Skip to main content

Asymptotic decay estimate of solutions to the generalized damped Bq equation

Abstract

In this paper, we investigate the Cauchy problem for the generalized damped Bq equation. By applying fixed point theorem, we prove the global existence and asymptotic decay estimate of solutions for all space dimensions n1 provided that the initial value is suitably small.

MSC:35L30, 35L75.

1 Introduction

We study asymptotic decay estimate of solution to the Cauchy problem for the generalized damped Bq equation:

u t t Δu2bΔ u t +α Δ 2 u=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, b>0 and α>0 are constants. The nonlinear term f(u) is a given smooth function of uR satisfying f(u)=O( u 2 ) for u0.

It is well known that the equation (called the classical Bq equation)

u t t +α Δ 2 uΔu=βΔ ( u 2 )
(1.3)

was derived by Boussinesq [1] in 1872 to describe shallow water waves, where u(x,t) is an elevation of the free surface of fluid and the constant coefficients α and β depend on the depth of fluid and the characteristic speed of long waves. It is interesting to note that this equation governs nonlinear string oscillations as well. For (1.3) and the generalized Bq equation, there are lots of important results (see [25]).

Equation (1.3) takes into account dispersion and nonlinearity, but in real processes viscosity also plays an important role. Varlamov considered the following damped Boussinesq equation (see [68]):

u t t Δu2bΔ u t +α Δ 2 u=βΔ ( u 2 ) ,
(1.4)

where b>0 and α>0 are constants. A classical solution to the Cauchy problem for (1.4) with small initial data was constructed by means of the application of both spectral and perturbation theories. Large time asymptotics of this solution was also obtained (see [6]). Varlamov [7] investigated the Cauchy problem for (1.4). For the cases of one, two and three space dimensions local in time existence and uniqueness of a solution is proved. The authors also showed that for discontinuous initial perturbations this solution is infinitely differentiable with respect to time t and space co-ordinates for t>0 on a bounded time interval. Varlamov [8] considered the Cauchy problem for (1.4) with small initial data in two space dimensions. Existence and uniqueness of the classical solution was proved and the solution was constructed in the form of a series. The major term of its long-time asymptotics is calculated explicitly and a uniform in space estimate of the residual term was given.

The main purpose of this paper is to establish the following asymptotic decay estimate of solutions to (1.1), (1.2) for n1:

x k u ( t ) H s + 2 k C E 0 ( 1 + t ) n 4 k 2 1 2

for 0ks+2 and smax{0,[ n 2 ]1}. Here E 0 := u 0 W ˙ 1 , 1 + u 1 W ˙ 2 , 1 + u 0 H s + 2 + u 1 H s .

Decay estimate of solutions to hyperbolic-type equations has been investigated by many authors. We refer to [9, 10] for hyperbolic equations, to [1115] for damped wave equation and to [1619] for various aspects of dissipation of the plate equation.

The paper is organized as follows. In Section 2 we derive the solution formula of our semi-linear problem. We study the decay property of the solution operators appearing in the solution formula in Section 3. In Section 4, asymptotic decay estimate of solutions to the cauchy problem (1.1), (1.2) is established by applying fixed point theorem.

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

u ˆ (ξ)=F[u]= R n e i ξ x u(x)dx,

and 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 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 homogeneous Sobolev space of 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 ; especially H s = W s , 2 , H ˙ s = W ˙ s , 2 . Moreover, we know that W s , p = L p W ˙ s , p for s0.

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

2 Solution formula

The aim of this section is to derive the solution formula for the problem (1.1), (1.2). We first investigate the linearized equation of (1.1).

u t t Δu2bΔ u t +α Δ 2 u=0

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

u ˆ t t +2b | ξ | 2 u ˆ t + ( | ξ | 2 + α | ξ | 4 ) u ˆ =0.
(2.1)

The corresponding initial values are given as

t=0: u ˆ = u ˆ 0 (ξ), u ˆ t = u ˆ 1 (ξ).
(2.2)

The characteristic equation of (2.1) is

λ 2 +2b | ξ | 2 λ+ | ξ | 2 +α | ξ | 4 =0.
(2.3)

Let λ= λ ± (ξ) be the corresponding eigenvalues of (2.3), we obtain

λ ± (ξ)=b | ξ | 2 ±|ξ| ( b 2 α ) | ξ | 2 1 .
(2.4)

The solution to the problem (2.1)-(2.2) 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)
(2.8)

and

H(x,t)= F 1 [ H ˆ ( ξ , t ) ] (x),
(2.9)

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 .
(2.10)

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.
(2.11)

3 Decay property

The aim of this section is to establish decay estimates of the solution operators G(t) and H(t) appearing in the solution formula (2.10).

Lemma 3.1 The solution of the problem (2.1), (2.2) satisfies

( | ξ | 2 + | ξ | 4 ) | u ˆ ( ξ , t ) | 2 + | u ˆ t ( ξ , t ) | 2 C e c | ξ | 2 t ( ( | ξ | 2 + | ξ | 4 ) | u ˆ 0 ( ξ ) | 2 + | u ˆ 1 ( ξ ) | 2 )
(3.1)

for ξ R n and t0.

Proof Multiplying (2.1) by u ˆ ¯ t and taking the real part yields

1 2 d d t { | u ˆ t | 2 + | ξ | 2 | u ˆ | 2 + α | ξ | 4 | u ˆ | 2 } +2b | ξ | 2 | u ˆ t | 2 =0.
(3.2)

Multiplying (2.1) by u ˆ ¯ and taking the real part, we obtain

1 2 d d t { 2 b | ξ | 2 | u ˆ | 2 + 2 Re ( u ˆ t u ˆ ¯ ) } + | ξ | 2 | u ˆ | 2 +α | ξ | 4 | u ˆ | 2 | u ˆ t | 2 =0.
(3.3)

Multiplying both sides of (3.3) by b | ξ | 2 and summing up the resulting equation and (3.2) yield

d d t E+F=0,
(3.4)

where

E= 1 2 | u ˆ t | 2 + 1 2 | ξ | 2 | u ˆ | 2 + α 2 | ξ | 4 | u ˆ | 2 + b 2 | ξ | 4 | u ˆ | 2 +b | ξ | 2 Re( u ˆ t u ˆ ¯ )

and

F=b | ξ | 2 | u ˆ t | 2 +b | ξ | 4 | u ˆ | 2 +bα | ξ | 6 | u ˆ | 2 .

A simple computation implies that

C E 0 EC E 0 ,
(3.5)

where

E 0 = ( | ξ | 2 + | ξ | 4 ) | u ˆ | 2 + | u ˆ t | 2 .

Note that

Fc | ξ | 2 E 0 .

It follows from (3.5) that

Fc | ξ | 2 E.
(3.6)

Using (3.4) and (3.6), we get

d d t E+c | ξ | 2 E0.

Thus

E(ξ,t) e c | ξ | 2 t E(ξ,0),

which together with (3.5) proves the desired estimates (3.1). Then we have completed the proof of the lemma. □

Lemma 3.2 Let G ˆ (ξ,t) and H ˆ (ξ,t) be the fundamental solutions of the linear equation of (1.1) in the Fourier space, which are given in (2.6) and (2.7), respectively. Then we have the estimates

( | ξ | 2 + | ξ | 4 ) | G ˆ ( ξ , t ) | 2 + | G ˆ t ( ξ , t ) | 2 C e c | ξ | 2 t
(3.7)

and

( | ξ | 2 + | ξ | 4 ) | H ˆ ( ξ , t ) | 2 + | H ˆ t ( ξ , t ) | 2 C ( | ξ | 2 + | ξ | 4 ) e c | ξ | 2 t
(3.8)

for ξ R n and t0.

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 (3.1) with u ˆ 0 (ξ)=0, we get (3.7).

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 (3.1) with u ˆ 1 (ξ)=0, we get the desired estimate (3.8). The lemma is proved. □

Lemma 3.3 Let G(x,t) and H(x,t) be the fundamental solutions of the linear equation of (1.1), which are given in (2.8) and (2.9), respectively. Then, for k0 and j0, we have

x k G ( t ) ϕ L 2 C ( 1 + t ) ( n 4 + k 2 + j 2 1 2 ) ϕ W ˙ j , 1 +C e c t x ( k 2 ) + ϕ L 2 ,
(3.9)
x k H ( t ) ϕ L 2 C ( 1 + t ) ( n 4 + k 2 + j 2 ) ϕ W ˙ j , 1 +C e c t x k ϕ L 2 ,
(3.10)
x k G t ( t ) ϕ L 2 C ( 1 + t ) ( n 4 + k 2 + j 2 ) ϕ W ˙ j , 1 +C e c t x k ϕ L 2 ,
(3.11)
x k H t ( t ) ϕ L 2 C ( 1 + t ) ( n 4 + k 2 + j 2 + 1 2 ) ϕ W ˙ j , 1 +C e c t x ( k + 2 ) ϕ L 2
(3.12)

and

x k G ( t ) Δ g L 2 C ( 1 + t ) ( n 4 + k 2 + 1 2 ) g L 1 +C e c t x k g L 2 ,
(3.13)
x k G t ( t ) Δ g L 2 C ( 1 + t ) ( n 4 + k 2 + 1 ) g L 1 +C e c t x ( k + 2 ) g L 2 ,
(3.14)

where ( k 2 ) + =max{0,k2} in (3.9).

Proof By the Plancherel theorem and (3.7), the Hausdorff-Young inequality, we obtain

x k G ( t ) ϕ L 2 2 = | ξ | R 0 | ξ | 2 k | G ˆ ( ξ , t ) | 2 | ϕ ˆ ( ξ ) | 2 d ξ + | ξ | R 0 | ξ | 2 k | G ˆ ( ξ , t ) | 2 | ϕ ˆ ( ξ ) | 2 d ξ | ξ | R 0 | ξ | 2 k 2 e c | ξ | 2 t | ϕ ˆ ( ξ ) | 2 d ξ + C e c t | ξ | R 0 | ξ | 2 k ( | ξ | 2 + | ξ | 4 ) 1 | ϕ ˆ ( ξ ) | 2 d ξ | ξ | R 0 | ξ | 2 k 2 + 2 j e c | ξ | 2 t | ξ | 2 j | ϕ ˆ ( ξ ) | 2 d ξ + C e c t x ( k 2 ) + ϕ L 2 2 C | ξ | j ϕ ˆ ( ξ ) L 2 | ξ | R 0 | ξ | 2 k 2 + 2 j e c | ξ | 2 t d ξ + C e c t x ( k 2 ) + ϕ L 2 2 C ( 1 + t ) ( n 2 + k + j 1 ) ( Δ ) j 2 ϕ L 1 2 + C e c t x ( k 2 ) + ϕ L 2 2 ,

where ( k 2 ) + =max{0,k2} and R 0 is a small positive constant. Thus (3.9) follows.

Similarly, using (3.7) and (3.8), respectively, we can prove (3.10)-(3.12).

In what follows, we prove (3.13). By the Plancherel theorem, (3.7) and the Hausdorff-Young inequality, we have

x k G ( t ) Δ g L 2 2 = | ξ | R 0 | ξ | 2 | k | | G ˆ ( ξ , t ) | 2 | ξ | 4 | g ˆ ( ξ ) | 2 d ξ + | ξ | R 0 | ξ | 2 k | G ˆ ( ξ , t ) | 2 | ξ | 4 | g ˆ ( ξ ) | 2 d ξ | ξ | R 0 | ξ | 2 k + 2 e c | ξ | 2 t | g ˆ ( ξ ) | 2 d ξ + C e c t | ξ | R 0 | ξ | 2 k | g ˆ ( ξ ) | 2 d ξ C g ˆ ( ξ ) L 2 | ξ | R 0 | ξ | 2 k + 2 e c | ξ | 2 t d ξ + C e c t x k g L 2 2 C ( 1 + t ) ( n 2 + k + 1 ) g L 1 2 + C e c t x k g L 2 2 ,

where R 0 is a small positive constant. Thus (3.13) follows. Similarly, we can prove (3.14). Thus we have completed the proof of the lemma. □

4 Asymptotic decay of solutions to (1.1), (1.2)

The purpose of this section is to prove asymptotic decay of solutions to the Cauchy problem (1.1), (1.2). We need the following lemma, which comes from [20] (see also [21]).

Lemma 4.1 Assume that f=f(v) is a smooth function. Suppose that f(v)=O( | v | 1 + θ ) (θ1 is an integer) when |v| ν 0 . Then, for 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

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 ,

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

In the previous section, we showed the decay estimates for the solution operators. With this preparation, we can prove the global existence and asymptotic decay of solutions to the integral equation (2.11) and hence to the problem (1.1), (1.2). So 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 4 + k 2 + 1 2 x k u ( t ) L 2 + l s ( 1 + t ) n 4 + l 2 + 1 x l u t ( t ) L 2 } .

For R>0, we define

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

Using Gagliardo-Nirenberg inequality, we obtain

u ( t ) L C x s 0 u ( t ) L 2 θ u ( t ) L 2 1 θ CR ( 1 + t ) n 4 1 2 ,
(4.1)

where

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

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

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

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

u C 0 ( [ 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 C E 0 ( 1 + t ) ( n 4 + k 2 + 1 2 )
(4.2)

and

x l u t ( t ) L 2 C E 0 ( 1 + t ) ( n 4 + l 2 + 1 )
(4.3)

for ks+2 and ls.

Proof Define the mapping

T(u)=G(t) u 1 +H(t) u 0 + 0 t G(tτ)Δf ( u ( τ ) ) dτ.
(4.4)

Using (3.9) and (3.10) with j=2,1, respectively, (3.13), Lemma 4.1 and (4.1), for ks+2, we obtain

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

Thus

( 1 + t ) n 4 + k 2 + 1 2 x k T ( u ) L 2 C E 0 +C R 2 .
(4.5)

It follows from (4.4) that

T ( u ) t = G t (t) u 1 + H t (t) u 0 + 0 t G t (tτ)Δf ( u ( τ ) ) dτ.
(4.6)

By using (3.11) and (3.12) with j=2,1, respectively, (3.14) Lemma 4.1 and (4.1), for ls, we have

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

Thus

( 1 + t ) n 4 + l 2 + 1 x l T ( u ) t L 2 C E 0 +C R 2 .
(4.7)

Combining (4.5), (4.7) and taking E 0 and R suitably small yields

T ( u ) X R.
(4.8)

For u ˜ , u ¯ X R , by using (4.4), we have

T( u ˜ )T( u ¯ )= 0 t G(tτ)Δ [ f ( u ˜ ) f ( u ¯ ) ] dτ.
(4.9)

Thanks to (4.9), (3.13) and Lemma 4.1, (4.1), for ks+2, we obtain

x k ( T ( u ˜ ) T ( 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 4 + k 2 + 1 2 ) ( f ( u ˜ ) f ( u ¯ ) ) L 1 d τ + C t 2 t ( 1 + t τ ) ( n 4 + 1 2 ) x k ( f ( u ˜ ) f ( u ¯ ) ) L 1 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 4 + k 2 + 1 2 ) ( u ˜ L 2 + u ¯ L 2 ) u ˜ u ¯ L 2 + C t 2 t ( 1 + t τ ) ( n 4 + 1 2 ) { ( x k u ˜ L 2 + x k u ˜ L 2 ) u ˜ u ¯ L 2 + ( 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 C R u ˜ u ¯ X 0 t 2 ( 1 + t τ ) ( n 4 + k 2 + 1 2 ) ( 1 + τ ) ( n 2 + 1 ) d τ + C R u ˜ u ¯ X t 2 t ( 1 + t τ ) ( n 4 + 1 2 ) ( 1 + τ ) ( n 2 + k 2 + 1 ) d τ + C C R u ˜ u ¯ X 0 t e c ( t τ ) ( 1 + τ ) ( 3 n 4 + k 2 + 1 ) d τ C R ( 1 + t ) ( n 4 + k 2 + 1 2 ) u ˜ u ¯ X ,

which implies

( 1 + t ) n 4 + k 2 + 1 2 x k ( T ( u ˜ ) T ( u ¯ ) ) L 2 CR u ˜ u ¯ X .
(4.10)

Similarly, for ls, from (4.6), (3.14) and (4.1), we have

x l ( T ( u ˜ ) T ( u ¯ ) ) t L 2 0 t x l G t ( t τ ) Δ [ f ( u ˜ ) f ( u ¯ ) ] L 2 d τ C 0 t 2 ( 1 + t τ ) n 4 l 2 1 ( f ( u ˜ ) f ( u ¯ ) ) L 1 d τ + C t 2 t ( 1 + t τ ) n 4 1 x l ( f ( u ˜ ) f ( u ¯ ) ) L 1 d τ + C 0 t e c ( t τ ) x l + 2 ( f ( u ˜ ) f ( u ¯ ) ) L 2 d τ C R ( 1 + t ) ( n 4 + l 2 + 1 ) u ˜ u ¯ X ,

which implies

( 1 + t ) n 4 + l 2 + 1 x l ( T ( u ˜ ) T ( u ¯ ) ) t L 2 CR u ˜ u ¯ X .
(4.11)

Combining (4.10), (4.11) and taking R suitably small yields

T ( u ˜ ) T ( u ¯ ) X 1 2 u ˜ u ¯ X .
(4.12)

From (4.8) and (4.12), we know that is a strictly contracting mapping. Consequently, we conclude that there exists a fixed point u X R of the mapping , which is a classical solution to (1.1), (1.2). We have completed the proof of the theorem. □

References

  1. Boussinesq J: Théorie des ondes et des remous qui se propagent le long dùn canal rectangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond. J. Math. Pures Appl. 1872, 17: 55–108.

    Google Scholar 

  2. Bona JL, Sachs RL: Global existence of smooth solutions and stability of solitary waves for a generalized Boussinesq equation. Commun. Math. Phys. 1988, 118: 15–29. 10.1007/BF01218475

    Article  MathSciNet  Google Scholar 

  3. Liu Y: Decay and scattering of small solutions of a generalized Boussinesq equation. J. Funct. Anal. 1997, 147: 51–68. 10.1006/jfan.1996.3052

    Article  MathSciNet  Google Scholar 

  4. Cho Y, Ozawa T: On small amplitude solutions to the generalized Boussinesq equations. Discrete Contin. Dyn. Syst. 2007, 17: 691–711.

    Article  MathSciNet  Google Scholar 

  5. Farah L, Wang H: Global solutions in lower order Sobolev spaces for the generalized Boussinesq equation. Electron. J. Differ. Equ. 2012, 2012(41):1–13.

    MathSciNet  Google Scholar 

  6. Varlamov VV: On the Cauchy problem for the damped Boussinesq equation. Differ. Integral Equ. 1996, 9: 619–634.

    MathSciNet  Google Scholar 

  7. Varlamov VV: Existence and uniqueness of a solution to the Cauchy problem for the damped Boussinesq equation. Math. Methods Appl. Sci. 1996, 19: 639–649. 10.1002/(SICI)1099-1476(19960525)19:8<639::AID-MMA786>3.0.CO;2-C

    Article  MathSciNet  Google Scholar 

  8. Varlamov VV: Asymptotic behavior of solutions of the damped Boussinesq equation in two space dimensions. Int. J. Math. Math. Sci. 1999, 22: 131–145. 10.1155/S016117129922131X

    Article  MathSciNet  Google Scholar 

  9. Dai W-R, Kong D-X: Global existence and asymptotic behavior of classical solutions of quasilinear hyperbolic systems with linearly degenerate characteristic fields. J. Differ. Equ. 2007, 235: 127–165. 10.1016/j.jde.2006.12.020

    Article  MathSciNet  Google Scholar 

  10. Kong D-X, Yang T: Asymptotic behavior of global classical solutions of quasilinear hyperbolic systems. Commun. Partial Differ. Equ. 2003, 28: 1203–1220. 10.1081/PDE-120021192

    Article  MathSciNet  Google Scholar 

  11. Nakao M, Ono K: Existence of global solutions to the Cauchy problem for the semilinear dissipative wave equations. Math. Z. 1993, 214: 325–342. 10.1007/BF02572407

    Article  MathSciNet  Google Scholar 

  12. Nishihara K: L p L q estimates of solutions to the damped wave equation in 3-dimensional space and their applications. Math. Z. 2003, 244: 631–649.

    MathSciNet  Google Scholar 

  13. Ono K: Global existence and asymptotic behavior of small solutions for semilinear dissipative wave equations. Discrete Contin. Dyn. Syst. 2003, 9: 651–662.

    Article  MathSciNet  Google Scholar 

  14. Wang W, Wang W: The pointwise estimates of solutions for semilinear dissipative wave equation in multi-dimensions. J. Math. Anal. Appl. 2010, 368: 226–241.

    Article  Google Scholar 

  15. Yang Z: Longtime behavior of the Kirchhoff type equation with strong damping on R n . J. Differ. Equ. 2007, 242: 269–286. 10.1016/j.jde.2007.08.004

    Article  Google Scholar 

  16. Sugitani Y, Kawashima S: Decay estimates of solution to a semi-linear dissipative plate equation. J. Hyperbolic Differ. Equ. 2010, 7: 471–501. 10.1142/S0219891610002207

    Article  MathSciNet  Google Scholar 

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

    Google Scholar 

  18. Wang Y-Z, Liu FG, Zhang YZ: 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 

  19. 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 

  20. Li TT, Chen YM: Nonlinear Evolution Equations. Scientific Press, Beijing; 1989.

    Google Scholar 

  21. Zheng SM Monographs and Surveys in Pure and Applied Mathematics 133. In Nonlinear Evolution Equations. Chapman Hall/CRC, Boca Raton; 2004.

    Chapter  Google Scholar 

Download references

Acknowledgements

This work was supported in part by the NNSF of China (Grant No. 11101144).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Yinxia Wang.

Additional information

Competing interests

The author declares that she has no competing interests.

Author’s contributions

The author completed the paper by herself. The author read and approved the final manuscript.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and permissions

About this article

Cite this article

Wang, Y. Asymptotic decay estimate of solutions to the generalized damped Bq equation. J Inequal Appl 2013, 323 (2013). https://doi.org/10.1186/1029-242X-2013-323

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/1029-242X-2013-323

Keywords