality f L∞ (R) f L2 (R) fx L2 (R) , it follows from (4.3) that i ∂ u(t, ·) ∂x i
i ∂ v(t, ·) + ∂x i
L∞ ([0,∞)×R)
L∞ ([0,∞)×R)
2Ci ,
(4.5)
for i = 0, 1, 2, . . . , k − 2. Next we are going to show that (4.2) holds for m = k. For this goal, we differentiate (2.5) k times with respect to x and multiply the first equation by ∂ k ψ(t, x)/∂x k and the second equation by c0 ∂ k θ (t, x)/∂x k and integrate the resulting identity over (x, t) ∈ R × [0, t] by parts to get the following inequality using Cauchy– Schwartz inequality
k k ∂ u(t, x) 2 ∂ v(t, x) 2 dx + c 0 ∂x k ∂x k R
t
+ 2(1 − α)
∂ k u(τ, x) ∂x k
2
+ c0
∂ k v(τ, x) ∂x k
2 dx dt
0 R
t
+ 2α
∂ k+1 u(τ, x) ∂x k+1
2
+ c0
∂ k+1 v(τ, x) ∂x k+1
2 dx dτ
0 R
k ∂ 2 k u0 (·), v0 (·) 2 ∂x L
2 + (2 − )(1 − α)
t
∂ k+1 θ¯ (τ, x) ∂x k+1
∂ k u(τ, x) ∂x k
0 R
t
+ α 0 R
t
∂ k u(τ, x) ∂x k
2 dx dτ
0 R
2 dx dτ
0 R
t
+ (1 − α)
1 + (2 − )(1 − α) 2 (R)
∂ k+1 u(τ, x) ∂x k+1
2
2
1 dx dτ + (1 − α)
c2 ν 2 dx dτ + 0 α
t
0 R
t
∂ k+1 v(τ, x) ∂x k+1
0 R
∂ k v(τ, x) ∂x k
2 dx dτ
2 dx dτ
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Z. Wang / J. Math. Anal. Appl. 319 (2006) 740–763
t − 2c0
∂ k+1 v(τ, x) ∂k u(τ, x)v(τ, x) dx dτ k ∂x ∂x k+1
0 R
t − 2c0
∂ k+1 v(τ, x) ∂k ¯ (τ, x)u(τ, x) θ dx dτ ∂x k ∂x k+1
0 R
t − 2c0
∂ k+1 v(τ, x) ∂k ¯ ψ(τ, x)v(τ, x) dx dτ ∂x k ∂x k+1
0 R
t + 2c0
∂k ∂ k v(τ, x) ¯ x + ψ¯ θ¯x + θ¯ ψ¯ x ) (ν ψ dx dτ. ∂x k ∂x k
(4.6)
0 R
Set c02 ν 2 1 and d3 = 2c0 α − . α (1 − α) Shuffling the terms in (4.6) and using Cauchy–Schwartz inequality yields that d1 = (2 − )(1 − α),
∂ k u(t, x) ∂x k
2
d2 = 2c0 (1 − α) −
+ c0
∂ k v(t, x) ∂x k
2
1 dx + d1 2
R
t
∂ k u(τ, x) ∂x k
2 dx dτ
0 R
t
+ d2
∂ k v(τ, x) 2 ∂x k
t
dx dt + (2 − )α
0 R
t
+ d3
t − 2c0
L (R)
− 2c0
∂ k+1 v(τ, x) ∂k u(τ, x)v(τ, x) dx dτ ∂x k ∂x k+1
0 R
∂ k+1 v(τ, x) ∂k θ¯ (τ, x)u(τ, x) dx dτ k ∂x ∂x k+1
0 R
t − 2c0
∂ k+1 v(τ, x) ∂k ¯ ψ(τ, x)v(τ, x) dx dτ k ∂x ∂x k+1
0 R
2 + d1
t
0 R t
+ 2c0 0 R
dx dτ
dx dτ
0 R
k ∂ 2 k u0 (·), v0 (·) 2 ∂x
2
0 R
∂ k+1 v(τ, x) 2 ∂x k+1
t
∂ k+1 u(τ, x) ∂x k+1
∂ k+1 θ¯ (τ, x) ∂x k+1
2 dx dτ
∂k ∂ k v(τ, x) ¯ x + ψ¯ x θ¯x + θ¯ ψ¯ x ) (ν ψ dx dτ ∂x k ∂x k
Z. Wang / J. Math. Anal. Appl. 319 (2006) 740–763
k ∂ 2 u (·), v (·) 0 ∂x k 0 2 L
+
+
+
+
+
6c02 d3 6c02 d3 6c02 d3 2 d1
t
d3 + 2 (R)
t
∂ k+1 v(τ, x) ∂x k+1
2 dx dτ
0 R
∂k u(τ, x)v(τ, x) k ∂x
2 dx dτ
0 R
t
∂k θ¯ (τ, x)u(τ, x) k ∂x
2 dx dτ
0 R
t
∂k ¯ ψ(τ, x)v(τ, x) k ∂x
2 dx dτ
0 R
t
¯ x) ∂ k+1 θ(τ, ∂x k+1
2 dx dτ +
d2 2
0 R
2c02 d2
751
t
t
∂ k v(τ, x) ∂x k
2 dx dτ
0 R
(ν ψ¯ x + ψ¯ θ¯x + θ¯ ψ¯ x )2 dx dτ.
(4.7)
0 R
Rearrangement of (4.7) gives that
∂ k u(t, x) ∂x k
2
+ c0
∂ k v(t, x) ∂x k
2 dx
R
1 + d1 2
t
∂ k u(τ, x) ∂x k
2
d2 dx dτ + 2
0 R
t
+ (2 − )α
∂ k+1 u(τ, x) ∂x k+1
2
d3 dx dτ + 2
0 R
L
6c2 + 0 d3
6c2 + 0 d3 (R)
t
t
t
∂ k+1 v(τ, x) ∂x k+1
∂k u(τ, x)v(τ, x) k ∂x
0 R
∂k θ¯ (τ, x)u(τ, x) k ∂x
2 dx dτ
0 R
0 R
dx dt 2
0 R
k ∂ 2 k u0 (·), v0 (·) 2 ∂x 6c2 + 0 d3
2
0 R
t
t
∂ k v(τ, x) ∂x k
∂k ¯ ψ(τ, x)v(τ, x) k ∂x
2 dx dτ
2 dx dτ
dx dτ
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Z. Wang / J. Math. Anal. Appl. 319 (2006) 740–763
2 + d1
t
∂ k+1 θ¯ (τ, x) ∂x k+1
2
0 R
k ∂ 2 u (·), v (·) 0 ∂x k 0 2
+
2c2 dx dτ + 0 d2 6c02
M1 +
d3
L (R)
t
(ν ψ¯ x + ψ¯ x θ¯x + θ¯ ψ¯ x )2 dx dτ
0 R
6c02 d3
M2 +
6c02 2c2 2 M3 + M4 + 0 M5 . d3 d1 d2 (4.8)
We proceed to show that M1 , M2 , M3 , M4 and M5 are bounded, respectively. Indeed, we derive from (4.3)–(4.5) that t
M1 =
∂k u(τ, x)v(τ, x) ∂x k
2 dx dτ
0 R
2 t k i k ∂ u(τ, x) ∂ k−i v(τ, x) = dx dτ i ∂x i ∂x k−i 0 R
t =
i=0
k−2 i k ∂ u(τ, x) ∂ k−i v(τ, x) ∂ k−1 u(τ, x) +k vx (τ, x) i k−i i ∂x ∂x ∂x k−1
0 R
i=0
2 ∂ k u(τ, x) + v(τ, x) dx dτ ∂x k
2 k−2 i ∂ u(t, ·) k 2 i ∂x i ∞
t
L ([0,∞)×R)
i=1
t
2 + k vx (t, ·) ∞
L ([0,∞)×R)
∂ k−i v(τ, x) ∂x k−i
2 dx dτ
0 R
∂ k−1 u(τ, x) ∂x k−1
2 dx dτ
0 R
t
2 + C v(t, ·) ∞
L ([0,∞)×R)
∂ k u(τ, x) ∂x k
2 dx dτ
0 R
Ck .
(4.9)
Furthermore, the application of Lemma 2.2 and the assumption (4.4) give the estimates t
M2 =
∂k θ¯ (τ, x)u(τ, x) k ∂x
2 dx dτ
0 R
=
2 t k i ¯ x) k ∂ u(τ, x) ∂ k−i θ(τ, 0 R
i=0
i
∂x i
∂x k−i
dx dτ
Z. Wang / J. Math. Anal. Appl. 319 (2006) 740–763
2 k k−i ¯ k ∂ θ (τ, x) 2 ∞ k−i i ∂x
t
L ([0,∞)×R)
i=1
∂ i u(τ, x) ∂x i
753
2 dx dτ
0 R
Ck .
(4.10)
Moreover, the same approach as above also shows that M3 is bounded. From Lemma 2.2, it is straightforward to verify that M4 and M5 are all bounded. Thus we show that (4.2) is true for m = k. By the mathematical induction, (4.2) is then proved. 2 Corollary 4.3. For any k ∈ N, the solution (u(t, x), v(t, x)) of system (2.5), (2.6) satisfies the following exponential decay estimates: k ∂ u(t, x), v(t, x) Ce−at (4.11) 2 ∂x k 2 L (R,R ) and
k ∂ ∂x k u(t, x), v(t, x)
L∞ (R,R 2 )
Ce−at/2 ,
(4.12)
where a < l can be chosen as close as to l as wished. Proof. Taking account of (2.9) and (4.2), we get by Ehrling–Browder’s inequality k ∂ ∂x k u(t, x), v(t, x) 2 L (R,R 2 ) q+k ∂ k/(q+k) u(t, x), v(t, x) q/(q+k) C q+k u(t, x), v(t, x) 2 L2 (R,R 2 ) ∂x L (R,R 2 ) Ce−qlt/2(q+k) . Thus (4.11) is obtained by defining a = ql/(2(q + k)) and letting q be big enough. And (4.12) is a consequence of (4.11) by the use of Sobolev inequality. We then complete the proof. 2 To prove Theorem 4.1, we prove the following result for later presentation. Lemma 4.4. Let γ and η be positive numbers, t > 0. Then t
(1 + t − s)−γ e−ηs ds C(1 + t)−γ .
0
Proof. Dividing the integral t
t
0 (1 + t
− s)−γ e−ηs ds = (1 + t)−γ
0 (1 + t
t
0
− s)−γ e−ηs ds by (1 + t)−γ , we derive that 1+t −s 1+t
−γ
e−ηs ds
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Z. Wang / J. Math. Anal. Appl. 319 (2006) 740–763
t
= 0
s 1+ 1+t −s
γ
which ends the proof with the fact that the integral ∞ γ −ηs ds. 2 0 (1 + s) e
e
−ηs
t
t ds
0 (1
(1 + s)γ e−ηs ds,
0
+ s)γ e−ηs ds is bounded by
The following Gronwall’s inequality is devised for the optimal decay rates. The proof may be performed by standard approach and is omitted here. Lemma 4.5. Assume that the nonnegative function g(t) satisfies −μ −rt
g(t) N1 (1 + t)
e
t
e−δs g(s) ds,
+ N2 0
with nonnegative constants N1 , N2 , μ, r and δ. Then it follows that ∞ g(t) N1 (1 + t)−μ e−rt exp N2
e−δs ds C(1 + t)−μ e−rt .
0
With the above lemmas in hand, we are in the position to prove Theorem 4.1. Proof of Theorem 4.1. We shall prove inequality (4.1) by using interpolation inequality. The proof may be spitted into four steps. In the first step, we derive the solution representation for system (2.5), (2.6) using Fourier analysis. In the second step, we deduce the L∞ -estimate of the solution. The L1 -estimate will be given in the third step. At the end, we use the interpolation inequality to get the Lp -estimate. Step 1. Taking the Fourier transform to (2.5) and (2.6), we obtain after the integration of the linear terms √ √ √ e−(1−α+αξ )t √νξ t vˆ0 + i ν uˆ 0 − e− νξ t vˆ0 − i ν uˆ 0 e √ 2i ν t √ √ 1 2 x ds + √ e−(1−α+αξ )(t−s) e νξ(t−s) − e− νξ(t−s) (uv) 2i ν 2
u(t, ˆ x) =
0
−
1 2
t
e−(1−α+αξ
2 )(t−s)
√ √νξ(t−s) + e− νξ(t−s) θˆ¯ x ds e
0
1 + √ 2i ν
t 0
and
e−(1−α+αξ
2 )(t−s)
√ √νξ(t−s) e − e− νξ(t−s)
¯ x + ν ψ¯ x + ψ¯ θ¯x + θ¯ ψ¯ x ds × F (θ¯ u)x + (ψv)
Z. Wang / J. Math. Anal. Appl. 319 (2006) 740–763
755
√ √ √ e−(1−α+αξ )t √νξ t e vˆ0 + i ν uˆ 0 + e− νξ t vˆ0 − i ν uˆ 0 v(t, ˆ x) = 2 t √ √ 1 2 x ds e−(1−α+αξ )(t−s) e νξ(t−s) + e− νξ(t−s) (uv) + 2 2
0
√ t √ √ i ν 2 e−(1−α+αξ )(t−s) e νξ(t−s) − e− νξ(t−s) θˆ¯ x ds − 2 0
+
1 2
t 0
e−(1−α+αξ
2 )(t−s)
√ √νξ(t−s) e + e− νξ(t−s)
¯ x + ν ψ¯ x + ψ¯ θ¯x + θ¯ ψ¯ x ds, × F (θ¯ u)x + (ψv)
where F denote the Fourier transform. With the convolution property, the solution is then obtained in the following form under the inverse Fourier transform: √ √ 1 u(t, x) = √ K− (t, x) ∗ v0 + i νu0 − K+ (t, x) ∗ v0 − i νu0 2i ν t ∂ 1 K− (t − s, x) − K+ (t − s, x) ∗ u(s, x)v(s, x) ds + √ ∂x 2i ν 0
−
1 2
t
∂ K− (t − s, x) − K+ (t − s, x) ∗ θ¯x (s, x) ds ∂x
0
1 + √ 2i ν
t
∂ ¯ + θ¯ u) ds K− (t − s, x) − K+ (t − s, x) ∗ (ψv ∂x
0
+
1 √ 2i ν
t
K− (t − s, x) − K+ (t − s, x) ∗ (ν ψ¯ x + ψ¯ θ¯x + θ¯ ψ¯ x ) ds
0
and v(t, x) =
√ √ 1 K− (t, x) ∗ v0 + i νu0 + K+ (t, x) ∗ v0 − i νu0 2 t ∂ 1 K− (t − s, x) + K+ (t − s, x) ∗ u(s, x)v(s, x) ds + 2 ∂x 0
√ t i ν K− (t − s, x) − K+ (t − s, x) ∗ θ¯x (s, x) ds + 2 0
(4.13)
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Z. Wang / J. Math. Anal. Appl. 319 (2006) 740–763
1 + 2
t
∂ ¯ + θ¯ u) ds K− (t − s, x) + K+ (t − s, x) ∗ (ψv ∂x
0
1 + 2
t
K− (t − s, x) + K+ (t − s, x) ∗ (ν ψ¯ x + ψ¯ θ¯x + θ¯ ψ¯ x ) ds.
(4.14)
0
Step 2 (L1 -estimate). By the solution expression (4.13) and (4.14), for i = 0, 1, 2, . . . , we obtain from Lemma 2.5 after some rearrangement i ∂ ∂x i u(t, ·), v(t, ·) 1 L (R,R 2 ) i ∂ u0 (·), v0 (·) 1 C i K+ (t, ·), K− (t, ·) L (R,R 2 ) ∂x L1 (R,R 2 ) t ∂ K +C (t − s, ·), K (t − s, ·) − ∂x +
L1 (R,R 2 )
0
t i+1 ∂ + C i+1 K+ (t − s, ·), K− (t − s, ·) ∂x 0
i ∂ u(s, ·)v(s, ·) ∂x i
ds L1 (R)
L1 (R,R 2 )
¯ x)v(s, ·) + θ¯ (s, ·)u(s, ·) 1 ds × ψ(s, L (R) t i ∂ + C i K+ (t, ·), K− (t, ·) ∂x 0
L1 (R,R 2 )
θ¯x + ν ψ¯ x + ψ¯ θ¯x + θ¯ ψ¯ x L1 (R) ds
= S1 + S 2 + S3 + S 4 ,
(4.15)
where S1 , S2 , S3 and S4 denote the terms on the right-hand side of inequality (4.5) from the first to the fourth term, respectively. We shall estimate them term by term next. Indeed, in view of the initial condition (u0 , v0 ) ∈ L1 (R, R 2 ), it is clear from (3.6) that ν
S1 Ce−(1−α− 4α )t .
(4.16)
Moreover, due to (3.6) and (4.12), it follows that t S2 C
e
i i ∂ u(s, ·) ∂x i
ν −(1−α− 4α )(t−s) − as 2
e
0
t C
e 0
− as 2
i i ∂ u(s, ·) ∂x i j =0
ds
L1 (R)
j =0
ds.
L1 (R)
Taking account of Lemma 2.4 and Minkowski inequality, we have that
(4.17)
Z. Wang / J. Math. Anal. Appl. 319 (2006) 740–763
t S3 C
757
ν ¯ ·) ∞ v(s, ·) 1 e−(1−α− 4α )(t−s) ψ(s, L (R) L (R)
0
+ θ¯ (s, ·)L∞ (R) u(s, ·)L1 (R) ds t C
e−(1−α)s u(s, ·), v(s, ·) L1 (R,R 2 ) ds.
(4.18)
0
Similarly, we deduce from Lemma 2.2 and (3.6) by Minkowski inequality that t S4 C
ν ¯ ·) ∞ θ¯x (s, ·) 1 e−(1−α− 4α )(t−s) θ¯x (s, ·)L1 (R) + ψ(s, L (R) L (R)
0
+ θ¯ (s, ·)L∞ (R) ψ¯ x (s, ·)L1 (R) ds Ce
ν −(1−α− 4α )t
Ce
ν −(1−α− 4α )t
t
ν
e− 4α s ds
0
(4.19)
,
here the last inequality results from the fact that t e
ν − 4α s
∞ ds
0
ν
e− 4α s ds < ∞.
0
Then the substitution of (4.16)–(4.19) into (4.15) yields i ∂ ∂x i u(t, ·), v(t, ·) 1 L (R,R 2 ) t i j ∂ − as ν −(1−α− 4α )t −(1−α)s 2 e Ce +C +e ds. ∂x j u(s, ·), v(s, ·) 1 L (R,R 2 ) j =0
0
(4.20)
Defining Y (t) =
k j ∂ u(t, ·), v(t, ·) ∂x j j =0
L1 (R,R 2 )
and summing (4.20) over i = 0, 1, 2, . . . , k, we have that Y (t) Ck e
ν −(1−α− 4α )t
t +C
− as e 2 + e−(1−α)s Y (s) ds.
0
Therefore, applying Gronwall’s inequality to (4.21) gives us that ν
Y (t) Ck e−(1−α− 4α )t .
(4.21)
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Z. Wang / J. Math. Anal. Appl. 319 (2006) 740–763
This implies that for any k ∈ N, it follows that k ∂ ν u(t, ·), v(t, ·) Ck e−(1−α− 4α )t . ∂x k 1 2 L (R,R )
(4.22)
Step 3 (L∞ -estimate). To derive L∞ -estimate, we first claim that for any Z(t, ·) ∈ it holds that k ∂ K± (t − s, ·) 1 ν ∗ Z(s, ·) C(1 + t − s)− 2 e−(1−α− 4α )(t−s) ∞ k ∂x L (R) × Z(s, ·)L1 (R) + Z(s, ·)L∞ (R) . (4.23)
L1 (R) ∩ L∞ (R),
Actually, from (3.6), we get the following estimate by using convolution inequality k k ∂ K± (t − s, ·) ∂ K± (t − s, ·) Z(s, ·) 1 ∗ Z(s, ·) C L (R) ∂x k ∂x k L∞ (R) L∞ (R) 1 ν C(t − s)− 2 e−(1−α− 4α )(t−s) Z(s, ·)L1 (R) . (4.24) On the other hand, if follows that k k ∂ K± (t − s, ·) ∂ K± (t − s, ·) Z(s, ·) ∞ ∗ Z(s, ·) C L (R) k k ∂x ∂x L∞ (R) L1 (R) ν Ce−(1−α− 4α )(t−s) Z(s, ·)L∞ (R) . (4.25) Then the combination of (4.24) and (4.25) implies (4.23). This proves the claim. Hence, from (4.13), (4.14) and (4.23), we deduce that i ∂ ∂x i u(t, ·), v(t, ·) ∞ L (R,R 2 ) ν − 12 −(1−α− 4α )t u0 (·), v0 (·) 1 u0 (·), v0 (·) ∞ C(1 + t) e 2 + t +C
ν − 12 −(1−α− 4α )(t−s)
(1 + t − s) 0
e
i ∂ + i u(s, ·)v(s, ·) ∂x
t +C
i ∂ ∂x i u(s, ·)v(s, ·)
L (R,R 2 )
L (R,R )
L1 (R)
L∞ (R)
ds
1 ν ¯ + θ¯ u L1 (R) + ψv ¯ + θ¯ u L∞ (R) ds (1 + t − s)− 2 e−(1−α− 4α )(t−s) ψv
0
t +C 0
1 ν (1 + t − s)− 2 e−(1−α− 4α )(t−s) θ¯x + ν ψ¯ x + ψ¯ θ¯x + θ¯ ψ¯ x L1 (R)
+ θ¯x + ν ψ¯ x + ψ¯ θ¯x + θ¯ ψ¯ x L∞ (R) ds
= I1 + I2 + I3 + I4 .
(4.26)
Z. Wang / J. Math. Anal. Appl. 319 (2006) 740–763
759
Next, we shall estimate I1 , I2 , I3 and I4 , respectively. In fact, due to the initial condition (u0 , v0 ) ∈ H 2 (R, R 2 ) and Sobolev inequality
∂u0 (·) ∂v0 (·) 1/2 u0 (·), v0 (·) ∞ u0 (·), v0 (·) 1/2 , 2 , L (R) L2 (R) ∂x ∂x L (R) we know that (u0 (·), v0 (·)) L∞ (R) is bounded. Note that (u0 , v0 ) ∈ L1 (R, R 2 ), we have 1
ν
I1 C(1 + t)− 2 e−(1−α− 4α )t .
(4.27)
Moreover, in view of (4.12) and (4.22), it follows that i i ∂ ∂ ∂x i u(s, ·)v(s, ·) 1 + ∂x i u(s, ·)v(s, ·) ∞ L (R) L (R) j j i i ∂ u(s, ·) ∂ v(s, ·) ∂x j ∞ ∂x j 1 L (R) L (R) j =0
j =0
i j ∂ u(s, ·)
+
j =0
i j ∂ v(s, ·) ∂x j
∂x j
L∞ (R) j =0
L∞ (R)
i j ∂ u(s, ·) −(1−α− ν )s − as 4α 2 C e +e ∂x j ∞ . L (R)
(4.28)
j =0
Therefore, I2 is bounded as t I2 C
1 ν ν as (1 + t − s)− 2 e−(1−α− 4α )(t−s) e−(1−α− 4α )s + e− 2
0
×
i j ∂ u(s, ·) ∂x j
L∞ (R)
j =0
t C
ν
ds
as
e−(1−α− 4α )s + e− 2
i j ∂ u(s, ·) ∂x j
L∞ (R)
j =0
0
ds.
Applying Minkowski inequality, we bound I3 from Lemma 2.2 and (4.22) as t I3 C
1
ν
(1 + t − s)− 2 e−(1−α− 4α )(t−s)
0
¯ L∞ (R) v L1 (R) + θ ¯ L∞ (R) u L1 (R) ds × ψ t +C 0
1
ν
(1 + t − s)− 2 e−(1−α− 4α )(t−s)
¯ L∞ (R) v L∞ (R) + θ ¯ L∞ (R) u L∞ (R) ds × ψ
(4.29)
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Z. Wang / J. Math. Anal. Appl. 319 (2006) 740–763
t C
1
ν
ν
(1 + t − s)− 2 e−(1−α− 4α )(t−s) e−(1−α− 4α )s e−(1−α)s ds
0
t +C
1
ν
(1 + t − s)− 2 e−(1−α− 4α )(t−s) e−(1−α)s
0
× v(s, ·)L∞ (R) + u(s, ·)L∞ (R) ds ν − 12 −(1−α− 4α )t
C(1 + t)
e
t +C
e−(1−α)s u(s, ·), v(s, ·) L∞ (R,R 2 ) ds. (4.30)
0
Similarly, we bound I4 from Lemmas 2.2 and 4.4 as t I4 C
1
ν
(1 + t − s)− 2 e−(1−α− 4α )(t−s) e−(1−α)s ds
0 1
ν
C(1 + t)− 2 e−(1−α− 4α )t .
(4.31)
Thus, inserting (4.27), (4.29), (4.30) and (4.31) into (4.26) yields i ∂ ∂x i u(t, ·), v(t, ·) ∞ L (R,R 2 ) ν − 12 −(1−α− 4α )t
C(1 + t)
e
t +C 0
t +C
i j −(1−α− ν )s a ∂ u(s, ·) − s 4α e +e 2 ∂x j ∞ ds L (R) j =0
e−(1−α)s u(s, ·), v(s, ·) L∞ (R,R 2 ) ds.
(4.32)
0
Summing (4.32) over i = 0, 1, 2, . . . , k and defining k j ∂ Y˜ (t) = u(t, ·), v(t, ·) , ∂x j ∞ L (R,R 2 ) j =1
we end up with Y˜ (t) C(1 + t)− 2 e−(1−α− 4α )t + C 1
ν
t
−(1−α− ν )s a 4α e + e− 2 s + e−(1−α)s Y˜ (s) ds.
0
Thus, the application of Gronwall’s inequality to (4.33) leads to Y˜ (t) C(1 + t)− 2 e−(1−α− 4α )t . 1
ν
That is, for any k ∈ N, it holds that
(4.33)
Z. Wang / J. Math. Anal. Appl. 319 (2006) 740–763
k ∂ ∂x k u(t, ·), v(t, ·)
1
L∞ (R,R 2 )
761
ν
C(1 + t)− 2 e−(1−α− 4α )t .
Step 4 (Lp -estimate). Finally, with the L1 -estimate and L∞ -estimate in hand, for any p ∈ [1, ∞], we use the interpolation inequality to derive the Lp -estimate in the form k ∂ u(t, ·), v(t, ·) ∂x k p L (R,R 2 ) k ∂ (p−1)/p ∂ k 1/p u(t, ·), v(t, ·) u(t, ·), v(t, ·) k ∞ 1 ∂x k L (R,R 2 ) ∂x L (R,R 2 ) 1 ν ν (p−1)/p −(1−α− )t 1/p 4α e C (1 + t)− 2 e−(1−α− 4α )t 1 ν − 12 + 2p −(1−α− 4α )t
C(1 + t)
e
This completes the proof.
.
2
5. Concluding comments For the special case where α = β and σ = 1, under the assumption ν < 4α(1 − α), Zhu et al. [16] established the global existence, non-stability and decay estimates of the solution to system (2.2) and (2.3) which are cited as Lemmas 2.3 and 2.4. In the present paper, the optimal decay rates (4.1) was obtained under the same assumption on coefficients. Due to the nonlinearity, the initial data is required to be small enough. Indeed, the same analysis as in [16] applies to the general system (1.1) with initial condition (1.4). But the optimal decay rates of the solution to the general system (1.1) is still unknown in the present paper. We summarize this in the following theorem. Theorem 5.1. Suppose that both δ = |ψ+ − ψ− | + |θ+ − θ− | and δ0 = u0 2H 2 (R) + v0 2H 2 (R) are small enough. For some positive constants C0 > 0 and ∈ (0, 2), if the coefficients satisfy the relations 2C0 β −
σ2 > 0, (σ − α)
1−β −
C0 ν 2 > 0, 2α
and initial data (ψ0 − ψ¯ 0 , θ0 − θ¯0 − θ˜0 ) ∈ H 2 (R, R 2 ) then for any ν subject to √ 4 αβ(1 − β)(σ − α) ν< σ
(5.1)
(5.2)
and constant
C0 ν 2 l = min (2 − )(σ − α), 2 1 − β − , 2α
(5.3)
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system (1.1) with initial condition (1.4) admits a unique global solution which decays exponentially for any nonnegative integers k in the L2 -norm form k 2 ∂ ¯ ¯ ˜ Ce−lt . ∂x k (ψ − ψ, θ − θ − θ )(t, x) 2 L (R,R 2 ) Remark 1. It is worthwhile to point out here that the set of constants C0 satisfying (5.1) is not empty. In fact, we may give an algorithm to determine it by the following steps: Step 1. Step 2. Step 3. Step 4.
√ Define k with 0 < k < 1 by νσ = k αβ(1 − β)(σ − α) . Choose with k2 < < 2. Choose λ with λ > 0. Define C0 by
σ2 1 2α(1 − β) C0 = . +λ 1 + λ 2β(σ − α) ν2
Remark 2. By taking into account (5.1) with ∈ (0, 2), it is straightforward to derive that √ 4 αβ(1 − β)(σ − α) , ν< σ which gives (5.2). As we perform the same procedure as in [16] to deduce the order of decay rates of the solution, the condition (5.3) will come up. Remark 3. For the general system (1.1), the optimal decay rates of the solution is not obtained yet since the calculation is too complicated to derive the desired results. It is still interesting to develop novel ideas or computational techniques to deal with such a problem.
References [1] R.A. Adams, Sobolev Space, Academic Press, New York, 1975. [2] J.H. Curry, J.R. Herring, J. Loncaric, S.A. Orszag, Order and disorder in two- and three-dimension Bérnard convection, J. Fluid Mech. 1 (1984) 1–38. [3] R. Duan, C.J. Zhu, Asymptotics of dissipative nonlinear evolution equations with ellipticity: Different end states, J. Math. Anal. Appl. 303 (2005) 15–35. [4] J. Duonandikoetxea, Fourier Analysis, Amer. Math. Soc., Providence, RI, 2001. [5] L. Hsiao, H.Y. Jian, Global smooth solutions to the spatically periodic Cauchy problem for dissipative nonlinear evolution equations, J. Math. Anal. Appl. 213 (1997) 262–274. [6] L. Hsiao, T.-P. Liu, Convergence to nonlinear diffusion waves for solutions of a system of hyperbolic conservation laws with damping, Comm. Math. Phys. 143 (1992) 599–605. [7] D.Y. Hsieh, On partial differential equations related to Lorenz system, J. Math. Phys. 28 (1987) 1589–1597. [8] H.Y. Jian, D.G. Chen, On the Cauchy problem for certain system of semilinear parabolic equations, Acta Math. Sinica (English ed.) 14 (1998) 17–34. [9] E.N. Lorenz, Deterministic non-periodic flows, J. Atmos. Sci. 20 (1963) 130–141. [10] K. Nishihara, Convergence rates to nonlinear diffusion waves for solutions of system of hyperbolic conservation laws with damping, J. Differential Equations 131 (1996) 171–188. [11] C. Sparrow, The Lorenz’s Equations: Bifurcations, Chaos, and Strange Attractors, Springer-Verlag, New York, 1982.
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[12] S.Q. Tang, Dissipative nonlinear evolution equation and chaos, PhD thesis, The Hong Kong University of Science and Technology, 1995. [13] S.Q. Tang, H.J. Zhao, Nonlinear stability for dissipative nonlinear evolution equations with ellipticity, J. Math. Anal. Appl. 233 (1999) 336–358. [14] H.J. Zhao, Large time decay estimates of solutions of nonlinear parabolic equations, Discrete Contin. Dynam. Syst. 8 (2002) 69–114. [15] C.J. Zhu, Z.A. Wang, Decay rates of solutions to dissipative nonlinear evolution equations with ellipticity, Z. Angew. Math. Phys. 55 (2004) 994–1014. [16] C.J. Zhu, Z.Y. Zhang, H. Yin, Convergence to diffusion waves for nonlinear evolution equations with ellipticity and damping, and with different end states, Acta Math. Sin. (Engl. Ser.), in press.
Optimal decay rates to diffusion wave for nonlinear evolution equations with ellipticity ✩ Zhian Wang Department of Mathematical and Statistical Sciences, 632 CAB, University of Alberta, Edmonton, Alberta, Canada T6G 2G1 Received 27 January 2005 Available online 2 August 2005 Submitted by D. O’Regan
Abstract We derive the optimal convergence rates to diffusion wave for the Cauchy problem of a set of nonlinear evolution equations with ellipticity and dissipative effects ψt = −(1 − α)ψ − θx + αψxx , θt = −(1 − α)θ + νψx + (ψθ)x + αθxx , subject to the initial data with end states (ψ, θ )(x, 0) = ψ0 (x), θ0 (x) → (ψ± , θ± )
as x → ±∞,
where α and ν are positive constants such that α < 1, ν < 4α(1 − α). Introducing the auxiliary function to avoid the difference of the end states, we show that the solutions to the reformulated problem decay as t → ∞ with the optimal decay order. The decay properties of the solution in the L2 -sense, which are not optimal, were already established in paper [C.J. Zhu, Z.Y. Zhang, H. Yin, Convergence to diffusion waves for nonlinear evolution equations with ellipticity and damping, and with different end states, Acta Math. Sinica (English ed.), in press]. The main element of this paper is to obtain the optimal decay order in the sense of Lp space for 1 p ∞, which is based on the application of Fourier analysis and interpolation inequality under some suitable restrictions on coefficients α and ν. Moreover, we discuss the asymptotic behavior of the solution to general system (1.1) at the end. However, the optimal decay rates of the solution to general system (1.1) remains unknown. ✩
The research was supported by the F.S. Chia Scholarship. E-mail addresses: [email protected], [email protected].
0022-247X/$ – see front matter © 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2005.06.046
Z. Wang / J. Math. Anal. Appl. 319 (2006) 740–763
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© 2005 Elsevier Inc. All rights reserved. Keywords: Evolution equation; Fourier transform; Optimal decay rate; Interpolation inequality
1. Introduction It is well known that Lorenz derived his famous equations in [9] from Rayleigh–Benard equations [2]. The rich contents of the Lorenz equations has been explored extensively [11]. Thus the Lorenz equations can actually stand alone without their historical association with the Rayleigh–Benard equations. However, it is still legitimate to ask the questions in what sense and to what degree the Lorenz equations represent a valid description of the original Rayleigh–Benard problem. Since most physical problems are formulated in terms of partial differential equations in a larger context, we may also ask in what sense the approximation with a finite number of modes by truncation or other means can reveal the behavior of the original problem. Numerical solutions of the Rayleigh–Benard problem [2] have shown that some of the salient features of the Lorenz system disappear when more modes are retained than the number Lorenz retained. It also seems that the occurrence of certain phenomenon depends on the number and which modes are retained. Since the Rayleigh–Benard problem is a system of two highly nonlinear partial differential equations with three independent variables, except for the linearized system, it is difficult to analyze those equations in any way besides the numerical computations. Therefore it will be useful to construct a manageable partial differential equation in order to make a comparative study of the original partial differential equation and its reduced system. Toward this end, Hsieh [7] proposed the following alternative system to try to yield more and new insight to Rayleigh–Benard equations ψt = −(σ − α)ψ − σ θx + αψxx , (1.1) θt = −(1 − β)θ + νψx + (ψθ )x + βθxx , where (t, x) ∈ [0, +∞) × R, α, β, σ and ν are all positive constants satisfying the relation α < σ and β < 1. With the change of variables √ ψ =√ − 2X(t) sin x, θ = 2Y (t) cos x + 2Z(t) cos 2x, the Lorenz equation can be derived from (1.1) by only retaining the coefficients of sin x, cos x and cos 2x. Then the rich contents of the Lorenz equations are presumably also contained in the solutions of system (1.1). More detailed physical background can be seen in [7,12,13]. The exploration of (1.1) is supplemented with the initial data (1.2) ψ(0, x), θ (0, x) = ψ0 (x), θ0 (x) . As a preliminary work, Jian and Chen [8] first established the global existence of solutions to system (1.1) when (ψ0 , θ0 ) ∈ H 1 (R, R 2 ) ∩ L1 (R, R 2 ). Hsiao and Jian [5] obtained the global existence of classical solutions for the initial boundary value problem of system
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(1.1) with initial condition (ψ0 , θ0 ) ∈ C 2,δ ([0, 1]) ∗ C 2,δ ([0, 1]) (0 < δ < 1) and periodic boundary condition (ψ0 )x , (θ0 )x (0) = (ψ0 )x , (θ0 )x (1), 0 t T . (ψ0 , θ0 )(0) = (ψ0 , θ0 )(1), Moreover, Tang and Zhao [13] studied the following slightly modified system ψt = −(σ − α)ψ − σ θx + αψxx , θt = −(1 − β)θ + νψx + 2ψθx + βθxx .
(1.3)
When (ψ0 , θ0 ) ∈ L2 (R, R 2 ), they established the global existence, nonlinear stability and optimal decay rate of the solution to (1.3) with suitable restrictions on coefficients α, β, σ and ν. Furthermore, if (ψ0 , θ0 ) ∈ L1 (R, R 2 ), they obtained the optimal decay rates of solutions for system (1.3). Zhu and Wang [15] recently extended the above results to more general case in which initial data (1.2) satisfy (1.4) ψ0 (x), θ0 (x) → (ψ± , θ± ) as x → ±∞, where (ψ+ − ψ− , θ+ − θ− ) = (0, 0). They derived correction functions for system (1.3) by using the idea of asymptotical analysis [6,10] and reformulated the original system (1.3) with (1.4) into a new system with initial data which go to zero as |x| → ∞. Based on the technique of a priori assumption and energy methods, they established the global existence, nonlinear stability and decay rates of the solution under suitable restrictions on coefficients. Moreover, Duan and Zhu [3] investigated the asymptotics of diffusion wave toward the solution of system (1.3). Using the same idea and method, Zhu et al. [16] recently obtained the global existence, nonlinear stability and decay rates of the solution to the diffusion wave for system (1.1). But the optimal decay estimates of the solution to diffusion wave were not derived in that paper. In the relevant exploration of PDEs, it is more essential to find the asymptotic profile than to find the decay rates of solutions. Here the decay rates of asymptotic profile is consistent with that of the solutions to linearized PDEs. Such decay rates are generally called optimal decay rates. However, the difference of the solution and asymptotic profile could satisfy higher decay rates. The elaborate statement about the optimal decay rates is presented in [14]. The aim of this paper is to show that the optimal decay rates of the diffusion wave may attain by using the Fourier analysis and interpolation inequality as did in [13]. Notation Most of the notations we adopt in this paper are standard. For the sake of reading, we specify some notations here. Throughout the paper, we use C to denote generic constants which can change from line to line. When the dependence of the constant on some index or a function is important, we highlight it in the notation. Lp (R) (1 p ∞) denotes usual Lebesgue space with the norm f Lp (R) = ( R |f (x)|p dx)1/p , 1 p < ∞, as well as f L∞ (R) = supR |f (x)|. Lp (R, R 2 ), for 1 p ∞, denotes the usual Lebesgue space of R 2 -valued functions equipped with the norm (u, v) Lp (R,R 2 ) = u Lp (R) + v Lp (R) . Moreover, we denote by H l (R) the usual lth order Sobolev space with its norm
Z. Wang / J. Math. Anal. Appl. 319 (2006) 740–763
f H l (R) =
l i 2 ∂ f x
743
1/2 .
i=0
Z+ denotes the class of all positive integers and N = Z+ ∪ {0}.
2. Asymptotic profile and preliminaries To make the analysis easier, as did in [13,15], we discuss the system (1.1) for the case α = β and σ = 1. That is, we consider the following system: ψt = −(1 − α)ψ − θx + αψxx , (2.1) θt = −(1 − α)θ + νψx + (ψθ )x + αθxx , with initial data (ψ0 (x), θ0 (x)) which satisfies initial condition (1.4). To study the asymptotic behavior of the solution to system (2.1), we first need to find the appropriate asymptotic profile. As in [16], the solution of the system is expected to behave as those of the following linear system: ψ¯ t = −(1 − α)ψ¯ + α ψ¯ xx , (2.2) θ¯t = −(1 − α)θ¯ + α θ¯xx . Note that both equations in (2.2) are independent and have the same form. It then suf¯ x) = φ(t, ¯ x)e−(1−α)t , fices to solve one of them, say (2.2)1 . With the transformation ψ(t, Eq. (2.2)1 becomes a heat equation φ¯ t = α φ¯ xx .
(2.3)
Note that the linear diffusion equation (2.3) is invariant under the transformation (t, x) → (c2 t, cx) for any c > 0 and the function f (t + 1, x) is a solution to (2.3) too if f (t, x) is a solution of (2.3). Then we easily observe that (2.3) possesses the solution of the form
x ¯ := p(ξ ), −∞ < ξ < +∞, (2.4) φ(t, x) = p √ 1+t √ subject to the boundary condition p(±∞) = ψ± , where ξ = x/ 1 + t. Substituting (2.4) into (2.3) and solving the resulting equation yields that ψ − ψ− ¯ t) = p(ξ ) = √ + φ(x, 4πα(1 + t) Therefore
x −∞
¯ x) = e−(1−α)t (ψ+ − ψ− ) ψ(t,
x
−∞
where
x2 exp − G(x, t) = √ 4αt 4παt 1
exp −
y2 dy + ψ− . 4α(1 + t)
G(t + 1, y) dy + ψ− ,
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is the heat kernel function. Performing the same procedure as above, one has that x θ¯ (t, x) = e−(1−α)t (θ+ − θ− )
G(t + 1, y) dy + θ− .
−∞
Then it is straightforward to get ¯ x) → ψ± e−(1−α)t ψ(t,
as x → ±∞,
and θ¯ (t, x) → θ± e−(1−α)t
as x → ±∞.
¯ ¯ t) and their derivatives in We now consider the asymptotic behavior of ψ(x, t), θ(x, Lp (R). First for the heat kernel function G(t, x), it has the following properties. Lemma 2.1. When 1 p +∞, 0 l, k < +∞, we have l k 1 1 k ∂ ∂ G(t, ·) p Ct − 2 (1− p )−l− 2 . t x L (R) Employing the above lemma, we easily get the following results by simple calculations. ¯ Lemma 2.2. The solutions ψ(x, t) and θ¯ (x, t) to (2.2) satisfy the following properties: ¯ ·) L∞ (R) Ce−(1−α)t , ∂tl θ¯ (t, ·) L∞ (R) Ce−(1−α)t , l = 0, 1, 2, . . . ; (i) ∂tl ψ(t, (ii) for any p with 1 p +∞, it holds that l k 1 k ∂ ∂ ψ(t, ¯ ·) p C|ψ+ − ψ− |e−(1−α)t (1 + t) 2p − 2 , t x L (R) k = 1, 2, . . . , l = 0, 1, 2, . . . , l k 1 k ∂ ∂ θ¯ (t, ·) p C|θ+ − θ− |e−(1−α)t (1 + t) 2p − 2 , t x L (R) k = 1, 2, . . . , l = 0, 1, 2, . . . . Now let ¯ x), u(t, x) = ψ(t, x) − ψ(t, ¯ v(t, x) = θ (t, x) − θ (t, x), then system (2.1) can be recasted to the following reformulated equations: ut = −(1 − α)u + αuxx − vx − θ¯x , ¯ x + H (ψ, ¯ θ¯ ) vt = −(1 − α)v + αvxx + νux + (uv)x + (θ¯ u)x + (ψv) with initial data ¯ 0) → 0, u(x, 0) = u0 (x) = ψ0 (x) − ψ(x, ¯ v(x, 0) = v0 (x) = θ0 (x) − θ (x, 0) → 0, where
x → ±∞, x → ±∞,
(2.5)
(2.6)
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745
¯ θ¯ ) = ν ψ¯ x + ψ¯ θ¯x + θ¯ ψ¯ x . H (ψ, Zhu et al. [16] established the global existence, nonlinear stability and decay rate of a solution to (2.2) and (2.3) with some suitable restrictions on coefficients. The aim of this paper is to derive the optimal decay estimates of solutions to (2.5), (2.6). First, we cite the main results obtained in [16] as follows for later presentation. Lemma 2.3. [16, Theorem 3.1] Suppose that both δ = |ψ+ − ψ− | + |θ+ − θ− | and δ0 = u0 2H 2 (R) + v0 2H 2 (R) are sufficiently small. If (u0 , v0 ) ∈ H 2 (R, R 2 ), then for any 0 < α < 1, ν < 4α(1 − α), problem (2.5), (2.6) admits a unique global solution (u(t, x), v(t, x)) := (u, v) ∈ X(0, T ) satisfying u(t, ·)2 2 + v(t, ·)2 2 + H (R) H (R)
t
u(τ, ·)2
H 3 (R)
2 + v(τ, ·)H 3 (R) dτ
0
C(δ + δ0 ), and
(2.7)
sup (u, v)(t, x) + (ux , vx )(t, x) → 0,
as t → ∞.
(2.8)
x∈R
That is, there exists a unique global solution (ψ(t, x), θ (t, x)) of (2.1), (1.4) satisfying ¯ θ − θ¯ ) ∈ X(0, T ) (ψ − ψ, and
sup (ψ, θ )(t, x) + (ψx , θx )(t, x) → 0,
as t → ∞,
x∈R
where the solution space X(0, T ) is defined by X(0, T ) = (u, v) | u, v ∈ L∞ 0, T ; H 2 (R, R 2 ) ∩ L2 0, T ; H 3 (R, R 2 ) . Lemma 2.4. [16, Theorem 4.2] Suppose that (u(t, x), v(t, x)) is a solution to problem (2.5), (2.6) under the assumptions imposed in Lemma 2.3, then for any ν < 4α(1 − α), we have k ∂ 2 1 u(t, ·), v(t, ·) C(δ + δ0 ) 4 e−lt , k = 0, 1, 2, (2.9) 2 ∂x k 2 L (R,R ) where l is determined by c0 ν 2 , l = min (2 − )(1 − α), 2(1 − α) − α and ∈ (0, 2), c0 > 0 satisfy the relation 1 > 0, 2c0 α − (1−α) 2(1 − α) −
c0 ν 2 α
> 0.
(2.10)
(2.11)
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What follows are some well-known results. We present them here for later use. Lemma 2.5 (Convolution inequality [4]). By ∗ denotes the convolution. If u ∈ Lp (R) and v ∈ Lq (R), then u ∗ v ∈ Lr (R), where 1/r = 1/p + 1/q − 1, 1 p, q ∞, and u ∗ v Lr (R) u Lp (R) v Lq (R) . In particular, it follows that u ∗ v L∞ (R) u L∞ (R) v L1 (R)
and u ∗ v L1 (R) u L1 (R) v L1 (R) .
Lemma 2.6 (Interpolation inequality in Lp -space [1]). For any f ∈ Lq (R) ∩ Lr (R), if 1 q p r ∞ and 1/p = λ/q + (1 − λ)/r, then f ∈ Lp (R) and satisfies the following inequality: 1−λ f Lp (R) f λLq (R) f L r (R) .
Lemma 2.7 (Ehrling–Browder’s inequality [1]). Let v ∈ Lq (R N , R n ) and D m v ∈ Lr (R N , R n ) with 1 q, r ∞. Then for any integer j with 0 j m, there exists a constant C = C(m, p, N ), such that 1−α D j v Lp (R N ,R n ) C D m v αLr (R N ,R n ) v L q (R N ,R n ) ,
where p is determined by
j 1 m 1−α 1 = +α − + , p N r N q
j α 1. m
3. Linear analysis To investigate the optimal decay rates of solution to system (2.6), we first study the homogeneous linearized system corresponding to (2.5) ut = −(1 − α)u − vx + αuxx , (3.1) vt = −(1 − α)v + νux + αvxx , with initial data u(0, x), v(0, x) = u0 (x), v0 (x) .
(3.2)
Taking Fourier transform to (3.1), it becomes the following ordinary differential equation iξ uˆ uˆ 1 − α + αξ 2 , (3.3) =− vˆ vˆ t −iνξ 1 − α + αξ 2 with initial data u(0, ˆ x), v(0, ˆ x) = uˆ 0 (x), vˆ0 (x) ,
(3.4)
where uˆ and vˆ denotes the Fourier transform with the frequency ξ . Using the standard approaches to solve ODEs, we get the fundamental matrix to (3.3) which reads
Z. Wang / J. Math. Anal. Appl. 319 (2006) 740–763
e−(1−α+αξ M= 2
2 )t
e
√ νξ t
√
+ e−
1 √ i ν
νξ t
√ √ √ i ν e νξ t − e− νξ t
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√ √νξ t e − e− νξ t
e
√
νξ t
√
+ e−
νξ t
.
Hence, the solution of (3.3), (3.4) takes the form ⎧ 2 √ √ √ ⎪ e−(1−α+αξ )t √νξ t ⎪ ⎪ (vˆ0 + i ν uˆ 0 ) − e− νξ t (vˆ0 − i ν uˆ 0 ) , e √ ⎨ uˆ = 2i ν 2 )t ⎪ −(1−α+αξ √ ⎪ −√νξ t √ √ ⎪ ⎩ vˆ = e e (vˆ0 − i ν uˆ 0 ) + e νξ t (vˆ0 + i ν uˆ 0 ) . 2 Therefore, the inverse Fourier transform gives the solution of (3.1), (3.2) in the form √ √ 1 K− (t, x) ∗ (v0 + i νu0 ) − K+ (t, x) ∗ (v0 − i νu0 ) , u(t, x) = 2i √ ν (3.5) √ √ v(t, x) = 12 K+ (t, x) ∗ (v0 − i νu0 ) + K− (t, x) ∗ (v0 + i νu0 ) , where the kernel functions K± (t, x) are given by √ 2 √ 2 K± (t, x) = F −1 e−(1−α+αξ ± νξ )t = e−(1−α)t F −1 e−αξ t∓ νξ t
√
i ν x2 1 ν t exp ∓ x exp − , =√ exp − 1 − α − 4α 2α 4αt 4παt here F −1 denotes the inverse Fourier transform. From the explicit expression of the solution (3.5) and kernel function K± (t, x), we observe that the system (3.1) is stable if and only if ν < 4α(1 − α). Moreover, when p ∈ [1, +∞] and t → +∞, it follows that
x2 ν K± (t, ·) p = √ 1 t exp − exp − 1 − α − 4παt L (R) 4α 4αt Lp (R)
ν x2 − 12 −(1−α− 4α )t = O(1)t e exp − 4αt p L (R) = O(1)t
1 ν − 12 + 2p −(1−α− 4α )t
e
.
Furthermore, from Lemma 2.1, it is easy to derive that
∂ 1 ν K± (t, ·) ∂x p = √4παt exp − 1 − α − 4α t L (R)
√ x2 ∂ x2 i ν exp − + exp − × ∓ p 2α 4αt ∂x 4αt L (R) Ct
1 ν − 12 + 2p −(1−α− 4α )t
Ct
1 − 12 + 2p
e e
ν −(1−α− 4α )t
+ Ct
1 ν −1+ 2p −(1−α− 4α )t
e
.
Actually, using mathematical induction, we may easily prove that the above optimal decay rate holds for (∂ k /∂x k )K± (t, x) for any k ∈ N and p ∈ [1, ∞]. That is k ∂ 1 ν − 12 + 2p K (t, ·) e−(1−α− 4α )t . (3.6) p Ct ∂x k ± L (R)
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From the solution expression (3.5), we find that the decay rate of the solution of (3.1), (3.2) is indicated by the decay rates of K± (t, x). Namely, (3.6) holds for (u(t, x), v(t, x)) as well. We summarize in the following theorem. Theorem 3.1. Suppose (u0 , v0 ) ∈ L1 (R, R 2 ) ∩ Lp (R, R 2 ), then there exists a unique smooth solution (u(t, x), v(t, x)) to system (3.1), (3.2), which decays if and only if ν < 4α(1 − α). Moreover, for any k ∈ N and p ∈ [1, ∞], the following decay rate is optimal: k ∂ ν −1+ 1 C(1 + t) 2 2p e−(1−α− 4α )t . ∂x k u(t, ·), v(t, ·) p L (R,R 2 ) 4. Main results and proof In this section, we are devoted to proving that the solution of nonlinear system (2.5) has the same decay rate as that of the corresponding homogeneous linearized system (3.1). That is, the solution of system (2.5), (2.6) may attain the optimal decay rates. The main result is given in the following theorem. Theorem 4.1. Assume that ν < 4α(1 − α) and (ψ0 , θ0 ) ∈ H 2 (R, R 2 ) ∩ L1 (R, R 2 ). If both δ = |ψ+ − ψ− | + |θ+ − θ− |
and δ0 = u0 2H 2 (R) + v0 2H 2 (R)
are small enough, then for any k ∈ N and p with 1 p ∞, the solution of (2.5), (2.6) obtained in Lemma 2.3 may decay with the following optimal decay rate: k ∂ ν −1+ 1 u(t, ·), v(t, ·) C(1 + t) 2 2p e−(1−α− 4α )t . (4.1) ∂x k p L (R,R 2 ) To prove Theorem 4.1, we need the following lemmas. Lemma 4.2. For any m ∈ N, there exists constants Cm dependent of m, such that the solution (ψ(t, x), θ (t, x)) of system (2.5), (2.6) satisfies
m m ∂ u(t, x) 2 ∂ v(t, x) 2 dx + ∂x m ∂x m R
t
+
∂ m u(τ, x) ∂x m
2
+
∂ m v(τ, x) ∂x m
2 dx dτ
0 R
t
+
∂ m+1 u(τ, x) ∂x m+1
2
∂ m+1 v(τ, x) + ∂x m+1
2 dx dτ Cm .
(4.2)
0 R
Proof. We shall prove Lemma 4.2 by performing the mathematical induction. Indeed, the inequality in Lemma 2.3 has given (4.2) for m = 0, 1, 2. So we only need to prove (4.2)
Z. Wang / J. Math. Anal. Appl. 319 (2006) 740–763
749
holds for m 3. To this end, we assume that (4.2) is true for m k − 1 with k 3. Under this assumption, it follows that
i i ∂ u(t, x) 2 ∂ v(t, x) 2 dx Ci , i = 0, 1, . . . , k − 1; + (4.3) ∂x i ∂x i R
and t
∂ i u(t, x) ∂x i
2
+
2
∂ i v(t, x) ∂x i
dx dt Ci ,
i = 0, 1, . . . , k.
(4.4)
0 R 1/2
1/2
By Sobolev inequality f L∞ (R) f L2 (R) fx L2 (R) , it follows from (4.3) that i ∂ u(t, ·) ∂x i
i ∂ v(t, ·) + ∂x i
L∞ ([0,∞)×R)
L∞ ([0,∞)×R)
2Ci ,
(4.5)
for i = 0, 1, 2, . . . , k − 2. Next we are going to show that (4.2) holds for m = k. For this goal, we differentiate (2.5) k times with respect to x and multiply the first equation by ∂ k ψ(t, x)/∂x k and the second equation by c0 ∂ k θ (t, x)/∂x k and integrate the resulting identity over (x, t) ∈ R × [0, t] by parts to get the following inequality using Cauchy– Schwartz inequality
k k ∂ u(t, x) 2 ∂ v(t, x) 2 dx + c 0 ∂x k ∂x k R
t
+ 2(1 − α)
∂ k u(τ, x) ∂x k
2
+ c0
∂ k v(τ, x) ∂x k
2 dx dt
0 R
t
+ 2α
∂ k+1 u(τ, x) ∂x k+1
2
+ c0
∂ k+1 v(τ, x) ∂x k+1
2 dx dτ
0 R
k ∂ 2 k u0 (·), v0 (·) 2 ∂x L
2 + (2 − )(1 − α)
t
∂ k+1 θ¯ (τ, x) ∂x k+1
∂ k u(τ, x) ∂x k
0 R
t
+ α 0 R
t
∂ k u(τ, x) ∂x k
2 dx dτ
0 R
2 dx dτ
0 R
t
+ (1 − α)
1 + (2 − )(1 − α) 2 (R)
∂ k+1 u(τ, x) ∂x k+1
2
2
1 dx dτ + (1 − α)
c2 ν 2 dx dτ + 0 α
t
0 R
t
∂ k+1 v(τ, x) ∂x k+1
0 R
∂ k v(τ, x) ∂x k
2 dx dτ
2 dx dτ
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Z. Wang / J. Math. Anal. Appl. 319 (2006) 740–763
t − 2c0
∂ k+1 v(τ, x) ∂k u(τ, x)v(τ, x) dx dτ k ∂x ∂x k+1
0 R
t − 2c0
∂ k+1 v(τ, x) ∂k ¯ (τ, x)u(τ, x) θ dx dτ ∂x k ∂x k+1
0 R
t − 2c0
∂ k+1 v(τ, x) ∂k ¯ ψ(τ, x)v(τ, x) dx dτ ∂x k ∂x k+1
0 R
t + 2c0
∂k ∂ k v(τ, x) ¯ x + ψ¯ θ¯x + θ¯ ψ¯ x ) (ν ψ dx dτ. ∂x k ∂x k
(4.6)
0 R
Set c02 ν 2 1 and d3 = 2c0 α − . α (1 − α) Shuffling the terms in (4.6) and using Cauchy–Schwartz inequality yields that d1 = (2 − )(1 − α),
∂ k u(t, x) ∂x k
2
d2 = 2c0 (1 − α) −
+ c0
∂ k v(t, x) ∂x k
2
1 dx + d1 2
R
t
∂ k u(τ, x) ∂x k
2 dx dτ
0 R
t
+ d2
∂ k v(τ, x) 2 ∂x k
t
dx dt + (2 − )α
0 R
t
+ d3
t − 2c0
L (R)
− 2c0
∂ k+1 v(τ, x) ∂k u(τ, x)v(τ, x) dx dτ ∂x k ∂x k+1
0 R
∂ k+1 v(τ, x) ∂k θ¯ (τ, x)u(τ, x) dx dτ k ∂x ∂x k+1
0 R
t − 2c0
∂ k+1 v(τ, x) ∂k ¯ ψ(τ, x)v(τ, x) dx dτ k ∂x ∂x k+1
0 R
2 + d1
t
0 R t
+ 2c0 0 R
dx dτ
dx dτ
0 R
k ∂ 2 k u0 (·), v0 (·) 2 ∂x
2
0 R
∂ k+1 v(τ, x) 2 ∂x k+1
t
∂ k+1 u(τ, x) ∂x k+1
∂ k+1 θ¯ (τ, x) ∂x k+1
2 dx dτ
∂k ∂ k v(τ, x) ¯ x + ψ¯ x θ¯x + θ¯ ψ¯ x ) (ν ψ dx dτ ∂x k ∂x k
Z. Wang / J. Math. Anal. Appl. 319 (2006) 740–763
k ∂ 2 u (·), v (·) 0 ∂x k 0 2 L
+
+
+
+
+
6c02 d3 6c02 d3 6c02 d3 2 d1
t
d3 + 2 (R)
t
∂ k+1 v(τ, x) ∂x k+1
2 dx dτ
0 R
∂k u(τ, x)v(τ, x) k ∂x
2 dx dτ
0 R
t
∂k θ¯ (τ, x)u(τ, x) k ∂x
2 dx dτ
0 R
t
∂k ¯ ψ(τ, x)v(τ, x) k ∂x
2 dx dτ
0 R
t
¯ x) ∂ k+1 θ(τ, ∂x k+1
2 dx dτ +
d2 2
0 R
2c02 d2
751
t
t
∂ k v(τ, x) ∂x k
2 dx dτ
0 R
(ν ψ¯ x + ψ¯ θ¯x + θ¯ ψ¯ x )2 dx dτ.
(4.7)
0 R
Rearrangement of (4.7) gives that
∂ k u(t, x) ∂x k
2
+ c0
∂ k v(t, x) ∂x k
2 dx
R
1 + d1 2
t
∂ k u(τ, x) ∂x k
2
d2 dx dτ + 2
0 R
t
+ (2 − )α
∂ k+1 u(τ, x) ∂x k+1
2
d3 dx dτ + 2
0 R
L
6c2 + 0 d3
6c2 + 0 d3 (R)
t
t
t
∂ k+1 v(τ, x) ∂x k+1
∂k u(τ, x)v(τ, x) k ∂x
0 R
∂k θ¯ (τ, x)u(τ, x) k ∂x
2 dx dτ
0 R
0 R
dx dt 2
0 R
k ∂ 2 k u0 (·), v0 (·) 2 ∂x 6c2 + 0 d3
2
0 R
t
t
∂ k v(τ, x) ∂x k
∂k ¯ ψ(τ, x)v(τ, x) k ∂x
2 dx dτ
2 dx dτ
dx dτ
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Z. Wang / J. Math. Anal. Appl. 319 (2006) 740–763
2 + d1
t
∂ k+1 θ¯ (τ, x) ∂x k+1
2
0 R
k ∂ 2 u (·), v (·) 0 ∂x k 0 2
+
2c2 dx dτ + 0 d2 6c02
M1 +
d3
L (R)
t
(ν ψ¯ x + ψ¯ x θ¯x + θ¯ ψ¯ x )2 dx dτ
0 R
6c02 d3
M2 +
6c02 2c2 2 M3 + M4 + 0 M5 . d3 d1 d2 (4.8)
We proceed to show that M1 , M2 , M3 , M4 and M5 are bounded, respectively. Indeed, we derive from (4.3)–(4.5) that t
M1 =
∂k u(τ, x)v(τ, x) ∂x k
2 dx dτ
0 R
2 t k i k ∂ u(τ, x) ∂ k−i v(τ, x) = dx dτ i ∂x i ∂x k−i 0 R
t =
i=0
k−2 i k ∂ u(τ, x) ∂ k−i v(τ, x) ∂ k−1 u(τ, x) +k vx (τ, x) i k−i i ∂x ∂x ∂x k−1
0 R
i=0
2 ∂ k u(τ, x) + v(τ, x) dx dτ ∂x k
2 k−2 i ∂ u(t, ·) k 2 i ∂x i ∞
t
L ([0,∞)×R)
i=1
t
2 + k vx (t, ·) ∞
L ([0,∞)×R)
∂ k−i v(τ, x) ∂x k−i
2 dx dτ
0 R
∂ k−1 u(τ, x) ∂x k−1
2 dx dτ
0 R
t
2 + C v(t, ·) ∞
L ([0,∞)×R)
∂ k u(τ, x) ∂x k
2 dx dτ
0 R
Ck .
(4.9)
Furthermore, the application of Lemma 2.2 and the assumption (4.4) give the estimates t
M2 =
∂k θ¯ (τ, x)u(τ, x) k ∂x
2 dx dτ
0 R
=
2 t k i ¯ x) k ∂ u(τ, x) ∂ k−i θ(τ, 0 R
i=0
i
∂x i
∂x k−i
dx dτ
Z. Wang / J. Math. Anal. Appl. 319 (2006) 740–763
2 k k−i ¯ k ∂ θ (τ, x) 2 ∞ k−i i ∂x
t
L ([0,∞)×R)
i=1
∂ i u(τ, x) ∂x i
753
2 dx dτ
0 R
Ck .
(4.10)
Moreover, the same approach as above also shows that M3 is bounded. From Lemma 2.2, it is straightforward to verify that M4 and M5 are all bounded. Thus we show that (4.2) is true for m = k. By the mathematical induction, (4.2) is then proved. 2 Corollary 4.3. For any k ∈ N, the solution (u(t, x), v(t, x)) of system (2.5), (2.6) satisfies the following exponential decay estimates: k ∂ u(t, x), v(t, x) Ce−at (4.11) 2 ∂x k 2 L (R,R ) and
k ∂ ∂x k u(t, x), v(t, x)
L∞ (R,R 2 )
Ce−at/2 ,
(4.12)
where a < l can be chosen as close as to l as wished. Proof. Taking account of (2.9) and (4.2), we get by Ehrling–Browder’s inequality k ∂ ∂x k u(t, x), v(t, x) 2 L (R,R 2 ) q+k ∂ k/(q+k) u(t, x), v(t, x) q/(q+k) C q+k u(t, x), v(t, x) 2 L2 (R,R 2 ) ∂x L (R,R 2 ) Ce−qlt/2(q+k) . Thus (4.11) is obtained by defining a = ql/(2(q + k)) and letting q be big enough. And (4.12) is a consequence of (4.11) by the use of Sobolev inequality. We then complete the proof. 2 To prove Theorem 4.1, we prove the following result for later presentation. Lemma 4.4. Let γ and η be positive numbers, t > 0. Then t
(1 + t − s)−γ e−ηs ds C(1 + t)−γ .
0
Proof. Dividing the integral t
t
0 (1 + t
− s)−γ e−ηs ds = (1 + t)−γ
0 (1 + t
t
0
− s)−γ e−ηs ds by (1 + t)−γ , we derive that 1+t −s 1+t
−γ
e−ηs ds
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Z. Wang / J. Math. Anal. Appl. 319 (2006) 740–763
t
= 0
s 1+ 1+t −s
γ
which ends the proof with the fact that the integral ∞ γ −ηs ds. 2 0 (1 + s) e
e
−ηs
t
t ds
0 (1
(1 + s)γ e−ηs ds,
0
+ s)γ e−ηs ds is bounded by
The following Gronwall’s inequality is devised for the optimal decay rates. The proof may be performed by standard approach and is omitted here. Lemma 4.5. Assume that the nonnegative function g(t) satisfies −μ −rt
g(t) N1 (1 + t)
e
t
e−δs g(s) ds,
+ N2 0
with nonnegative constants N1 , N2 , μ, r and δ. Then it follows that ∞ g(t) N1 (1 + t)−μ e−rt exp N2
e−δs ds C(1 + t)−μ e−rt .
0
With the above lemmas in hand, we are in the position to prove Theorem 4.1. Proof of Theorem 4.1. We shall prove inequality (4.1) by using interpolation inequality. The proof may be spitted into four steps. In the first step, we derive the solution representation for system (2.5), (2.6) using Fourier analysis. In the second step, we deduce the L∞ -estimate of the solution. The L1 -estimate will be given in the third step. At the end, we use the interpolation inequality to get the Lp -estimate. Step 1. Taking the Fourier transform to (2.5) and (2.6), we obtain after the integration of the linear terms √ √ √ e−(1−α+αξ )t √νξ t vˆ0 + i ν uˆ 0 − e− νξ t vˆ0 − i ν uˆ 0 e √ 2i ν t √ √ 1 2 x ds + √ e−(1−α+αξ )(t−s) e νξ(t−s) − e− νξ(t−s) (uv) 2i ν 2
u(t, ˆ x) =
0
−
1 2
t
e−(1−α+αξ
2 )(t−s)
√ √νξ(t−s) + e− νξ(t−s) θˆ¯ x ds e
0
1 + √ 2i ν
t 0
and
e−(1−α+αξ
2 )(t−s)
√ √νξ(t−s) e − e− νξ(t−s)
¯ x + ν ψ¯ x + ψ¯ θ¯x + θ¯ ψ¯ x ds × F (θ¯ u)x + (ψv)
Z. Wang / J. Math. Anal. Appl. 319 (2006) 740–763
755
√ √ √ e−(1−α+αξ )t √νξ t e vˆ0 + i ν uˆ 0 + e− νξ t vˆ0 − i ν uˆ 0 v(t, ˆ x) = 2 t √ √ 1 2 x ds e−(1−α+αξ )(t−s) e νξ(t−s) + e− νξ(t−s) (uv) + 2 2
0
√ t √ √ i ν 2 e−(1−α+αξ )(t−s) e νξ(t−s) − e− νξ(t−s) θˆ¯ x ds − 2 0
+
1 2
t 0
e−(1−α+αξ
2 )(t−s)
√ √νξ(t−s) e + e− νξ(t−s)
¯ x + ν ψ¯ x + ψ¯ θ¯x + θ¯ ψ¯ x ds, × F (θ¯ u)x + (ψv)
where F denote the Fourier transform. With the convolution property, the solution is then obtained in the following form under the inverse Fourier transform: √ √ 1 u(t, x) = √ K− (t, x) ∗ v0 + i νu0 − K+ (t, x) ∗ v0 − i νu0 2i ν t ∂ 1 K− (t − s, x) − K+ (t − s, x) ∗ u(s, x)v(s, x) ds + √ ∂x 2i ν 0
−
1 2
t
∂ K− (t − s, x) − K+ (t − s, x) ∗ θ¯x (s, x) ds ∂x
0
1 + √ 2i ν
t
∂ ¯ + θ¯ u) ds K− (t − s, x) − K+ (t − s, x) ∗ (ψv ∂x
0
+
1 √ 2i ν
t
K− (t − s, x) − K+ (t − s, x) ∗ (ν ψ¯ x + ψ¯ θ¯x + θ¯ ψ¯ x ) ds
0
and v(t, x) =
√ √ 1 K− (t, x) ∗ v0 + i νu0 + K+ (t, x) ∗ v0 − i νu0 2 t ∂ 1 K− (t − s, x) + K+ (t − s, x) ∗ u(s, x)v(s, x) ds + 2 ∂x 0
√ t i ν K− (t − s, x) − K+ (t − s, x) ∗ θ¯x (s, x) ds + 2 0
(4.13)
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Z. Wang / J. Math. Anal. Appl. 319 (2006) 740–763
1 + 2
t
∂ ¯ + θ¯ u) ds K− (t − s, x) + K+ (t − s, x) ∗ (ψv ∂x
0
1 + 2
t
K− (t − s, x) + K+ (t − s, x) ∗ (ν ψ¯ x + ψ¯ θ¯x + θ¯ ψ¯ x ) ds.
(4.14)
0
Step 2 (L1 -estimate). By the solution expression (4.13) and (4.14), for i = 0, 1, 2, . . . , we obtain from Lemma 2.5 after some rearrangement i ∂ ∂x i u(t, ·), v(t, ·) 1 L (R,R 2 ) i ∂ u0 (·), v0 (·) 1 C i K+ (t, ·), K− (t, ·) L (R,R 2 ) ∂x L1 (R,R 2 ) t ∂ K +C (t − s, ·), K (t − s, ·) − ∂x +
L1 (R,R 2 )
0
t i+1 ∂ + C i+1 K+ (t − s, ·), K− (t − s, ·) ∂x 0
i ∂ u(s, ·)v(s, ·) ∂x i
ds L1 (R)
L1 (R,R 2 )
¯ x)v(s, ·) + θ¯ (s, ·)u(s, ·) 1 ds × ψ(s, L (R) t i ∂ + C i K+ (t, ·), K− (t, ·) ∂x 0
L1 (R,R 2 )
θ¯x + ν ψ¯ x + ψ¯ θ¯x + θ¯ ψ¯ x L1 (R) ds
= S1 + S 2 + S3 + S 4 ,
(4.15)
where S1 , S2 , S3 and S4 denote the terms on the right-hand side of inequality (4.5) from the first to the fourth term, respectively. We shall estimate them term by term next. Indeed, in view of the initial condition (u0 , v0 ) ∈ L1 (R, R 2 ), it is clear from (3.6) that ν
S1 Ce−(1−α− 4α )t .
(4.16)
Moreover, due to (3.6) and (4.12), it follows that t S2 C
e
i i ∂ u(s, ·) ∂x i
ν −(1−α− 4α )(t−s) − as 2
e
0
t C
e 0
− as 2
i i ∂ u(s, ·) ∂x i j =0
ds
L1 (R)
j =0
ds.
L1 (R)
Taking account of Lemma 2.4 and Minkowski inequality, we have that
(4.17)
Z. Wang / J. Math. Anal. Appl. 319 (2006) 740–763
t S3 C
757
ν ¯ ·) ∞ v(s, ·) 1 e−(1−α− 4α )(t−s) ψ(s, L (R) L (R)
0
+ θ¯ (s, ·)L∞ (R) u(s, ·)L1 (R) ds t C
e−(1−α)s u(s, ·), v(s, ·) L1 (R,R 2 ) ds.
(4.18)
0
Similarly, we deduce from Lemma 2.2 and (3.6) by Minkowski inequality that t S4 C
ν ¯ ·) ∞ θ¯x (s, ·) 1 e−(1−α− 4α )(t−s) θ¯x (s, ·)L1 (R) + ψ(s, L (R) L (R)
0
+ θ¯ (s, ·)L∞ (R) ψ¯ x (s, ·)L1 (R) ds Ce
ν −(1−α− 4α )t
Ce
ν −(1−α− 4α )t
t
ν
e− 4α s ds
0
(4.19)
,
here the last inequality results from the fact that t e
ν − 4α s
∞ ds
0
ν
e− 4α s ds < ∞.
0
Then the substitution of (4.16)–(4.19) into (4.15) yields i ∂ ∂x i u(t, ·), v(t, ·) 1 L (R,R 2 ) t i j ∂ − as ν −(1−α− 4α )t −(1−α)s 2 e Ce +C +e ds. ∂x j u(s, ·), v(s, ·) 1 L (R,R 2 ) j =0
0
(4.20)
Defining Y (t) =
k j ∂ u(t, ·), v(t, ·) ∂x j j =0
L1 (R,R 2 )
and summing (4.20) over i = 0, 1, 2, . . . , k, we have that Y (t) Ck e
ν −(1−α− 4α )t
t +C
− as e 2 + e−(1−α)s Y (s) ds.
0
Therefore, applying Gronwall’s inequality to (4.21) gives us that ν
Y (t) Ck e−(1−α− 4α )t .
(4.21)
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Z. Wang / J. Math. Anal. Appl. 319 (2006) 740–763
This implies that for any k ∈ N, it follows that k ∂ ν u(t, ·), v(t, ·) Ck e−(1−α− 4α )t . ∂x k 1 2 L (R,R )
(4.22)
Step 3 (L∞ -estimate). To derive L∞ -estimate, we first claim that for any Z(t, ·) ∈ it holds that k ∂ K± (t − s, ·) 1 ν ∗ Z(s, ·) C(1 + t − s)− 2 e−(1−α− 4α )(t−s) ∞ k ∂x L (R) × Z(s, ·)L1 (R) + Z(s, ·)L∞ (R) . (4.23)
L1 (R) ∩ L∞ (R),
Actually, from (3.6), we get the following estimate by using convolution inequality k k ∂ K± (t − s, ·) ∂ K± (t − s, ·) Z(s, ·) 1 ∗ Z(s, ·) C L (R) ∂x k ∂x k L∞ (R) L∞ (R) 1 ν C(t − s)− 2 e−(1−α− 4α )(t−s) Z(s, ·)L1 (R) . (4.24) On the other hand, if follows that k k ∂ K± (t − s, ·) ∂ K± (t − s, ·) Z(s, ·) ∞ ∗ Z(s, ·) C L (R) k k ∂x ∂x L∞ (R) L1 (R) ν Ce−(1−α− 4α )(t−s) Z(s, ·)L∞ (R) . (4.25) Then the combination of (4.24) and (4.25) implies (4.23). This proves the claim. Hence, from (4.13), (4.14) and (4.23), we deduce that i ∂ ∂x i u(t, ·), v(t, ·) ∞ L (R,R 2 ) ν − 12 −(1−α− 4α )t u0 (·), v0 (·) 1 u0 (·), v0 (·) ∞ C(1 + t) e 2 + t +C
ν − 12 −(1−α− 4α )(t−s)
(1 + t − s) 0
e
i ∂ + i u(s, ·)v(s, ·) ∂x
t +C
i ∂ ∂x i u(s, ·)v(s, ·)
L (R,R 2 )
L (R,R )
L1 (R)
L∞ (R)
ds
1 ν ¯ + θ¯ u L1 (R) + ψv ¯ + θ¯ u L∞ (R) ds (1 + t − s)− 2 e−(1−α− 4α )(t−s) ψv
0
t +C 0
1 ν (1 + t − s)− 2 e−(1−α− 4α )(t−s) θ¯x + ν ψ¯ x + ψ¯ θ¯x + θ¯ ψ¯ x L1 (R)
+ θ¯x + ν ψ¯ x + ψ¯ θ¯x + θ¯ ψ¯ x L∞ (R) ds
= I1 + I2 + I3 + I4 .
(4.26)
Z. Wang / J. Math. Anal. Appl. 319 (2006) 740–763
759
Next, we shall estimate I1 , I2 , I3 and I4 , respectively. In fact, due to the initial condition (u0 , v0 ) ∈ H 2 (R, R 2 ) and Sobolev inequality
∂u0 (·) ∂v0 (·) 1/2 u0 (·), v0 (·) ∞ u0 (·), v0 (·) 1/2 , 2 , L (R) L2 (R) ∂x ∂x L (R) we know that (u0 (·), v0 (·)) L∞ (R) is bounded. Note that (u0 , v0 ) ∈ L1 (R, R 2 ), we have 1
ν
I1 C(1 + t)− 2 e−(1−α− 4α )t .
(4.27)
Moreover, in view of (4.12) and (4.22), it follows that i i ∂ ∂ ∂x i u(s, ·)v(s, ·) 1 + ∂x i u(s, ·)v(s, ·) ∞ L (R) L (R) j j i i ∂ u(s, ·) ∂ v(s, ·) ∂x j ∞ ∂x j 1 L (R) L (R) j =0
j =0
i j ∂ u(s, ·)
+
j =0
i j ∂ v(s, ·) ∂x j
∂x j
L∞ (R) j =0
L∞ (R)
i j ∂ u(s, ·) −(1−α− ν )s − as 4α 2 C e +e ∂x j ∞ . L (R)
(4.28)
j =0
Therefore, I2 is bounded as t I2 C
1 ν ν as (1 + t − s)− 2 e−(1−α− 4α )(t−s) e−(1−α− 4α )s + e− 2
0
×
i j ∂ u(s, ·) ∂x j
L∞ (R)
j =0
t C
ν
ds
as
e−(1−α− 4α )s + e− 2
i j ∂ u(s, ·) ∂x j
L∞ (R)
j =0
0
ds.
Applying Minkowski inequality, we bound I3 from Lemma 2.2 and (4.22) as t I3 C
1
ν
(1 + t − s)− 2 e−(1−α− 4α )(t−s)
0
¯ L∞ (R) v L1 (R) + θ ¯ L∞ (R) u L1 (R) ds × ψ t +C 0
1
ν
(1 + t − s)− 2 e−(1−α− 4α )(t−s)
¯ L∞ (R) v L∞ (R) + θ ¯ L∞ (R) u L∞ (R) ds × ψ
(4.29)
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Z. Wang / J. Math. Anal. Appl. 319 (2006) 740–763
t C
1
ν
ν
(1 + t − s)− 2 e−(1−α− 4α )(t−s) e−(1−α− 4α )s e−(1−α)s ds
0
t +C
1
ν
(1 + t − s)− 2 e−(1−α− 4α )(t−s) e−(1−α)s
0
× v(s, ·)L∞ (R) + u(s, ·)L∞ (R) ds ν − 12 −(1−α− 4α )t
C(1 + t)
e
t +C
e−(1−α)s u(s, ·), v(s, ·) L∞ (R,R 2 ) ds. (4.30)
0
Similarly, we bound I4 from Lemmas 2.2 and 4.4 as t I4 C
1
ν
(1 + t − s)− 2 e−(1−α− 4α )(t−s) e−(1−α)s ds
0 1
ν
C(1 + t)− 2 e−(1−α− 4α )t .
(4.31)
Thus, inserting (4.27), (4.29), (4.30) and (4.31) into (4.26) yields i ∂ ∂x i u(t, ·), v(t, ·) ∞ L (R,R 2 ) ν − 12 −(1−α− 4α )t
C(1 + t)
e
t +C 0
t +C
i j −(1−α− ν )s a ∂ u(s, ·) − s 4α e +e 2 ∂x j ∞ ds L (R) j =0
e−(1−α)s u(s, ·), v(s, ·) L∞ (R,R 2 ) ds.
(4.32)
0
Summing (4.32) over i = 0, 1, 2, . . . , k and defining k j ∂ Y˜ (t) = u(t, ·), v(t, ·) , ∂x j ∞ L (R,R 2 ) j =1
we end up with Y˜ (t) C(1 + t)− 2 e−(1−α− 4α )t + C 1
ν
t
−(1−α− ν )s a 4α e + e− 2 s + e−(1−α)s Y˜ (s) ds.
0
Thus, the application of Gronwall’s inequality to (4.33) leads to Y˜ (t) C(1 + t)− 2 e−(1−α− 4α )t . 1
ν
That is, for any k ∈ N, it holds that
(4.33)
Z. Wang / J. Math. Anal. Appl. 319 (2006) 740–763
k ∂ ∂x k u(t, ·), v(t, ·)
1
L∞ (R,R 2 )
761
ν
C(1 + t)− 2 e−(1−α− 4α )t .
Step 4 (Lp -estimate). Finally, with the L1 -estimate and L∞ -estimate in hand, for any p ∈ [1, ∞], we use the interpolation inequality to derive the Lp -estimate in the form k ∂ u(t, ·), v(t, ·) ∂x k p L (R,R 2 ) k ∂ (p−1)/p ∂ k 1/p u(t, ·), v(t, ·) u(t, ·), v(t, ·) k ∞ 1 ∂x k L (R,R 2 ) ∂x L (R,R 2 ) 1 ν ν (p−1)/p −(1−α− )t 1/p 4α e C (1 + t)− 2 e−(1−α− 4α )t 1 ν − 12 + 2p −(1−α− 4α )t
C(1 + t)
e
This completes the proof.
.
2
5. Concluding comments For the special case where α = β and σ = 1, under the assumption ν < 4α(1 − α), Zhu et al. [16] established the global existence, non-stability and decay estimates of the solution to system (2.2) and (2.3) which are cited as Lemmas 2.3 and 2.4. In the present paper, the optimal decay rates (4.1) was obtained under the same assumption on coefficients. Due to the nonlinearity, the initial data is required to be small enough. Indeed, the same analysis as in [16] applies to the general system (1.1) with initial condition (1.4). But the optimal decay rates of the solution to the general system (1.1) is still unknown in the present paper. We summarize this in the following theorem. Theorem 5.1. Suppose that both δ = |ψ+ − ψ− | + |θ+ − θ− | and δ0 = u0 2H 2 (R) + v0 2H 2 (R) are small enough. For some positive constants C0 > 0 and ∈ (0, 2), if the coefficients satisfy the relations 2C0 β −
σ2 > 0, (σ − α)
1−β −
C0 ν 2 > 0, 2α
and initial data (ψ0 − ψ¯ 0 , θ0 − θ¯0 − θ˜0 ) ∈ H 2 (R, R 2 ) then for any ν subject to √ 4 αβ(1 − β)(σ − α) ν< σ
(5.1)
(5.2)
and constant
C0 ν 2 l = min (2 − )(σ − α), 2 1 − β − , 2α
(5.3)
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Z. Wang / J. Math. Anal. Appl. 319 (2006) 740–763
system (1.1) with initial condition (1.4) admits a unique global solution which decays exponentially for any nonnegative integers k in the L2 -norm form k 2 ∂ ¯ ¯ ˜ Ce−lt . ∂x k (ψ − ψ, θ − θ − θ )(t, x) 2 L (R,R 2 ) Remark 1. It is worthwhile to point out here that the set of constants C0 satisfying (5.1) is not empty. In fact, we may give an algorithm to determine it by the following steps: Step 1. Step 2. Step 3. Step 4.
√ Define k with 0 < k < 1 by νσ = k αβ(1 − β)(σ − α) . Choose with k2 < < 2. Choose λ with λ > 0. Define C0 by
σ2 1 2α(1 − β) C0 = . +λ 1 + λ 2β(σ − α) ν2
Remark 2. By taking into account (5.1) with ∈ (0, 2), it is straightforward to derive that √ 4 αβ(1 − β)(σ − α) , ν< σ which gives (5.2). As we perform the same procedure as in [16] to deduce the order of decay rates of the solution, the condition (5.3) will come up. Remark 3. For the general system (1.1), the optimal decay rates of the solution is not obtained yet since the calculation is too complicated to derive the desired results. It is still interesting to develop novel ideas or computational techniques to deal with such a problem.
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