In this article, we investigate the initial-value problem for the sixth-order Boussinesq equation with fourth order dispersion term

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ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

ASYMPTOTIC BEHAVIOR OF THE SIXTH-ORDER BOUSSINESQ EQUATION WITH FOURTH-ORDER

DISPERSION TERM

YU-ZHU WANG, YANSHUO LI, QINHUI HU Communicated by Hongjie Dong

Abstract. In this article, we investigate the initial-value problem for the sixth-order Boussinesq equation with fourth order dispersion term. Existence of a a global solution and asymptotic behavior in Morrey spaces are established under suitable conditions. The proof is mainly based on the decay properties of the solutions operator in Morrey spaces and the contraction mapping principle.

1. Introduction

In this article, we investigate the initial-value problem for the sixth-order Boussi- nesq equation with fourth-order dispersion term

u_tt−a∆u_tt−2b∆u_t−α∆³u+β∆²u−∆u= ∆f(u) (1.1) with initial value

u(x,0) =u0(x), ut(x,0) = ΛU1(x), x∈Rⁿ. (1.2) Here u = u(x, t) is the unknown function of x = (x1, . . . , xn) ∈ Rⁿ and t > 0,

∆utt denotes the dispersion term,a, b, α, β are positive constants. The nonlinear termf(u) =O(u²) and Λ = (−∆)^1/2. The initial valueu0(x) andU1(x) are given functions.

Zhang et al. [25] investigated the first initial boundary value problem for (1.1) withf(u) =u²in a unit circle. The existence and the uniqueness of strong solution were established and the solution was constructed in the form of series in the small parameter present in the initial conditions. The long-time asymptotics was also obtained in the explicit form. The author considered the initial-boundary value problem for (1.1) in the unit ballB ⊂R³, similar results were established in [8].

Wang and Wang [18] proved the global existence and asymptotic decay estimates of solutions to problem (1.1), (1.2) withL¹initial data. Their proof is based on the contraction mapping principle and makes use of the sharp decay estimates for the linearized problem. When n≥ 2, global existence and optimal decay estimate of solutions to (1.1), (1.2) withL²data were established by Wang and Zhang [19]. By constructing a class of special initial value, Wang [16] proved that global existence

2010Mathematics Subject Classification. 35L30, 35B40.

Key words and phrases. Sixth order Boussinesq equation; Morrey spaces; global solution;

decay estimate.

2018 Texas State University.c

Submitted November 28, 2017. Published September 6, 2018.

1

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and faster decay estimate of solutions to (1.1), (1.2). [12] proved that the Cauchy problem for (1.1) is globally well-posed. Under certain conditions, they also proved that the global solution decays exponentially to zero in the infinite time limit.

Very recently, the well-posedness of global solutions and blow-up of solutions were obtained by Wang [17]. Moreover, the asymptotic behavior of the solution was established by the multiplier method.

If the fourth-order dispersion term ∆u_ttis neglected, (1.1) is reduced to the sixth- order Boussinesq equation with damped term. Guo and Fang [6] established global existence and pointwise estimates of classical solutions by virtue of the Fourier analysis and Greens function. If the damped term ∆utis also neglected, then it is reduced to the classical sixth-order Boussinesq equation that models the nonlinear lattice dynamics in elastic crystals [10]. The study of the classical sixth-order Boussinesq equation has a long history and lots of interesting results have been established, we may refer to [1, 2, 3, 4, 13, 14] for local well-posedness, global well-posedness, stability of solitary waves and blow-up and so on. For other type of higher order hyperbolic equation, we may refer to [15, 20, 21, 22, 23, 24] and references therein.

It is well known that the Morrey spaceMp,q generalizes the Lebesgue spaceL^q. For (1.1), there are few results about global existence and asymptotic behavior in Morrey spaceMp,q. Our main purpose is to investigate global existence and decay estimates of solutions to problem (1.1), (1.2) in Morrey spaces. More precisely, we prove that problem (1.1), (1.2) has a unique global solution in Morrey spaces under suitable conditions. Moreover, decay estimates of this solution are also established.

We state our main results as follows:

Theorem 1.1. Let 1 ≤p≤q1 ≤n < mq1, q1 ≤q2 and m is a positive integer.

Assume that for0≤k≤m,∇^ku0,∇^(k−1)⁺U1∈Mp,q₁∩Mp,q₂. Put E0=

m

X

k=0

hk∇^ku0k_M_p,q

1∩Mp,q2 +k∇^(k−1)⁺U1k_M_p,q

1∩Mp,q2

i .

Then there exists ₀ > 0, such that if E₀ ≤₀, problem (1.1), (1.2) has a unique global solution usuch that

∇^ku∈C([0,∞);Mp,q1∩ Mp,q2), ∇^l∂_tu∈C([0,∞);Mp,q1∩ Mp,q2).

Moreover, we have the following decay estimates:

k∇^ku(t)k_M_p,q

1 ≤CE₀(1 +t)^−k/2, (1.3) k∇^ku(t)k_M_p,q

2 ≤CE0(1 +t)⁻^k²⁻ⁿ²⁽^q¹¹⁻^q¹²⁾, (1.4) k∇^l∂tu(t)k_M_p,q

1 ≤CE0(1 +t)⁻^l+1² , (1.5) k∇^l∂tu(t)kMp,q2 ≤CE0(1 +t)⁻^l+1² ⁻ⁿ²⁽^q¹¹⁻^q¹²⁾, (1.6) where0≤l≤m−2in (1.5)and (1.6).

Remark 1.2. If p=q1 =q2 = 2, thenMp,q₁ =Mp,q₂ =L², therefore Theorem 1.1 reduces to [19, Theorem 1]. Ifp=q1= 1, then Theorem 1.1 gives the existence and decay estimate of global solutions to problem (1.1), (1.2) in L¹ space. In a word, the results obtained in this paper generalize the results in [18] and [19].

There are two goals in this paper. Firstly, study the wave equation with damped term in Morrey spaces. To the best of our knowledge, there are only a few results

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in this setting, since the classical energy method used in Sobolev space can not be applied in Morrey spaces. Secondly, we hope that the method used here provide an idea for studying hyperbolic equations with damping terms and related models in Morrey spaces.

The plan of the paper is as follows. Firstly, we recall the definition of Mor- rey spaces and state some useful lemmas in Section 2. Section 3 is devoted to establish the decay estimate of the solutions operator in Morrey spaces. Finally, global existence and decay estimates are proved by Banach fixed point theorem in Section 4.

Notation: The Fourier transform of a functionuis defined by bu(ξ) =F[u](ξ) :=

Z

Rⁿ

e^−iξ·xu(x)dx.

We denote its inverse transform byF⁻¹.

The Morrey spaceM_p,q(1≤p≤q≤ ∞(q=∞, p < q)) is defined as the set of functionsf ∈L^p(Rⁿ) such that

kfk_M_p,q= sup

z∈Rⁿ

sup

r>0

rⁿ⁽¹^q⁻¹^p⁾Z

B(z,r)

|f(y)|^pdy1/p

= sup

z∈Rⁿ

sup

r>0

rⁿ⁽¹^q⁻¹^p⁾kfk_Lp(B(z,r)) <∞.

(1.7)

HereB(z, r) denotes the open ball inRⁿ with radiusrcentered atz.

2. Some lemmas

In this section, we state some useful results, such as interpolation inequalities in Morrey spaces, which may be found in [5, 7, 9, 11].

Lemma 2.1. Let 1≤p≤q≤ ∞(q=∞, p < q). Then (1) Mp,q is a Banach space,

(2) Mp,p∼=L^p, (3) Mp,∞∼=L^∞.

Lemma 2.2. Let 1≤p≤q <∞,λ >0 and let ϑλ(x) =ϑ(x

λ). (2.1)

Then

kϑλkM_p,q =λ^n/qkϑkM_p,q. (2.2) Proof. Owing to the definition of Morrey spaces and direct computation, we obtain

(2.2). Here we omit the details.

Remark 2.3. LetX be a homogeneous Banach space. The smoothness degree of X is defined as

deg(X) := log_λ−1Λ(λ), ∀λ >0, where

Λ(λ) = kϑλ(·)kX

kϑ(·)kX

, ∀λ >0, andϑis a nonzero function inX. Then deg(Mp,q) =−n/q.

From the definition of Morrey spaces we have the following interpolation result.

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Lemma 2.4. Let ¹_q = ^1−θ_q

1 + _q^θ

2 and 0 < θ < 1. If f ∈ Mp,q₁ ∩ Mp,q₂, then f ∈ Mp,q and

kfk_M_p,q ≤Ckfk^1−θ_M

p,q1kfk^θ_M

p,q2. (2.3)

Lemma 2.5. Let 1 ≤p≤q < ∞ and n < mq. If f ∈ M_p,q and ∇^mf ∈ M_p,q, thenf ∈ M_p,∞ and

kfk_M_p,∞ ≤Ckfk^1−θ_M

p,qk∇^mfk^θ_M_p,q (2.4) withθ=n/(mq).

Proof. By Lemma 2.2, the interpolation result (2.4) may be established. For the details, we may refer to [5]. Here we omit the details.

Letβ ∈[0, n). Assume that$(ξ) is smooth on Rⁿ and homogeneous of degree

−β inξ. We call$(ξ)∈P−β

1 (Rⁿ) if$(ξ) satisfies

|D^α$(ξ)| ≤Cα|ξ|^−β−α. Lemma 2.6 ([11]). If$(ξ)∈P0

1(Rⁿ), and1< p≤q <∞, then T =$(D) :Mp,q−→ Mp,q

is a bounded operator.

3. Decay properties of the solution operator We investigate the linearized equation of (1.1),

utt−a∆utt−2b∆ut−α∆³u+β∆²u−∆u= 0. (3.1) Taking the Fourier transform of (3.1), (1.2) yields

(1 +a|ξ|²)ubtt+ 2b|ξ|²ubt+ (|ξ|²+β|ξ|⁴+α|ξ|⁶)bu= 0, (3.2) bu(ξ,0) =ub0(ξ), ubt(ξ,0) =|ξ|Ub1(ξ). (3.3) The characteristic equation is

(1 +a|ξ|²)λ²+ 2b|ξ|²λ+ (|ξ|²+β|ξ|⁴+α|ξ|⁶) = 0.

Letλ=λ±(ξ) be the corresponding eigenvalues. Solving the characteristic equation, we arrive at

λ_±(ξ) =−b|ξ|²± |ξ|p

−1−(a+β−b²)|ξ|²−(α+aβ)|ξ|⁴−aα|ξ|⁶

1 +a|ξ|² . (3.4)

Then the solution to problem (3.2), (3.3) is

bu(ξ, t) =G(ξ, t)|ξ|b Ub1(ξ) +G(ξ, t)b ub0(ξ), (3.5) where

G(ξ, t) =b 1 λ₊(ξ)−λ₋(ξ)

e^λ⁺^(ξ)t−e^λ⁻^(ξ)t ,

G(ξ, t) =b 1 λ₊(ξ)−λ₋(ξ)

λ₊(ξ)e^λ⁻^(ξ)t−λ₋(ξ)e^λ⁺^(ξ)t .

(3.6)

LetG(x, t) =F⁻¹[G(·, t)](x) andb G(x, t) =F⁻¹[G(·, t)](x), whereb F⁻¹ denotes the inverse Fourier transform. Then, we apply F⁻¹ to (3.5) and obtain the solution formula to problem (3.1), (1.2):

u(t) =G(t)∗ΛU₁+G(t)∗u₀. (3.7)

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Owing to Duhamel principle, we obtain the solution formula to problem (1.1), (1.2):

u(t) =G(t)∗ΛU1+G(t)∗u0+ Z t

0

G(t−τ)∗(I−a∆)⁻¹∆f(u)(τ)dτ. (3.8) To establish decay properties of solutions operator in Morrey spaces, we shall make analysis for G and G by the Fourier splitting frequency technique. To this end, let

χ(ξ) =

(1, |ξ|< r, 0, |ξ|>2r.

be smooth cut-off functions, where 0< r <1 is constant. Define Gb_l(ξ, t) =χ(ξ)G(ξ, t),b Gb_h(ξ, t) = (1−χ(ξ))G(ξ, t),b

Gbl(ξ, t) =χ(ξ)G(ξ, t),b Gbh(ξ) = (1−χ(ξ))G(ξ, t)b Then

Gl(x, t) =χ(D)G(x, t), Gh(x, t) = (1−χ(D))G(x, t), Gl(x, t) =χ(D)G(x, t), Gh(x, t) = (1−χ(D))G(x, t), G(x, t) =G_l(x, t) +G_h(x, t), G(x, t) =G_l(x, t) +G_h(x, t), where the operatorχ(D) is defined by

χ(D) =F⁻¹[χ(ξ)].

The following energy estimate in the Fourier space has been obtained in [18, 16], which may be derived by the energy method in the Fourier space.

Lemma 3.1. The solution of problem (3.2),(3.3)satisfies

|ξ|²(1 +|ξ|²)|u(ξ, t)|b ²+|bu_t(ξ, t)|²

≤Ce^−cω(ξ)t(|ξ|²(1 +|ξ|²)|bu0(ξ)|²+|ξ|²|Ub1(ξ)|²), (3.9) forξ∈Rⁿ andt≥0, where ω(ξ) = _1+|ξ|^|ξ|²2.

The above lemma and the solution formula (3.5) imply that the decay estimates of solution operatorsG(t) andG(t) hold.

Lemma 3.2. Let Gb and Gb be the fundamental solutions of (3.1) in the Fourier space, which are given explicitly in (3.6). Then we have

|ξ|²(1 +|ξ|²)|G(ξ, t)|b ²+|Gbt(ξ, t)|²≤Ce^−cω(ξ)t, (3.10)

|ξ|²(1 +|ξ|²)|G(ξ, t)|b ²+|Gbt(ξ, t)|²≤C|ξ|²(1 +|ξ|²)e^−cω(ξ)t (3.11) forξ∈Rⁿ andt≥0, where ω(ξ) = _1+|ξ|^|ξ|²2.

Lemma 3.2 implies that

|ξ|Gbl(ξ, t)∼e^−c|ξ|²^t, Gbl(ξ, t)∼e^−c|ξ|²^t, (3.12)

|ξ|Gb_h(ξ, t)∼e^−ct, Gb_h(ξ, t)∼e^−ct. (3.13)

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Lemma 3.3. Let 1 ≤ p≤ q1 ≤q2 ≤ ∞, and let k be nonnegative integer. The following decay properties hold for solution operator:

k∇^kG(t)∗ΛφkMp,q2 ≤C(1 +t)⁻^k²⁻ⁿ²⁽^q¹¹⁻^q¹²⁾kφkMp,q1

+Ce^−ctk∇^k−1+lφk_M_p,q

2,

(3.14) k∇^kG(t)∗ψkM_p,q₂ ≤C(1 +t)⁻^k²⁻ⁿ²⁽^q¹¹⁻^q¹²⁾kψkM_p,q₁

+Ce^−ctk∇^k+lψk_M_p,q

2,

(3.15) k∇^k∂tG(t)∗ΛφkMp,q2 ≤C(1 +t)⁻^k+1² ⁻ⁿ²⁽^q¹¹⁻^q¹²⁾kφkMp,q1

+Ce^−ctk∇^k+1+lφk_M_p,q

2

(3.16) k∇^k∂_tG(t)∗ψkMp,q2 ≤C(1 +t)⁻^k+1² ⁻ⁿ²⁽^q¹¹⁻^q¹²⁾kψkMp,q1

+Ce^−ctk∇^k+2+lψk_M_p,q

2.

(3.17) Proof. The proofs of (3.14)–(3.17) is similar, we only prove (3.14). Obviously, it holds that

k∇^kG(t)∗Λφk_M_p,q

≤ k∇^kG_l(t)∗Λφk_M_p,q+k∇^kG_h(t)∗Λφk_M_p,q =:I1+I2. (3.18) By using (3.12), we have

|∇^k+1Gl(t)| ≤C(1 +t)⁻^n+k² e^−c^k^|x|

2 t , which implies

k∇^kG_l(t)∗Λφk_Lp(B(z,R))

≤C(1 +t)⁻^n+k² ^p Z

Rⁿ

χZ,R(x)|e^−c^k^|x|

2

t ∗φ|^p(x)dx

≤C(1 +t)⁻^k²^pkφk^p_Lp(B(z,R)).

Then the above inequality and the definition ofMp,q₁ implies

k∇^kG_l(t)∗ΛφkMp,q1 ≤C(1 +t)^−k/2kφkMp,q1 (3.19) Thanks to (3.12) and H¨old inequality, we arrive at

|∇^kG_l(t)∗Λφ| ≤C(1 +t)⁻^n+k² Z

Rⁿ

e^−c^|x−y|

2

t |φ(y)|dy

≤C(1 +t)⁻^n+k² Z 1

0

ds Z

B(x,|ctlogs|^1/2)

|φ(y)|dy

≤C(1 +t)⁻^n+k² Z 1

0

|tlogs|ⁿ²⁽¹⁻^p¹⁾kφk_Lp(B(x,|ctlogs|^1/2))ds

≤C(1 +t)⁻^n+k² Z 1

0

|tlogs|ⁿ²⁽¹⁻^q¹¹⁾dskφkM_p,q₁

≤C(1 +t)⁻^k²⁻^2qⁿ¹kφk_M_p,q

1, which implies

k∇^kG_l(t)∗ΛφkMp,∞ ≤C(1 +t)⁻^k²⁻^2qⁿ¹kφkMp,q1. (3.20)

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Lemma 2.4, (3.19) and (3.20) yield I1≤ k∇^kG_l(t)∗Λφk¹⁻

q1 q2

Mp,∞k∇^kG_l(t)∗Λφk

q1 q2

Mp,q2

≤C(1 +t)⁻^k²⁻ⁿ²⁽^q¹¹⁻^q¹²⁾kφk_M_p,q

1.

(3.21) Set$(ξ) = 1, (3.15) gives

|ξ|²|Gbh(ξ, t)| ≤Ce^−ct$(ξ), which together with Lemma 2.6 with$(ξ) = 1 yields

I2≤Ce^−ctk∇^kφkMp,q2. (3.22) Inserting (3.21) and (3.22) into (3.18) immediately yields (3.14). The proof is

complete.

Noting that the boundness of the operator (I−a∆)⁻¹ in Morrey spaces, the following lemma immediately follows from (3.14) and (3.16).

Lemma 3.4. Let 1 ≤ p≤ q1 ≤q2 ≤ ∞, and let k be nonnegative integer. The following decay properties of solution operator hold:

k∇^kG(t)∗(I−a∆)⁻¹∆fk_M_p,q

2

≤C(1 +t)⁻^k+1² ⁻ⁿ²⁽^q¹¹⁻^q¹²⁾kfkM_p,q₁+Ce^−ctk∇^k−2+lfkM_p,q₂,

(3.23) and

k∇^k∂tG(t)∗(I−a∆)⁻¹∆fkM_p,q₂

≤C(1 +t)⁻^k+2² ⁻ⁿ²⁽^q¹¹⁻^q¹²⁾kfk_M_p,q

1 +Ce^−ctk∇^k+lfk_M_p,q

2.

(3.24) 4. Proof of main results

In this section, our main purpose is to prove Theorem 1.1. For this purpose, we need the following lemma(see [26]).

Lemma 4.1. Assume thatf =f(v)is a smooth function satisfyingf(v) =O(v^1+σ) forv→0, where σ≥1 is an integer. Let v∈L^∞ andkvkL^∞ ≤M0 for a positive constant M0. Let 1 ≤p, q, r ≤+∞ and ¹_p = ¹_q +¹_r, and let k ≥0 be an integer.

Then we have

k∂_x^kf(v)k_Lp≤Ckvk^σ−1_L∞kvk_Lqk∂_x^kvk_Lr, Furthermore,

k∂_x^α(f(v1)−f(v2))kL^p≤Cn

(k∂^α_xv1kL^q+k∂_x^αv2kL^q)kv1−v2kL^r+ (kv1kL^r

+kv2kL^r)k∂_x^α(v1−v2)kL^q

o(kv1kL^∞+kv2kL^∞)^σ−1. whereC=C(M0)is a constant depending onM0.

Proof of Theorem 1.1. To prove existence and decay estimate of global solutions to problem (1.1), (1.2), we define the mapping by (3.8)

T(u) =G(t)∗ΛU1+G(t)∗u0+ Z t

0

G(t−τ)∗(I−a∆)⁻¹∆f(u)(τ)dτ. (4.1) Based on the decay properties of solutions operator, we define the function space

X =

∇^ku∈C([0,∞);Mp,q₁∩ Mp,q₂), k= 0,1, . . . , m

kukX <∞ ,

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where

kukX = sup

t≥0 m

X

k=0

n(1 +t)^k²k∇^ku(t)kMp,q1

+ (1 +t)^k²⁺ⁿ²⁽^q¹¹⁻^q¹²⁾k∇^ku(t)k_M_p,q

2

o .

(4.2)

ForR >0, let

Y ={u∈X :kukX≤R}.

ThenY is a closed set of X. Hence,Y is also a Banach space. To prove existence and decay estimate of global solutions to problem (1.1), (1.2), it is suffice to prove that the mappingT has a unique fixed point in the function spaceY.

By (4.1), Minkowski inequaity, (3.14), (3.15) and (3.23), we have k∇^kT(u)(t)k_M_p,q

1

≤ k∇^kG(t)∗ΛU₁k_M_p,q

1+k∇^kG(t)∗u₀k_M_p,q

1

+ Z t

0

k∇^kG(t−τ)∗(I−a∆)⁻¹∆f(u)(τ)kMp,q1dτ

≤C(1 +t)^−k/2(kU1k_M_p,q

1+ku0k_M_p,q

1) +Ce^−ct(k∇^(k−1)⁺U₁k_M_p,q

1 +k∇^ku₀k_M_p,q

1) +C

Z t/2 0

(1 +t−τ)⁻^k+1² kf(u)(τ)k_M_p,q

1dτ

+C Z t

t/2

(1 +t−τ)^−1/2k∇^kf(u)(τ)kMp,q1dτ +C

Z t 0

e^−c(t−τ)k∇^kf(u)(τ)k_M_p,q

1dτ.

(4.3)

Lemma 2.5 implies

k∇^jukL^∞ ≤Ck∇^juk^1−θ_M

p,q1k∇^j+muk^θ_M_p,q

1,

whereθ=n/(mq1), which together with (4.2) gives

k∇^ju(τ)kL^∞ ≤CkukX(1 +τ)⁻^j²⁻^2qⁿ¹. (4.4) It follows from the definition of Morrey spaces, Lemma 4.1, (4.4) and (4.2) that

kf(u)(τ)kMp,q1 ≤Cku(τ)kL^∞ku(τ)kMp,q1 ≤Ckuk²_X(1 +τ)⁻^2qⁿ¹, (4.5) k∇^kf(u)(τ)k_M_p,q

1 ≤ ku(τ)k_L^∞k∇^ku(τ)k_M_p,q

1 ≤Ckuk²_X(1 +τ)⁻^k²⁻^2qⁿ¹. (4.6)

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We insert (4.5) and (4.6) into (4.3) and arrive at k∇^kT(u)(t)k_M_p,q

1

≤C(1 +t)^−k/2(kU1kMp,q1 +ku0kMp,q1

+k∇^(k−1)⁺U1kMp,q1 +k∇^ku0kMp,q1) +Ckuk²_X

Z t/2 0

(1 +t−τ)⁻^k+1² (1 +τ)⁻^2qⁿ¹dτ +Ckuk²_X

Z t t/2

(1 +t−τ)^−1/2(1 +τ)⁻^k²⁻^2qⁿ¹dτ +Ckuk²_X

Z t 0

e^−c(t−τ)(1 +τ)⁻^k²⁻^2qⁿ¹dτ

≤C(1 +t)^−k/2(kU1kM_p,q₁ +ku0kM_p,q₁

+k∇^(k−1)⁺U1k_M_p,q

1 +k∇^ku0k_M_p,q

1)

+Ckuk²_X(1 +t)^−k/2%(t) +Ckuk²_X(1 +t)⁻^k²⁻^2qⁿ¹⁺¹²,

(4.7)

where

%(t) =







(1 +t)^−1/2, n >2q1, (1 +t)^−1/2log(2 +t), n= 2q₁, (1 +t)¹²⁻^2qⁿ¹, n <2q₁. Similarly, we have

k∇^kT(u)(t)k_M_p,q

2

≤ k∇^kG(t)∗ΛU1kM_p,q₂ +k∇^kG(t)∗u0kM_p,q₂

+ Z t

0

k∇^kG(t−τ)∗(I−a∆)⁻¹∆f(u)(τ)k_M_p,q

2dτ

≤C(1 +t)⁻^k²⁻ⁿ²⁽^q¹¹⁻^q¹²⁾(kU₁k_M_p,q

1+ku₀k_M_p,q

1) +Ce^−ct(k∇^(k−1)⁺U₁k_M_p,q

2+k∇^ku₀k_M_p,q

2) +C

Z t/2 0

(1 +t−τ)⁻^k+1² ⁻ⁿ²⁽^q¹¹⁻^q¹²⁾kf(u)(τ)k_M_p,q

1dτ

+C Z t

t/2

(1 +t−τ)^−1/2k∇^kf(u)(τ)kMp,q2dτ +C

Z t 0

e^−c(t−τ)k∇^kf(u)(τ)k_M_p,q

2dτ

≤C(1 +t)⁻^k²⁻ⁿ²⁽^q¹¹⁻^q¹²⁾E0+Ckuk²_X(1 +t)⁻^k²⁻ⁿ²⁽^q¹¹⁻^q¹²⁾%(t) +Ckuk²_X(1 +t)⁻^k²⁻ⁿ²⁽^q¹¹⁻^q¹²⁾⁻^2qⁿ¹⁺¹²

≤C(1 +t)⁻^k²⁻ⁿ²⁽^q¹¹⁻^q¹²⁾E0+Ckuk²_X(1 +t)⁻^k²⁻ⁿ²⁽^q¹¹⁻^q¹²⁾.

(4.8)

where we have used

k∇^kf(u)(τ)kMp,q2 ≤CkukL^∞k∇^ku(τ)kMp,q2

≤C(1 +τ)⁻^k²⁻ⁿ²⁽^q¹¹⁻^q¹²⁾⁻^2qⁿ¹kuk²_X,

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which may be derived from Lemma 4.1 and (4.2), (4.4).

Combining (4.7) and (4.8) yields

kT(u)kX ≤CE0+Ckuk²_X

Thus, we arrive at

kT(u)kX≤CE0≤R, (4.9)

provided that takingR= 2CE0 andE0 suitably small.

The definition of Morrey spaces, Lemma 4.1, Lemma 2.5 and (4.2) yield

k∇^k(f(¯u)−f(˜u))(τ)kMp,q1

≤C

k¯u−uk˜ L^∞(k∇^ku(τ)k¯ _M_p,q

1 +k∇^ku(τ˜ )k_M_p,q

1) + (k¯ukL^∞+kuk˜ L^∞)k∇^k(¯u−u)(τ)k˜ M_p,q₁

≤C(1 +τ)⁻^k²⁻^2qⁿ¹Rk¯u−uk˜ X.

(4.10)

Using (4.1), Minkowski inequality and (4.10), we obtain

k∇^k(T(¯u)−T(˜u))(t)kM_p,q₁

≤ Z t

0

k∇^kG(t−τ)∗(I−a∆)⁻¹∆(f(¯u)−f(˜u))(τ)k_M_p,q

1dτ

≤C Z t/2

0

(1 +t−τ)⁻^k+1² k(f(¯u)−f(˜u))(τ)k_M_p,q

1dτ

+C Z t

t/2

(1 +t−τ)^−1/2k∇^k(f(¯u)−f(˜u))(τ)kMp,q1dτ +C

Z t 0

e^−c(t−τ)k∇^k(f(¯u)−f(˜u))(τ)k_M_p,q

1dτ

≤CRk¯u−uk˜ X

Z t/2 0

(1 +t−τ)⁻^k+1² (1 +τ)⁻^2qⁿ¹dτ +CRku¯−uk˜ X

Z t t/2

(1 +t−τ)^−1/2(1 +τ)⁻^k²⁻^2qⁿ¹dτ +CRku¯−uk˜ X

Z t 0

e^−c(t−τ)(1 +τ)⁻^k²⁻^2qⁿ¹dτ

≤CRk¯u−uk˜ _X(1 +t)^−k/2%(t) +CRku₁−u₂k_X(1 +t)⁻^k²⁻^2qⁿ¹⁺¹²

≤CRk¯u−uk˜ _X(1 +t)^−k/2.

(4.11)

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Similarly,

k∇^k(T(¯u)−T(˜u))(t)k_M_p,q

2

≤ Z t

0

k∇^kG(t−τ)∗(I−a∆)⁻¹∆(f(¯u)−f(˜u))(τ)k_M_p,q

2dτ

≤C Z t/2

0

(1 +t−τ)⁻^k+1² ⁻ⁿ²⁽^q¹¹⁻^q¹²⁾k(f(¯u)−f(˜u))(τ)k_M_p,q

1dτ

+C Z t

t/2

(1 +t−τ)^−1/2k∇^k(f(¯u)−f(˜u))(τ)k_M_p,q

2dτ

+C Z t

0

e^−c(t−τ)k∇^k(f(¯u)−f(˜u))(τ)k_M_p,q

2dτ

≤CRk¯u−uk˜ _X Z t/2

0

(1 +t−τ)⁻^k+1² ⁻ⁿ²⁽^q¹¹⁻^q¹²⁾(1 +τ)⁻^2qⁿ¹dτ +CRku¯−uk˜ X

Z t t/2

(1 +t−τ)^−1/2(1 +τ)⁻^k²⁻^qⁿ¹⁺^2qⁿ²dτ +CRku¯−uk˜ _X

Z t 0

e^−c(t−τ)(1 +τ)⁻^k²⁻^qⁿ¹⁺^2qⁿ²dτ

≤CRk¯u−uk˜ X(1 +t)⁻^k²⁻ⁿ²⁽^q¹¹⁻^q¹²⁾,

(4.12)

where we have used

k∇^k(f(¯u)−f(˜u))(τ)kMp,q2

≤Cn

ku¯−uk˜ L^∞(k∇^ku(τ)k¯ M_p,q₂ +k∇^ku(τ)k˜ M_p,q₂) + (k¯ukL^∞+kuk˜ L^∞)k∇^k(¯u−u)(τ)k˜ Mp,q2

o

≤C(1 +τ)⁻^k²⁻^qⁿ¹⁺^2qⁿ²Rku¯−˜ukX. Combining (4.11) and (4.12) yields

kT(¯u)−T(˜u)kX ≤CRk¯u−uk˜ X

Thus, we arrive at

kT(¯u)−T(˜u)kX ≤1

2k¯u−uk˜ X. (4.13) Inequalities (4.9) and (4.13) imply that T is a strictly contracting mapping.

The contraction mapping principle implies that the mappingT has a unique fixed pointu∈Y, which is a global solution to problem (1.1), (1.2). Moreover,uverifies decay estimates (1.3) and (1.4).

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In what follows, we prove the decay estimates (1.5) and (1.6). Owing to (4.1), Minkowski inequality, (3.16), (3.17), (3.24), (4.5), (4.6) and (1.3), we arrive at

k∇^l∂_tu(t)k_M_p,q

1

≤ k∇^l∂tG(t)∗ΛU1k_M_p,q

1 +k∇^l∂tG(t)∗u0k_M_p,q

1

+ Z t

0

k∇^l∂tG(t−τ)∗(1−a∆)⁻¹∆f(u)(τ)k_M_p,q

1dτ

≤C(1 +t)⁻^l+1² (kU1kMp,q1 +ku0kMp,q1) +Ce^−ct(k∇^l+1U1kM_p,q₁ +k∇^l+2u0k_M_p,q

1) +C

Z t/2 0

(1 +t−τ)⁻^l+2² kf(u)(τ)kM_p,q₁dτ +C

Z t t/2

(1 +t−τ)^−1/2k∇^l+1f(u)(τ)kMp,q1dτ +C

Z t 0

e^−c(t−τ)k∇^lf(u)(τ)kM_p,q₁dτ

≤CE0(1 +t)⁻^l+1² +CE₀² Z t/2

0

(1 +t−τ)⁻^l+2² (1 +τ)⁻^2qⁿ¹dτ +CE₀²

Z t t/2

(1 +t−τ)^−1/2(1 +τ)⁻^l+1² ⁻^2qⁿ¹dτ +CE₀²

Z t 0

e^−c(t−τ)(1 +τ)⁻²^l⁻^2qⁿ¹dτ

≤CE₀(1 +t)⁻^l+1² .

(4.14)

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Similarly,

k∇^l∂tu(t)k_M_p,q

2

≤ k∇^l∂_tG(t)∗ΛU₁kMp,q2+k∇^l∂_tG(t)∗u₀kMp,q2

+ Z t

0

k∇^l∂tG(t−τ)∗(I−a∆)⁻¹∆f(u)(τ)kMp,q2dτ

≤C(1 +t)⁻^l+1² ⁻ⁿ²⁽^q¹¹⁻^q¹²⁾(kU1kMp,q1 +ku0kMp,q1) +Ce^−ct(k∇^l+1U1k_M_p,q

2 +k∇^l+2u0k_M_p,q

2) +C

Z t/2 0

(1 +t−τ)⁻^l+2² ⁻ⁿ²⁽^q¹¹⁻^q¹²⁾kf(u)(τ)kMp,q1dτ +C

Z t t/2

(1 +t−τ)^−1/2k∇^l+1f(u)(τ)k_M_p,q

2dτ

+C Z t

0

e^−c(t−τ)k∇^lf(u)(τ)kMp,q2dτ

≤CE0(1 +t)⁻^l+1² ⁻ⁿ²⁽^q¹¹⁻^q¹²⁾ +CE₀²

Z t/2 0

(1 +t−τ)⁻^l+2² ⁻ⁿ²⁽^q¹¹⁻^q¹²⁾(1 +τ)⁻^2qⁿ¹dτ +CE₀²

Z t t/2

(1 +t−τ)^−1/2(1 +τ)⁻^l+1² ⁻ⁿ²⁽^q¹¹⁻^q¹²⁾⁻^2qⁿ¹dτ +CE₀²

Z t 0

e^−c(t−τ)(1 +τ)⁻^l²⁻ⁿ²⁽^q¹¹⁻^q¹²⁾⁻^2qⁿ¹dτ

≤CE0(1 +t)⁻^l+1² ⁻ⁿ²⁽^q¹¹⁻^q¹²⁾.

(4.15)

Inequalities (4.14) and (4.15) imply that (1.5) and (1.6) hold. Moreover, (4.14) and (4.15) also imply that∇^l∂tu∈C([0,∞);Mp,q₁∩Mp,q₂). The proof is complete.

Acknowledgements. This research was supported by the NNSF of China (Grant No. 11101144), by the Program for Science & Technology Innovation Talents in Universities of Henan Province (Grant No. 14HASTIT041), and by the Plan For Scientific Innovation Talent of Henan Province(Grant No. 154100510012).

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Yu-Zhu Wang

School of Mathematics and Statistics, North China University of Water Resources and Electric Power, Zhengzhou 450011, China

E-mail address:[email protected]

Yanshuo Li

Qinhui Hu