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Global existence and asymptotic behavior of solutions to a class of fourth-order wave equations

Abstract

This paper is concerned with the Cauchy problem for a class of fourth-order wave equations in an n-dimensional space. Based on the decay estimate of solutions to the corresponding linear equation, a solution space is defined. We prove the global existence and optimal decay estimate of the solution in the corresponding Sobolev spaces by the contraction mapping principle provided that the initial value is suitably small.

MSC:35L30, 35L75.

1 Introduction

We investigate the Cauchy problem for a class of fourth-order wave equations

a u t t + Δ 2 u+ u t =Δf(u)
(1.1)

with the initial value

t=0:u= u 0 (x), u t = u 1 (x).
(1.2)

Here u=u(x,t) is the unknown function of x=( x 1 ,, x n ) R n and t>0, and a>0 is a constant. The nonlinear term f(u) is a smooth function with f(u)=O( u 2 ) for u0.

Equation (1.1) is reduced to the classical Cahn-Hilliard equation if a=0 (see [1]), which has been widely studied by many authors. Galenko et al. [25] proposed to add inertial term a u t t to the classical Cahn-Hilliard equation in order to model non-equilibrium decompositions caused by deep supercooling in certain glasses. For more background, we refer to [46] and references therein. It is obvious that (1.1) is a fourth-order wave equation. For global existence and asymptotic behavior of solutions to more higher order wave equations, we refer to [714] and references therein.

Very recently, global existence and asymptotic behavior of solutions to the problem (1.1), (1.2) were established by Wang and Wei [7] under smallness condition on the initial data. When u 0 H s + 2 L 1 , u 1 H s L 1 , they obtained the following decay estimate:

x k u ( t ) H s + 2 k C ( u 0 L 1 H s + 2 + u 1 L 1 H s ) ( 1 + t ) n 8 k 4
(1.3)

for 0ks+2 and s[n/2]+5. The main purpose of this paper is to refine the result in [7] and prove the following decay estimate for the solution to the problem (1.1), (1.2) for n1 with L 1 data,

x k u ( t ) H s + 2 k CE 0 ( 1 + t ) n 8 k 4 1 2 ,
(1.4)

for 0ks+2 and smax{0,[n/2]1}. Here E 0 := u 0 W ˙ 2 , 1 H s + 2 + u 1 W ˙ 2 , 1 H s is assumed to be small. We also establish the decay estimate for the solution to the problem (1.1), (1.2) for n1 with L 2 data,

x k u ( t ) H s + 2 k CE 1 ( 1 + t ) k 4 1 2 ,
(1.5)

for 0ks+2 and smax{0,[n/2]1}. Here E 1 := u 0 H ˙ 2 H s + 2 + u 1 H ˙ 2 H s is assumed to be small. Compared to the result in [7], we obtain a better decay estimate of solutions for small initial data.

The paper is organized as follows. In Section 2, we study the decay property of the solution operators appearing in the solution formula. We prove global existence and asymptotic behavior of solutions for the Cauchy problem (1.1), (1.2) in Sections 3 and 4, respectively.

Notations We introduce some notations which are used in this paper. Let F[u] denote the Fourier transform of u defined by

u ˆ (ξ)=F[u](ξ):= R n e i ξ x u(x)dx.

We denote its inverse transform by F 1 . For 1p, L p = L p ( R n ) denotes the usual Lebesgue space with the norm L p . The usual Sobolev space of order s is defined by W s , p = ( I Δ ) s 2 L p with the norm f W s , p = ( I Δ ) s 2 f L p . The corresponding homogeneous Sobolev space of order s is defined by W ˙ s , p = ( Δ ) s 2 L p with the norm f W ˙ s , p = ( Δ ) s 2 f L p ; when p=2, we write H s = W s , 2 and H ˙ s = W ˙ s , 2 . We note that W s , p = L p W ˙ s , p for s0.

For a nonnegative integer k, x k denotes the totality of all the k th order derivatives with respect to x R n . Also, for an interval I and a Banach space X, C k (I;X) denotes the space of k-times continuously differential functions on I with values in X.

Throughout the paper, we denote every positive constant by the same symbol C or c without confusion. [] is the Gauss symbol.

2 Decay property

The aim of this section is to derive the solution formula to the Cauchy problem (1.1), (1.2). Without loss of generality, we take a=1. We first study the linearized equation of (1.1),

u t t + Δ 2 u+ u t =0,
(2.1)

with the initial data in (1.2). Taking the Fourier transform, we have

u ˆ t t + u ˆ t + | ξ | 4 u ˆ =0.
(2.2)

The corresponding initial value are given as

t=0: u ˆ = u ˆ 0 (ξ), u ˆ t = u ˆ 1 (ξ).
(2.3)

The characteristic equation of (2.2) is

λ 2 +λ+ | ξ | 4 =0.

Let λ= λ ± (ξ) be the corresponding eigenvalues, we obtain

λ ± (ξ)= 1 ± 1 4 | ξ | 4 2 .
(2.4)

The solution to the problem (2.2), (2.3) is given in the form

u ˆ (ξ,t)= G ˆ (ξ,t) u ˆ 1 (ξ)+ H ˆ (ξ,t) u ˆ 0 (ξ),
(2.5)

where

G ˆ (ξ,t)= 1 λ + ( ξ ) λ ( ξ ) ( e λ + ( ξ ) t e λ ( ξ ) t )
(2.6)

and

H ˆ (ξ,t)= 1 λ + ( ξ ) λ ( ξ ) ( λ + ( ξ ) e λ ( ξ ) t λ ( ξ ) e λ + ( ξ ) t ) .
(2.7)

We define G(x,t) and H(x,t) by G(x,t)= F 1 [ G ˆ (ξ,t)](x) and H(x,t)= F 1 [ H ˆ (ξ,t)](x), respectively, where F 1 denotes the inverse Fourier transform. Then, applying F 1 to (2.5), we obtain

u(t)=G(t) u 1 +H(t) u 0 .
(2.8)

By the Duhamel principle, we obtain the solution formula to (1.1), (1.2)

u(t)=G(t) u 1 +H(t) u 0 + 0 t G(tτ)Δf(u)(τ)dτ.
(2.9)

The aim of this section is to establish decay estimates of the solution operators G(t) and H(t) appearing in the solution formula (2.8).

Lemma 2.1 The solution of the problem (2.2), (2.3) verifies the estimate

( 1 + | ξ | 2 ) 2 | u ˆ ( ξ , t ) | 2 + | u ˆ t ( ξ , t ) | 2 C e c ω ( ξ ) t ( ( 1 + | ξ | 2 ) 2 | u ˆ 0 ( ξ ) | 2 + | u ˆ 1 ( ξ ) | 2 ) ,
(2.10)

for ξ R n and t0, where ω(ξ)= | ξ | 4 ( 1 + | ξ | 2 ) 2 .

Proof We apply the energy method in the Fourier space to prove (2.10). Such an energy method was first developed in [15]. We multiply (2.2) by u ˆ ¯ t and take the real part. This yields

1 2 d d t { | u ˆ t | 2 + | ξ | 4 | u ˆ | 2 } + | u ˆ t | 2 =0.
(2.11)

Multiplying (2.2) by u ˆ ¯ and taking the real part, we obtain

1 2 d d t { | u ˆ | 2 + 2 Re ( u ˆ t u ˆ ¯ ) } + | ξ | 4 | u ˆ | 2 | u ˆ t | 2 =0.
(2.12)

Combining (2.11) and (2.12) yields

d d t E+F=0,
(2.13)

where

E= | u ˆ t | 2 + [ 1 2 + | ξ | 4 ] | u ˆ | 2 +Re( u ˆ t u ˆ ¯ )

and

F= | ξ | 4 | u ˆ | 2 + | u ˆ t | 2 .

A simple computation implies that

CE 0 E CE 0 ,
(2.14)

where

E 0 = ( 1 + | ξ | 2 ) 2 | u ˆ | 2 + | u ˆ t | 2 .

Note that

F | ξ | 4 ( 1 + | ξ | 2 ) 2 E 0 .

It follows from (2.14) that

Fcω(ξ)E,
(2.15)

where

ω(ξ)= | ξ | 4 ( 1 + | ξ | 2 ) 2 .

Using (2.13) and (2.15), we get

d d t E+cω(ξ)E0.

Thus

E(ξ,t) e c w ( ξ ) t E(ξ,0),

which together with (2.14) proves the desired estimates (2.10). Then we have completed the proof of the lemma. □

Lemma 2.2 Let G ˆ (ξ,t) and H ˆ (ξ,t) be the fundamental solutions to (2.1) in the Fourier space, which are given in (2.6) and (2.7), respectively. Then we have the estimates

( 1 + | ξ | 2 ) 2 | G ˆ ( ξ , t ) | 2 + | G ˆ t ( ξ , t ) | 2 C e c ω ( ξ ) t
(2.16)

and

( 1 + | ξ | 2 ) 2 | H ˆ ( ξ , t ) | 2 + | H ˆ t ( ξ , t ) | 2 C ( 1 + | ξ | 2 ) 2 e c ω ( ξ ) t
(2.17)

for ξ R n and t0, where ω(ξ)= | ξ | 4 ( 1 + | ξ | 2 ) 2 .

Proof If u ˆ 0 (ξ)=0, from (2.5), we obtain

u ˆ (ξ,t)= G ˆ (ξ,t) u ˆ 1 (ξ), u ˆ t (ξ,t)= G ˆ t (ξ,t) u ˆ 1 (ξ).

Substituting the equalities into (2.10) with u ˆ 0 (ξ)=0, we get (2.16).

In what follows, we consider u ˆ 1 (ξ)=0, it follows from (2.5) that

u ˆ (ξ,t)= H ˆ (ξ,t) u ˆ 0 (ξ), u ˆ t (ξ,t)= H ˆ t (ξ,t) u ˆ 0 (ξ).

Substituting the equalities into (2.10) with u ˆ 1 (ξ)=0, we get (2.17). The lemma is proved. □

Lemma 2.3 Let G ˆ (ξ,t) and H ˆ (ξ,t) be the fundamental solutions to (2.1) in the Fourier space, which are given in (2.6) and (2.7), respectively. Then there exists a small positive number R 0 such that if |ξ| R 0 and t0, we have the following estimate:

| G ˆ t ( ξ , t ) | C | ξ | 4 e c | ξ | 4 t +C e c t
(2.18)

and

| H ˆ t ( ξ , t ) | C | ξ | 4 e c | ξ | 4 t +C e c t .
(2.19)

Proof For sufficiently small ξ, using the Taylor formula, we get

λ + (ξ)= | ξ | 4 +O ( | ξ | 8 ) , λ (ξ)=1+ | ξ | 4 +O ( | ξ | 8 )
(2.20)

and

1 λ + ( ξ ) λ ( ξ ) =1+2 | ξ | 4 +O ( | ξ | 8 ) .
(2.21)

It follows from (2.6) and (2.7) that

{ G ˆ t ( ξ , t ) = λ + ( ξ ) e λ + t λ ( ξ ) e λ t λ + ( ξ ) λ ( ξ ) , H ˆ t ( ξ , t ) = 1 λ + ( ξ ) λ ( ξ ) ( λ + ( ξ ) λ ( ξ ) e λ ( ξ ) t λ ( ξ ) λ + ( ξ ) e λ + ( ξ ) t ) .
(2.22)

Equations (2.18) and (2.19) follow from (2.20)-(2.22). The proof of Lemma 2.3 is completed. □

Lemma 2.4 Let 1p2 and k0. Then we have

x k G ( t ) ϕ L 2 C ( 1 + t ) n 4 ( 1 p 1 2 ) k + l 4 ϕ W ˙ l , p +C e c t x ( k 2 ) + ϕ L 2 ,
(2.23)
x k H ( t ) ϕ L 2 C ( 1 + t ) n 4 ( 1 p 1 2 ) k + l 4 ϕ W ˙ l , p +C e c t x k ϕ L 2 ,
(2.24)
x k G t ( t ) ϕ L 2 C ( 1 + t ) n 4 ( 1 p 1 2 ) k + l 4 1 ϕ W ˙ l , p +C e c t x k ϕ L 2 ,
(2.25)

and

x k H t ( t ) ϕ L 2 C ( 1 + t ) n 4 ( 1 p 1 2 ) k + l 4 1 ϕ W ˙ l , p +C e c t x k + 2 ϕ L 2 ,
(2.26)
x k G ( t ) Δ g L 2 C ( 1 + t ) n 4 ( 1 p 1 2 ) k 4 1 2 g L p +C e c t x k g L 2 ,
(2.27)
x k G t ( t ) Δ g L 2 C ( 1 + t ) n 4 ( 1 p 1 2 ) k 4 3 2 g L p +C e c t x k + 2 g L 2 ,
(2.28)

where ( k 2 ) + =max{0,k2} in (2.23).

Proof By the property of the Fourier transform and (2.16), we obtain

x k G ( t ) ϕ L 2 2 = R n | ξ | 2 k | G ˆ ( ξ , t ) | 2 | ϕ ˆ ( ξ ) | 2 d ξ | ξ | R 0 | ξ | 2 k ( 1 + | ξ | 2 ) 2 e c ω ( ξ ) t | ϕ ˆ ( ξ ) | 2 d ξ + C | ξ | R 0 | ξ | 2 k ( 1 + | ξ | 2 ) 2 e c ω ( ξ ) t | ϕ ˆ ( ξ ) | 2 d ξ C | ξ | R 0 | ξ | 2 k e c | ξ | 4 t | ϕ ˆ ( ξ ) | 2 d ξ + C | ξ | R 0 | ξ | 2 k | ξ | 4 e c t | ϕ ˆ ( ξ ) | 2 d ξ C | ξ | 2 k + 2 l e c | ξ | 4 t L q | ξ | l ϕ ˆ ( ξ ) L p 2 + C e c t x ( k 2 ) + ϕ L 2 ,
(2.29)

where R 0 is a positive constant in Lemma 2.3, and 1 q + 2 p =1 and 1 p + 1 p =1.

By a straight computation, we get

| ξ | 2 k + 2 l e c | ξ | 4 t L q ( | ξ | R 0 ) C ( 1 + t ) n 4 q k + l 2 C ( 1 + t ) n 4 ( 2 p 1 ) k + l 2 .
(2.30)

It follows from the Hausdorff-Young inequality that

| ξ | l ϕ ˆ ( ξ ) L p C ( Δ ) l 2 ϕ L p C ϕ W ˙ l , p .
(2.31)

Combining (2.29)-(2.31) yields (2.23). Similarly, we can prove (2.24)-(2.28). Thus we have completed the proof of the lemma. □

3 Global existence and decay estimate (I)

The purpose of this section is to prove global existence and asymptotic behavior of solutions to the Cauchy problem (1.1), (1.2) with L 1 data. We need the following lemma, which comes from [16] (see also [17]).

Lemma 3.1 Assume that f=f(v) is smooth, where v=( v 1 ,, v n ) is a vector function. Suppose that f(v)=O( | v | 1 + θ ) (θ1 is an integer) when |v| ν 0 . Then, for the integer m0, if v,w W m , q ( R n ) L p ( R n ) L ( R n ) and v L ν 0 , w L ν 0 , then f(v)f(w) W m , r ( R n ). Furthermore, the following inequalities hold:

x m f ( v ) L r C v L p x m v L q v L θ 1
(3.1)

and

x m ( f ( v ) f ( w ) ) L r C { ( x m v L q + x m w L q ) v w L p + + ( v L p + w L p x m ( v w ) L q ) } ( v L + w L ) θ 1 ,
(3.2)

where 1 r = 1 p + 1 q , 1p,q,r+.

Based on the decay estimates of solutions to the linear problem (2.1), (1.2), we define the following solution space:

X= { u C ( [ 0 , ) ; H s + 2 ( R n ) ) C 1 ( [ 0 , ) ; H s ( R n ) ) : u X < } ,

where

u X = sup t 0 { k s + 2 ( 1 + t ) n 8 + k 4 + 1 2 x k u ( t ) L 2 + h s ( 1 + t ) n 8 + h 4 + 3 2 x h u t ( t ) L 2 } .

For R>0, we define

X R = { u X : u X R } .

The Gagliardo-Nirenberg inequality gives

u ( t ) L C u ( t ) L 2 1 n 2 s 0 x s 0 u ( t ) L 2 n 2 s 0 C ( 1 + t ) ( n 4 + 1 2 ) u X ,
(3.3)

where s 0 =[ n 2 ]+1, s 0 s+2 (i.e., s[ n 2 ]1).

Theorem 3.1 Assume that u 0 W ˙ 2 , 1 ( R n ) H s + 2 ( R n ), u 1 W ˙ 2 , 1 ( R n ) H s ( R n ) (smax{0,[n/2]1}). Put

E 0 = u 0 W ˙ 2 , 1 H s + 2 + u 1 W ˙ 2 , 1 H s .

If E 0 is suitably small, the Cauchy problem (1.1), (1.2) has a unique global solution u(x,t) satisfying

uC ( [ 0 , ) ; H s + 2 ( R n ) ) C 1 ( [ 0 , ) ; H s ( R n ) ) .

Moreover, the solution satisfies the decay estimate

x k u ( t ) L 2 CE 0 ( 1 + t ) n 8 k 4 1 2
(3.4)

and

x h u t ( t ) L 2 CE 0 ( 1 + t ) n 8 h 4 3 2
(3.5)

for 0ks+2 and 0hs.

Proof Let us define the mapping

Φ(u)=G(t) u 1 +H(t) u 0 + 0 t G(tτ)Δf ( u ( τ ) ) dτ.
(3.6)

Using (2.23), (2.24), (2.27), (3.1) and (3.3), for 0ks+2, we obtain

x k Φ ( u ) L 2 x k G ( t ) u 1 L 2 + C x k H ( t ) u 0 L 2 + C 0 t x k G ( t τ ) Δ f ( u ( τ ) ) L 2 d τ C ( 1 + t ) n 8 k 4 1 2 ( u 0 W ˙ 2 , 1 H s + 2 + u 1 W ˙ 2 , 1 H s ) + C 0 t 2 ( 1 + t τ ) n 8 k 4 1 2 f ( u ) L 1 d τ + C t 2 t ( 1 + t τ ) 1 2 x k f ( u ) L 2 d τ + C 0 t e c ( t τ ) x k f ( u ) L 2 d τ C ( 1 + t ) n 8 k 4 1 2 ( u 0 W ˙ 2 , 1 H s + 2 + u 1 W ˙ 2 , 1 H s ) + C 0 t 2 ( 1 + t τ ) n 8 k 4 1 2 u L 2 2 d τ + C t 2 t ( 1 + t τ ) 1 2 x k u L 2 u L d τ + C 0 t e c ( t τ ) x k u L 2 u L d τ C ( 1 + t ) n 8 k 4 1 2 ( u 0 W ˙ 2 , 1 H s + 2 + u 1 W ˙ 2 , 1 H s ) + CR 2 0 t 2 ( 1 + t τ ) n 8 k 4 1 2 ( 1 + τ ) n 4 1 d τ + CR 2 t 2 t ( 1 + t τ ) 1 2 ( 1 + τ ) 3 n 8 k 4 1 d τ + CR 2 0 t e c ( t τ ) ( 1 + τ ) 3 n 8 k 4 1 d τ C ( 1 + t ) n 8 k 4 1 2 { ( u 0 W ˙ 2 , 1 H s + 2 + u 1 W ˙ 2 , 1 H s ) + R 2 } .

Thus

( 1 + t ) n 8 + k 4 + 1 2 x k Φ ( u ) L 2 CE 0 + CR 2 .
(3.7)

It follows from (3.6) that

Φ ( u ) t = G t (t) u 1 + H t (t) u 0 + 0 t G t (tτ)Δf ( u ( τ ) ) dτ.
(3.8)

By exploiting (3.8), (2.25), (2.26), (2.28), (3.1) and (3.3), for hs, we have

x h Φ ( u ) t L 2 C x h G t ( t ) u 1 L 2 + C x h H t ( t ) u 0 L 2 + C 0 t x h G t ( t τ ) Δ f ( u ( τ ) ) L 2 d τ C ( 1 + t ) n 8 h 4 3 2 ( u 0 W ˙ 2 , 1 H s + 2 + u 1 W ˙ 2 , 1 H s ) + C 0 t 2 ( 1 + t τ ) n 8 h 4 3 2 f ( u ) L 1 d τ + C t 2 t ( 1 + t τ ) 3 2 x h f ( u ) L 2 d τ + C 0 t e c ( t τ ) x h f ( u ) L 2 d τ C ( 1 + t ) n 8 h 4 3 2 { ( u 0 W ˙ 2 , 1 H s + 2 + u 1 W ˙ 2 , 1 H s ) + R 2 } .

The above inequality implies

( 1 + t ) n 8 + h 4 + 3 2 x h Φ ( u ) t L 2 CE 0 + CR 2 .
(3.9)

Combining (3.7) and (3.9) and taking E 0 and R suitably small, we get

Φ ( u ) X R.
(3.10)

For u ˜ , u ¯ X R , (3.6) gives

Φ( u ˜ )Φ( u ¯ )= 0 t G(tτ)Δ [ f ( u ˜ ) f ( u ¯ ) ] dτ.
(3.11)

By (2.27), (3.2) and (3.3), for 0ks+2, we infer that

x k ( Φ ( u ˜ ) Φ ( u ¯ ) ) L 2 0 t x k G ( t τ ) Δ ( f ( u ˜ ) f ( u ¯ ) ) L 2 d τ C 0 t 2 ( 1 + t τ ) n 8 k 4 1 2 f ( u ˜ ) f ( u ¯ ) L 1 d τ + C t 2 t ( 1 + t τ ) 1 2 x k ( f ( u ˜ ) f ( u ¯ ) ) L 2 d τ + C 0 t e c ( t τ ) x k ( f ( u ˜ ) f ( u ¯ ) ) L 2 d τ C 0 t 2 ( 1 + t τ ) n 8 k 4 1 2 ( u ˜ L 2 + u ¯ L 2 ) u ˜ u ¯ L 2 d τ + C t 2 t ( 1 + t τ ) 1 2 { ( x k u ˜ L 2 + x k u ˜ L 2 ) u ˜ u ¯ L + ( u ˜ L 2 + u ¯ L 2 ) x k ( u ˜ u ¯ ) L 2 } d τ + C 0 t e c ( t τ ) { ( x k u ˜ L 2 + x k u ˜ L 2 ) u ˜ u ¯ L + ( u ˜ L + u ¯ L ) x k ( u ˜ u ¯ ) L 2 } d τ CR u ˜ u ¯ X 0 t 2 ( 1 + t τ ) n 8 k 4 1 2 ( 1 + τ ) n 4 1 2 d τ + CR u ˜ u ¯ X t 2 t ( 1 + t τ ) n 8 1 2 ( 1 + τ ) 3 n 8 k 4 1 d τ + CR | u ˜ u ¯ X 0 t e c ( t τ ) ( 1 + τ ) 3 n 8 k 4 1 d τ CR ( 1 + t ) n 8 k 4 1 2 u ˜ u ¯ X ,

which implies

( 1 + t ) n 8 + k 4 + 1 2 x k ( Φ ( u ˜ ) Φ ( u ¯ ) ) L 2 CR u ˜ u ¯ X .
(3.12)

Similarly, for 0hs, from (3.11), (2.28) and (3.2), (3.3), we deduce that

x h ( Φ ( u ˜ ) Φ ( u ¯ ) ) t L 2 0 t x h G t ( t τ ) Δ [ f ( u ˜ ) f ( u ¯ ) ] L 2 d τ C 0 t 2 ( 1 + t τ ) n 8 h 4 3 2 ( f ( u ˜ ) f ( u ¯ ) ) L 1 d τ + C t 2 t ( 1 + t τ ) 3 2 x h ( f ( u ˜ ) f ( u ¯ ) ) L 2 d τ + C 0 t e c ( t τ ) x h ( f ( u ˜ ) f ( u ¯ ) ) L 2 d τ CR ( 1 + t ) n 8 h 4 3 2 u ˜ u ¯ X ,

which gives

( 1 + t ) n 8 + h 4 + 3 2 x h ( Φ ( u ˜ ) Φ ( u ¯ ) ) t L 2 CR u ˜ u ¯ X .
(3.13)

Combining (3.12) and (3.13) and taking R suitably small yields

Φ ( u ˜ ) Φ ( u ¯ ) X 1 2 u ˜ u ¯ X .
(3.14)

From (3.10) and (3.14), we know that Φ is a strictly contracting mapping. Consequently, we conclude that there exists a fixed point u X R of the mapping Φ, which is a solution to (1.1), (1.2). We have completed the proof of the theorem. □

4 Global existence and decay estimate (II)

In the previous section, we have proved global existence and asymptotic behavior of solutions to the Cauchy problem (1.1), (1.2) with L 1 data. The purpose of this section is to establish global existence and asymptotic behavior of solutions to the Cauchy problem (1.1), (1.2) with L 2 data. Based on the decay estimates of solutions to the linear problem (2.1), (1.2), we define the following solution space:

X= { u C ( [ 0 , ) ; H s + 2 ( R n ) ) C 1 ( [ 0 , ) ; H s ( R n ) ) : u X < } ,

where

u X = sup t 0 { k s + 2 ( 1 + t ) k 4 + 1 2 x k u ( t ) L 2 + h s ( 1 + t ) h 4 + 3 2 x h u t ( t ) L 2 } .

For R>0, we define

X R = { u X : u X R } .

Thanks to the Gagliardo-Nirenberg inequality, we get

u ( t ) L C ( 1 + t ) ( n 8 + 1 2 ) u X .
(4.1)

Theorem 4.1 Suppose that u 0 H ˙ 2 ( R n ) H s + 2 ( R n ), u 1 H ˙ 2 ( R n ) H s ( R n ) (smax{0,[n/2]1}). Put

E 1 = u 0 H ˙ 2 H s + 2 + u 1 H ˙ 2 H s .

If E 0 is suitably small, the Cauchy problem (1.1), (1.2) has a unique global solution u(x,t) satisfying

uC ( [ 0 , ) ; H s + 2 ( R n ) ) C 1 ( [ 0 , ) ; H s ( R n ) ) .

Moreover, the solution satisfies the decay estimate

x k u ( t ) L 2 CE 1 ( 1 + t ) k 4 1 2
(4.2)

and

x h u t ( t ) L 2 CE 1 ( 1 + t ) h 4 3 2
(4.3)

for 0ks+2 and 0hs.

Proof Let the mapping Φ be defined in (3.6).

For ks+2, (2.23), (2.24), (2.27), (3.1) and (4.1) give

x k Φ ( u ) L 2 x k G ( t ) u 1 L 2 + C x k H ( t ) u 0 L 2 + C 0 t x k G ( t τ ) Δ f ( u ( τ ) ) L 2 d τ C ( 1 + t ) k 4 1 2 ( u 0 H ˙ 2 H s + 2 + u 1 H ˙ 2 H s ) + C 0 t 2 ( 1 + t τ ) n 8 k 4 1 2 f ( u ) L 1 d τ + C t 2 t ( 1 + t τ ) 1 2 x k f ( u ) L 2 d τ + C 0 t e c ( t τ ) x k f ( u ) L 2 d τ C ( 1 + t ) k 4 1 2 ( u 0 H ˙ 2 H s + 2 + u 1 H ˙ 2 H s ) + C 0 t 2 ( 1 + t τ ) n 8 k 4 1 2 u L 2 2 d τ + C t 2 t ( 1 + t τ ) 1 2 x k u L 2 u L d τ + C 0 t e c ( t τ ) x k u L 2 u L d τ C ( 1 + t ) k 4 1 2 ( u 0 H ˙ 2 H s + 2 + u 1 H ˙ 2 H s ) + CR 2 0 t 2 ( 1 + t τ ) n 8 k 4 1 2 ( 1 + τ ) 1 d τ + CR 2 t 2 t ( 1 + t τ ) 1 2 ( 1 + τ ) n 8 k 4 1 d τ + CR θ + 1 0 t e c ( t τ ) ( 1 + τ ) n 8 k 4 1 d τ C ( 1 + t ) k 4 1 2 { ( u 0 H ˙ 2 H s + 2 + u 1 H ˙ 2 H s ) + R 2 } .

Thus we get

( 1 + t ) k 4 + 1 2 x k Φ ( u ) L 2 CE 1 + CR 2 .
(4.4)

Applying t to (3.6), we obtain

Φ ( u ) t = G t (t) u 1 + H t (t) u 0 + 0 t G t (tτ)Δf ( u ( τ ) ) dτ.
(4.5)

By using (2.25), (2.26), (2.28), (3.1), (4.1), for 0hs, we have

x h Φ ( u ) t L 2 C x h G t ( t ) u 1 L 2 + C x h H t ( t ) u 0 L 2 + C 0 t x h G t ( t τ ) Δ f ( u ( τ ) ) L 2 d τ C ( 1 + t ) h 4 3 2 ( u 0 H ˙ 2 H s + 2 + u 1 H ˙ 2 H s ) + C 0 t 2 ( 1 + t τ ) n 8 h 4 3 2 f ( u ) L 1 d τ + C t 2 t ( 1 + t τ ) 3 2 x h f ( u ) L 2 d τ + C 0 t e c ( t τ ) x h f ( u ) L 2 d τ C ( 1 + t ) h 4 3 2 { ( u 0 H ˙ 2 H s + 2 + u 1 H ˙ 2 H s ) + R 2 } .

This yields

( 1 + t ) h 4 + 3 2 x h Φ ( u ) t L 2 CE 1 + CR 2 .
(4.6)

Combining (4.4) and (4.6) and taking E 1 and R suitably small, we obtain

Φ ( u ) X R.
(4.7)

For u ˜ , u ¯ X R , by using (3.6), we have

Φ( u ˜ )Φ( u ¯ )= 0 t G(tτ)Δ [ f ( u ˜ ) f ( u ¯ ) ] dτ.
(4.8)

It follows from (2.27), (3.2) and (4.1) for 0ks+2 that

x k ( Φ ( u ˜ ) Φ ( u ¯ ) ) L 2 0 t x k G ( t τ ) Δ ( f ( u ˜ ) f ( u ¯ ) ) L 2 d τ C 0 t 2 ( 1 + t τ ) n 8 k 4 1 2 f ( u ˜ ) f ( u ¯ ) L 1 d τ + C t 2 t ( 1 + t τ ) 1 2 x k ( f ( u ˜ ) f ( u ¯ ) ) L 2 d τ + C 0 t e c ( t τ ) x k ( f ( u ˜ ) f ( u ¯ ) ) L 2 d τ CR ( 1 + t ) k 4 1 2 u ˜ u ¯ X ,

which implies

( 1 + t ) k 4 + 1 2 x k ( Φ ( u ˜ ) Φ ( u ¯ ) ) L 2 CR u ˜ u ¯ X .
(4.9)

Similarly, for 0hs, from (4.5), (2.28), (3.2) and (4.1), we infer that

x h ( Φ ( u ˜ ) Φ ( u ¯ ) ) t L 2 0 t x h G t ( t τ ) Δ [ f ( u ˜ ) f ( u ¯ ) ] L 2 d τ C 0 t 2 ( 1 + t τ ) n 8 h 4 3 2 ( f ( u ˜ ) f ( u ¯ ) ) L 1 d τ + C t 2 t ( 1 + t τ ) 3 2 x h ( f ( u ˜ ) f ( u ¯ ) ) L 2 d τ + C 0 t e c ( t τ ) x h ( f ( u ˜ ) f ( u ¯ ) ) L 2 d τ CR ( 1 + t ) h 4 3 2 u ˜ u ¯ X ,

which implies

( 1 + t ) h 4 + 3 2 x h ( Φ ( u ˜ ) Φ ( u ¯ ) ) t L 2 CR u ˜ u ¯ X .
(4.10)

Using (4.9) and (4.10) and taking R suitably small yields

Φ ( u ˜ ) Φ ( u ¯ ) X 1 2 u ˜ u ¯ X .
(4.11)

It follows from (4.7) and (4.11) that Φ is a strictly contracting mapping. Consequently, we infer that there exists a fixed point u X R of the mapping Φ, which is a solution to (1.1), (1.2). We have completed the proof of the theorem. □

References

  1. Cahn JW, Hilliard JE: Free energy of a nonuniform system, I. Interfacial free energy. J. Chem. Phys. 1958, 28: 258–267. 10.1063/1.1744102

    Article  Google Scholar 

  2. Galenko P: Phase-field model with relaxation of the diffusion flux in nonequilibrium solidification of a binary system. Phys. Lett. A 2001, 287: 190–197. 10.1016/S0375-9601(01)00489-3

    Article  Google Scholar 

  3. Galenko P, Jou D: Diffuse-interface model for rapid phase transformations in nonequilibrium systems. Phys. Rev. E 2005., 71: Article ID 046125

    Google Scholar 

  4. Galenko P, Lebedev V: Analysis of the dispersion relation in spinodal decomposition of a binary system. Philos. Mag. Lett. 2007, 87: 821–827. 10.1080/09500830701395127

    Article  Google Scholar 

  5. Galenko P, Lebedev V: Local nonequilibrium effect on spinodal decomposition in a binary system. Int. J. Appl. Thermodyn. 2008, 11: 21–28.

    Google Scholar 

  6. Gatti S, Grasselli M, Miranville A, Pata V: Hyperbolic relaxation of the viscous Cahn-Hilliard equation in 3D. Math. Models Methods Appl. Sci. 2005, 15: 165–198. 10.1142/S0218202505000327

    Article  MathSciNet  Google Scholar 

  7. Wang Y-X, Wei Z: Global existence and asymptotic behavior of solution to Cahn-Hilliard equation with inertial term. Int. J. Math. 2012., 23: Article ID 1250087

    Google Scholar 

  8. Wang Y-X: Existence and asymptotic behavior of solutions to the generalized damped Boussinesq equation. Electron. J. Differ. Equ. 2012, 2012(96):1–11.

    Google Scholar 

  9. Wang W, Wu Z: Optimal decay rate of solutions for Cahn-Hilliard equation with inertial term in multi-dimensions. J. Math. Anal. Appl. 2012, 387: 349–358. 10.1016/j.jmaa.2011.09.016

    Article  MathSciNet  Google Scholar 

  10. Wang Y: Global existence and asymptotic behaviour of solutions for the generalized Boussinesq equation. Nonlinear Anal. 2009, 70: 465–482. 10.1016/j.na.2007.12.018

    Article  MathSciNet  Google Scholar 

  11. Wang Y, Wang Y: Existence and nonexistence of global solutions for a class of nonlinear wave equations of higher order. Nonlinear Anal. 2010, 72: 4500–4507. 10.1016/j.na.2010.02.025

    Article  MathSciNet  Google Scholar 

  12. Wang Y, Wang Y: Global existence and asymptotic behavior of solutions to a nonlinear wave equation of fourth-order. J. Math. Phys. 2012., 53: Article ID 013512

    Google Scholar 

  13. Wang Y, Liu F, Zhang Y: Global existence and asymptotic of solutions for a semi-linear wave equation. J. Math. Anal. Appl. 2012, 385: 836–853. 10.1016/j.jmaa.2011.07.010

    Article  MathSciNet  Google Scholar 

  14. Wang S, Xu H: On the asymptotic behavior of solution for the generalized IBq equation with hydrodynamical damped term. J. Differ. Equ. 2012, 252: 4243–4258. 10.1016/j.jde.2011.12.016

    Article  Google Scholar 

  15. Umeda T, Kawashima S, Shizuta Y: On the decay of solutions to the linearized equations of electro-magneto-fluid dynamics. Jpn. J. Appl. Math. 1984, 1: 435–457. 10.1007/BF03167068

    Article  MathSciNet  Google Scholar 

  16. Li TT, Chen YM: Nonlinear Evolution Equations. Science Press, Beijing; 1989. (in Chinese)

    Google Scholar 

  17. Zheng SM: Nonlinear Evolution Equations. CRC Press, Boca Raton; 2004.

    Book  Google Scholar 

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Zhuang, Z., Zhang, Y. Global existence and asymptotic behavior of solutions to a class of fourth-order wave equations. Bound Value Probl 2013, 168 (2013). https://doi.org/10.1186/1687-2770-2013-168

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