Solvability of the Cauchy problem of nonlinear beam equations in Besov spaces

Nonlinear Analysis 65 (2006) 802–824 www.elsevier.com/locate/na

Solvability of the Cauchy problem of nonlinear beam equations in Besov spaces Ai Guo a,b,∗ , Shangbin Cui a a Department of Mathematics, Sun Yat-Sen University, Guangzhou, Guangdong 510275, People’s Republic of China b School of Mathematical Sciences, South China University of Technology, Guangzhou, Guangdong 510633,

People’s Republic of China Received 14 July 2005; accepted 3 October 2005

Abstract In this paper we study solvability of the Cauchy problem of the nonlinear beam equation ∂t2 u + 2 u = ±u p with initial data in Besov spaces. We prove that, for any 1 ≤ q < ∞, the Cauchy problem of this sp s (R n ), where s = n − 4 and equation is locally well-posed in the Besov spaces B˙ 2,q (R n ) and B2,q p 2 p−1 s > s p , and globally well-posed in these spaces if initial data are small. Moreover we obtain scattering sp results in B˙ 2,q (R n ). c 2005 Elsevier Ltd. All rights reserved.  Keywords: Beam equations; Cauchy problem; Well-posedness; Besov space

1. Introduction and main result This paper is concerned with the Cauchy problem of the nonlinear beam equation ∂t2 u(x, t) + 2 u(x, t) = ±u p (x, t)

for x ∈ R n , t > 0,

(1.1)

u(x, 0) = u 0 (x), ∂t u(x, 0) = u 1 (x)

for x ∈ R n ,

(1.2)

where  is the Laplacian on R n , p is a positive integer, p ≥ 2, and u 0 , u 1 are given functions. Beam equations arise from elasticity theory. We refer the reader to [12,18] for the physical background of such equations. Since beam equations were derived from mechanics long ago, ∗ Corresponding author at: School of Mathematical Sciences, South China University of Technology, Guangzhou, Guangdong 510633, People’s Republic of China. Tel.: +86 208 711 4594; fax: +86 2087 1100 37. E-mail addresses: [email protected] (A. Guo), [email protected] (S. Cui).

c 2005 Elsevier Ltd. All rights reserved. 0362-546X/$ - see front matter  doi:10.1016/j.na.2005.10.002

A. Guo, S. Cui / Nonlinear Analysis 65 (2006) 802–824

803

there has been a lot of work on them in the literature concerning various different topics, including the solvability of initial and initial-boundary value problems, the blow-up of solutions, the existence of periodic solutions, and so on. Here we refer the reader to [2,9–11,20] for recent work on beam equations. In this paper we want to establish local and global solvability for the problem (1.1) and (1.2) sp s (R n ), where s = n − 4 , s > s , and 1 ≤ q ≤ ∞. (R n ) and B2,q in Besov spaces B˙ 2,q p p 2 p−1 sp s n s n n s n p ˙ ˙ Note that B2,2 (R ) = H (R ) and B2,2 (R ) = H (R ). Our interest in this topic is motivated 4 by the following observation: s p = n2 − p−1 is the critical index of the Sobolev space H˙ s (R n ) for Eq. (1.1) under a standard scaling argument. More precisely, if u(x, t) solves (1.1) then, 4

for any λ > 0, u λ (x, t) = λ p−1 u(λx, λ2 t) also solves this equation, with initial data u λ (x, 0) satisfying 4

u λ (·, 0) H˙ s = λ p−1

− n2 +s

u(·, 0) H˙ s .

4 The exponent p−1 − n2 + s is zero if and only if s = s p . Thus, as for other nonlinear dispersive equations (cf. [4–7]), we can conjecture that if s > s p then (1.1) and (1.2) is well-posed at least locally in H s (R n ), and if s < s p then it is not well-posed in H s (R n ). In this paper we shall prove the following results. For any n ≥ 1, if p > 1 + n8 then (1.1) and (1.2) is locally well-posed in sp 4 B˙ 2,q (R n ) (1 ≤ q < ∞) and, for n ≥ 3, if s > s p and either n = 3 and p > 1 + n−2 or n ≥ 4 s and p > 1 + n8 then (1.1) and (1.2) is locally well-posed in B2,q (R n ) (1 ≤ q < ∞). If the initial data are small, then the corresponding global well-posedness results hold. Besides, we shall also sp (R n ). prove some scattering results in B˙ 2,q Here, we particularly mention Ref. [11], where the author studied the following initial-value problem:

for x ∈ R n , t > 0, ∂t2 u + 2 u + u = ± f (u) ∂t u(x, 0) = u 1 (x) for x ∈ R n , u(x, 0) = u 0 (x),

(1.3) (1.4)

where f (u) is a nonlinear function that behaves like u p . It was proved that, if either 1 ≤ n ≤ 4 2 n and p > 1 or n ≥ 5 and 1 < p < n+4 n−4 , then local well-posedness holds in H (R ). Note that the condition p < n+4 n−4 (for n ≥ 5) is equivalent to 2 > s p . Hence our result on local well-posedness is compatible with that of Levandosky [11]. We also refer the reader to the related work on nonlinear wave equations (cf. [8,17,19]) and nonlinear Schr¨odinger equations by Planchon [15, 16] and, on generalized KdV equations and generalized Benjamin–Ono equations, Molinet and Ribaud [13,14]. This work is to a large extent inspired by those references. Before stating our main results, we first introduce some notation. For 1 ≤ p < ∞ and q p 1 ≤ q < ∞, we denote by L t L x the function space on R n × [0, +∞) consisting of measurable n functions f = f (x, t) on R × [0, +∞) satisfying the following condition:
 1/q    f  L qt L xp =

+∞

q/ p

Rn

0 q

p

| f (x, t)| p dx

dt

< ∞;

for q = ∞, the space L t L x is defined similarly by replacing the corresponding integral with q p the supremum norm. For 0 < T < ∞, 1 ≤ p < ∞ and 1 ≤ q < ∞, we denote by L T L x the n n function spaces on R × [0, T ] consisting of measurable functions f = f (x, t) on R × [0, T ] satisfying the following condition:

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A. Guo, S. Cui / Nonlinear Analysis 65 (2006) 802–824


  f  L q L xp = T

T

Rn q

| f (x, t)| dx p

0

1/q

q/ p



< ∞;

dt

p

for q = ∞, the space L T L x is defined similarly by replacing the corresponding integral with the supremum norm. We shall use both F and ˆ to denote the Fourier transform in the space variable, namely u(ξ ˆ ) = (Fu)(ξ ), where ξ ∈ R n , and use ∗ to denote convolution. For s ∈ R we denote s by H˙ (R n ) the homogeneous Sobolev space with index s. For 1 ≤ r ≤ ∞, we denote by r the s dual number of r , i.e., 1r + r1 = 1. For s ∈ R and 1 ≤ q ≤ ∞, we use the notations B˙ 2,q (R n ) s and B2,q (R n ) to denote, respectively, the homogeneous and nonhomogeneous Besov spaces with index (s, 2, q), which are defined as follows. First, take a function ψ ∈ S(R n ) such that  ˆ − j ξ ) = 1, ∀ξ ∈ R n \ {0}, ψ(2 and suppψˆ = {ξ ∈ R n : 2−1 ≤ |ξ | ≤ 2} j ∈Z

 ˆ − j ξ )). Next, for where Z represents the set of all integers. Let ψ0 (x) = F −1 (1 − j ≥1 ψ(2 every j ∈ Z , we denote by  j and S0 , respectively, the following convolution operators: ˆ − j ξ ) fˆ(ξ )),  j f (x) = F −1 (ψ(2

∀ f ∈ S (R n ),

S0 f (x) = F −1 (ψˆ 0 (2− j ξ ) fˆ(ξ )),

∀ f ∈ S (R n ).

Then s B˙ 2,q (R n ) =

 

f ∈ S (R n ) :  f  B˙ s = 2,q  

s (R n ) = B˙ 2,∞

and s B2,q (R n ) =




  

f ∈ S (R n ) :  f  B˙ s

2,∞

 j ∈Z

(2 j s   j f  L 2 )q

 

< ∞ , 1 ≤ q < ∞,  

= sup(2 j s   j f  L 2 ) < ∞ , j ∈Z


s = f ∈ S (R n ) :  f  B2,q

1/q

q S0 f  L 2

 + (2 j s   j f  L 2 )q j ≥1

1/q

 

< ∞ , 1 ≤ q < ∞. 

s (R n ) and B s (R n ) We shall frequently abbreviate the notations L p (R n ), H˙ s (R n ), B˙ 2,q 2,q s s and B2,q . respectively as L p , H˙ s , B˙ 2,q 4 As before, we denote s p = n2 − p−1 , and shall use this notation throughout the whole paper. The first main result of this paper is the following local well-posedness result.

Theorem 1.1. (1) Let n ≥ 1, p ∈ N, p > 1 + n8 and 1 ≤ q < ∞. For any (u 0 , u 1 ) ∈ sp s p −2 ( B˙ 2,q , B˙ 2,q ) there exists a corresponding T > 0 such that the problem (1.1) and (1.2) has a unique solution u in R n × [0, T ] satisfying s p −2 u t ∈ C([0, T ]; B˙ 2,q ).

sp ), u ∈ C([0, T ]; B˙ 2,q

4 or n ≥ 4 (2) Let n ≥ 3, p ∈ N and 1 ≤ q < ∞. If either n = 3 and p > 1 + n−2  s−2 8 s −2 ˙ ˙ H ) and s p < s < s, there exists a and p > 1 + n then, for any (u 0 , u 1 ) ∈ (B2,q , B2,q corresponding T = T (u 0  B s + D −2 u 1  B s ) > 0 such that the problem (1.1) and (1.2) has 2,q

2,q

A. Guo, S. Cui / Nonlinear Analysis 65 (2006) 802–824

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a unique solution u in R n × [0, T ] satisfying    s−2 s ), u t ∈ C [0, T ]; B˙ 2,q u ∈ C([0, T ]; B2,q H˙ −2 . The second main result is the following global well-posedness result for the small initial-value problem. Theorem 1.2. (1) Let n ≥ 1, p ∈ N and p > 1 + n8 . Then there exists a corresponding δ > 0 sp s p −2 such that, for any (u 0 , u 1 ) ∈ ( B˙ 2,∞ , B˙ 2,∞ ) satisfying u 0  B˙ s p + u 1  ˙ s p −2 ≤ δ, the problem 2,∞

B2,∞

(1.1) and (1.2) has a unique global solution u satisfying s p −2 u t ∈ C([0, +∞); B˙ 2,∞ ).

sp u ∈ C([0, +∞); B˙ 2,∞ ),

(2) Let n ≥ 1, p ∈ N, p > 1 + n8 and 1 ≤ q < ∞. Then there exists a corresponding δ > 0 s s −2 such that, for any (u 0 , u 1 ) ∈ ( B˙ p , B˙ p ) satisfying u 0  ˙ s p + u 1  s p −2 ≤ δ, the problem 2,q

2,q

B2,∞

B˙ 2,∞

(1.1) and (1.2) has a unique global solution u satisfying sp u ∈ C([0, +∞); B˙ 2,q ),

s p −2 u t ∈ C([0, +∞); B˙ 2,q ).

4 (3) Let n ≥ 3, p ∈ N and 1 ≤ q < ∞. If either n = 3 and p > 1 + n−2 or n ≥ 4 and p > s−2  ˙ −2 8 s , B˙ 2,q 1 + n , then there exists a corresponding δ > 0, such that for any (u 0 , u 1 ) ∈ (B2,q H ) and s > s p satisfying u 0  B˙ s p +u 1  ˙ s p −2 ≤ δ, the problem (1.1) and (1.2) has a unique global 2,∞

B2,∞

solution u, satisfying

   s−2 s ), u t ∈ L ∞ [0, +∞); B˙ 2,q u ∈ L ∞ ([0, +∞); B2,q H˙ −2 .

We also have the following scattering result. s −2

p p Theorem 1.3. Let n ≥ 1, p ∈ N, p > 1 + n8 and 1 ≤ q < ∞. Let (u 0 , u 1 ) ∈ ( B˙ 2,q , B˙ 2,q ) satisfying u 0  B˙ s p + u 1  ˙ s p −2 ≤ δ, where δ is as in Theorem 1.2(1), and let u(x, t) be the

s

2,∞

B2,∞

+ + ˙ s p ˙ s p −2 solution of (1.1) and (1.2). Then there exists (u + 0 , u 1 ) ∈ ( B2,q , B2,q ) such that, if u L is the solution to the following free beam equation 2 + ∂t2 u + for x ∈ R n , t > 0, L +  uL = 0 + + + ∂t u + for x ∈ R n , u L (x, 0) = u 0 , L (x, 0) = u 1

(1.5) (1.6)

then s lim u(t) − u + L (t) B˙ p = 0.

t →+∞

(1.7)

2,q

The organization of this paper is as follows. Section 2 is devoted to establishing some basic estimates for corresponding linear problems. Section 3 aims to establish nonlinear estimates. In the last section, Section 4, we present the proofs of the above results. 2. Linear estimates Consider the following initial-value problem:

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A. Guo, S. Cui / Nonlinear Analysis 65 (2006) 802–824

∂t2 u + 2 u = f (x, t) u(x, 0) = u 0 (x),

for x ∈ R n , t > 0, ∂t u(x, 0) = u 1 (x)

(2.1)

for x ∈ R n

(2.2)

where f , u 0 and u 1 are given functions. For t = 0 let  1 it |ξ |2 2 (e + e−it |ξ | )eixξ dξ, G 0 (x, t) = (2π)−n n 2 R 1 2 2 −n G 1 (x, t) = (2π) (eit |ξ | − e−it |ξ | )eixξ dξ. 2 R n 2|ξ | i It is clear that ∂ G 1 (x, t) . ∂t

G 0 (x, t) =

For every t > 0, let W (t) be the operator on S (R n ) defined by W (t)ϕ(x) = G 1 (x, t) ∗ ϕ(x)

for ϕ ∈ S (R n ).

Then, for suitable f , u 0 and u 1 , the unique solution of the problem (2.1) and (2.2) is given by u = v + w, where v is the solution of the problem ∂t2 v + 2 v = 0 v(x, 0) = u 0 (x),

for x ∈ R n , t > 0, ∂t v(x, 0) = u 1 (x)

for x ∈ R , n

(2.3) (2.4)

namely v(x, t) = W˙ (t)u 0 (x) + W (t)u 1 (x),

(2.5)

where W˙ (t) is the operator ∂ G 1 (x, t) ∗ ϕ(x), W˙ (t)ϕ(x) = ∂t and w is the solution of the problem ∂t2 w + 2 v = f (x, t) w(x, 0) = 0,

for x ∈ R n , t > 0,

∂t w(x, 0) = 0

(2.6)

for x ∈ R , n

(2.7)

namely  w(x, t) =

t

W (t − τ ) f (x, τ )dτ.

(2.8)

0

Therefore, u(x, t) = W˙ (t)u 0 (x) + W (t)u 1 (x) +



t

W (t − τ ) f (x, τ )dτ.

(2.9)

0

Note that ∂t v(x, t) = W (t)(− 2 u 0 (x)) + W˙ (t)u 1 (x),  t ∂t w(x, t) = W˙ (t − τ ) f (x, τ )dτ. 0

(2.10) (2.11)

A. Guo, S. Cui / Nonlinear Analysis 65 (2006) 802–824

807

Definition 2.1. We say that a pair (γ (r ), r ) is admissible if   1 n 1 1 = − γ (r ) 2 2 r and 2≤r ≤

2n (2 ≤ r ≤ ∞ if n = 1, 2 ≤ r < ∞ if n = 2). n−2

Lemma 2.2 ([3]). Let U (t)ϕ(x) = F −1 (eit |ξ | ϕ(ξ ˆ )) for t > 0 and ξ ∈ R n . We have the following Strichartz estimates: 2

U (t)ϕ L γ (r) L r ≤ Cϕ L 2 , t x  t     U (t − τ ) f (x, τ )dτ    γ (r) 0

Lt

(2.12) ≤ C f  L rx

γ (q)

Lt

q

Lx

,

(2.13)

U (t)ϕ L γ (r) B˙ s ≤ Cϕ H˙ s , t r,2   t    U (t − τ ) f (x, τ )dτ   γ (r) ˙ s ≤ C f  L γt (q) B˙ s , 

(2.14)

U (t)ϕ L γ (r) B s ≤ Cϕ H s , t r,2  t     U (t − τ ) f (x, τ )dτ   

(2.16)

0

Lt

γ (r)

0

Lt

s Br,2

(2.15)

q ,2

Br,2

≤ C f 

γ (q)

Lt

Bqs ,2

,

(2.17)

where s ∈ R, and (γ (r ), r ), (γ (q), q) are admissible pairs, and · L γ (r) B˙ s , · L γ (r) B s represent t

r,2

s s ), L γ (r) ([0, ∞); Br,2 ), respectively. the norm in the spaces L γ (r) ([0, ∞); B˙ r,2

t

r,2

Since s p > 0, we can take a real number s0 > 0 sufficiently small such that s0 < min(2, s p ). 2(n+2) 2n(n+2) n n ∗ Let α = 2(n+2) n , β = n+4 and β = n 2 +(8−2s )n+8−4s , namely β ∗ = β + 2 − s0 . 0

0

Corollary 2.3. The following estimates hold: ˙ (t)u 0 + W (t)u 1  L α L α ≤ C(u 0  L 2 + u 1  ˙ −2 ), (2.18) W˙ (t)u 0 + W (t)u 1  L ∞ 2 + W H t x t Lx  t     W (t − τ ) f (x, τ )dτ    ∞ 2 0 Lt Lx  t     + W (t − τ ) f (x, τ )dτ ≤ C( f  L β H˙ −s0 ). (2.19)   L αt L αx

0

t

β∗

Proof. Taking r = 2 in (2.12) and r = 2, q = β in (2.13), we get, respectively, U (t)ϕ L ∞ 2 ≤ Cϕ L 2 , t Lx  t x    U (t − τ ) f (x, τ )dτ    0

2 L∞ t Lx

≤ C f  L β L β .

(−t ) ˙ Since W (t) = (−∆)−1 U (t )+U , W (t) = 2i

W˙ (t)u 0  L ∞ 2 ≤ Cu 0  L 2 , t Lx

t

x

U (t )−U (−t ) , 2

we have (2.20)

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A. Guo, S. Cui / Nonlinear Analysis 65 (2006) 802–824

W (t)u 1  L ∞ 2 ≤ Cu 1  H ˙ −2 , t Lx  t     W (t − τ ) f (x, τ )dτ    ∞ 2 ≤ C f  L βt H˙β−2 ≤ C f  L βt H˙ β−s∗ 0 . 0

(2.21) (2.22)

Lt Lx

−s The last inequality follows from the Sobolev embedding: H˙ β ∗ 0 ⊂ H˙ β−2. Next, taking r = α in (2.12), (2.13), and q = β in (2.13), we have

U (t)ϕ L αt L αx ≤ Cϕ L 2 ,  t     U (t − τ ) f (x, τ )dτ    0

≤ C f  L β L β .

L αt L αx

t

x

Therefore W˙ (t)u 0  L αt L αx ≤ Cu 0  L 2 , W (t)u 1  L αt L αx ≤ Cu 1  H˙ −2 ,  t     W (t − τ ) f (x, τ )dτ    α α ≤ C f  L βt H˙ β−2 ≤ C f  L βt H˙ β−s∗ 0 . 0

(2.23) (2.24) (2.25)

Lt Lx

Combining (2.20)–(2.25), we obtain the desired estimates (2.18) and (2.19).




3. Nonlinear estimates We shall use the spaces E q , Fq , G q,θ and E q,T , Fq,T , G q,θ,T as our working spaces, which are respectively defined by p E q = C([0, ∞); B˙ 2,q ),

s

Fq = {u(x, t) ∈ S (R n ) : {2 j s p   j u L αt L αx }l q < ∞}, G q,θ = {u(x, t) ∈ S (R n ) : {2 j s p   j u L p¯ L q¯ }l q < ∞}, t

x

and p E q,T = C([0, T ]; B˙ 2,q ),

s

Fq,T = {u(x, t) ∈ S (R n ) : {2 j s p   j u L αT L αx }l q < ∞}, G q,θ,T = {u(x, t) ∈ S (R n ) : {2 j s p   j u L p¯ L q¯ }l q < ∞}, T

θ 2

x

= + and θ = 1 − Clearly 0 < θ < 1. where = Note that, for p > q ≥ 1, the following holds: 1 p¯




 j ∈Z

1−θ 1 α , q¯




1

p

|a j | p

1−θ α ,

≤



n+4 np .

1 q

|a j |q

j ∈Z

Thus, G q,θ ⊆ G ∞,θ for 1 ≤ q < ∞. We claim that  E∞ F∞ ⊂ G ∞,θ .

.

A. Guo, S. Cui / Nonlinear Analysis 65 (2006) 802–824

809

In fact, from 2 2 jsp  j u ∈ L∞ t Lx,

2 j s p  j u ∈ L αt L αx , p¯

q¯

by interpolation with parameter θ , we immediately get 2 j s p  j u ∈ L t L x . Lemma 3.1. Let u ∈ G ∞,θ . Then the following inequality holds: sup 2 j (s p −s0 )   j (u p ) L β L β ∗ ≤ C sup(2 j s p   j u L p¯ L q¯ ) p . x

t

j

t

j

(3.1)

x

∞ ˆ −l Proof. For every j ∈ Z , we denote ψ j (x) = F −1 (1 − l= j +1 ψ(2 ξ ))  j ˆ −l ξ )), and define an operator S j : S (R) → S(R) as follows: F −1 ( l=−∞ ψ(2 S j f (x) = F −1 (ψˆ j (ξ ) fˆ(ξ )),

=

∀ f ∈ S (R).

Then, from [13,16], we know that the following identity holds: ∞ 

u p (x) =

l+1 (u)

l=−∞

p−1 

(Sl+1 (u))k (Sl (u)) p−k−1 .

k=0

 p−1 Since the support of frequencies of the function l+1 (u) k=0 (Sl+1 (u))k (Sl (u)) p−k−1 is contained in the ball |ξ | ≤ 2 p+l , we get   p−1 ∞   p k p−k−1  j (u ) =  j l+1 (u) (Sl+1 (u)) (Sl (u)) l= j − p

 = j



k=0

l (u)(Sl (u))

 + the rest terms

p−1

l≥ j

≡ I + II.

(3.2)

Using the uniform boundedness of the operators  j in L p (R) (1 ≤ p ≤ ∞) and the H¨older inequality, we have   l (u)Sl (u) p−1  L β L β ∗ 2 j (s p −s0 ) I L β L β ∗ ≤ C2 j (s p −s0 ) t

x

≤ C2 j (s p −s0 )

 l (u) L p¯ L q¯ Sl (u) t

l≥ j

Since Sl u =

l r=−∞

 r (u)

r (u) and

β ∗ q¯ ( p−1) q−β ¯ ∗ Lx

≤ C2

β ∗ q¯ ∗ (p q−β ¯

x

t

l≥ j



p−1

x

Lt

β p¯ ( p−1) p−β ¯

β ∗ q¯ ∗ ( p−1) q−β ¯

.

(3.3)

Lx

− 1) > q, ¯ then from the Bernstein inequality ∗

¯ nr( q1¯ − β ∗q−β ) q( ¯ p−1)

 r (u) L q¯ , x

we have Sl (u) Lt

β p¯ ( p−1) p−β ¯

β ∗ q¯ ( p−1) q−β ¯ ∗ Lx

≤C

l 

 r (u)

r=−∞

≤C

l  r=−∞

Lt

β p¯ ( p−1) p−β ¯ ∗

2

¯ nr( q1¯ − β ∗q−β ) q( ¯ p−1)

β ∗ q¯ ∗ ( p−1) q−β ¯

Lx

 r u L p¯ L q¯ , t

x

(3.4)

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A. Guo, S. Cui / Nonlinear Analysis 65 (2006) 802–824

where we have used the fact that θ = 1 −

n+4 np ,

namely

β p¯ (p p−β ¯

− 1) = p. ¯ Note that

−s0 n n(q¯ − β ∗ ) + − ∗ = sp. p − 1 q¯ β q( ¯ p − 1) Therefore Sl (u) Lt

β p¯ ( p−1) p−β ¯

β ∗ q¯ ( p−1) q−β ¯ ∗ Lx

l 

≤C

r

2

s0 p−1

2rs p  r (u) L p¯ L q¯ t

r=−∞ l

= C2

l

= C2

s0 p−1

s0 p−1

l 

−s

2

r=−∞ ∞ 

0 (l−r) p−1 rs p

2

 r (u) L p¯ L q¯ t

s

2

x

0 ) (l−r)(− p−1

x

χ(l−r≥0) 2rs p  r (u) L p¯ L q¯ , (3.5) t

r=−∞

x

where χ(l−r≥0) = 1 for l ≥ r , and χ(l−r≥0) = 0 for l < r . From (3.3) it follows that  ∞  2( j −l)(s p −s0 ) 2ls p  l (u) p¯ q¯ 2 j (s p −s0 ) I L β L β ∗ ≤ C L L x

t

t

l= j


·

∞ 

s

2

0 ) (l−r)(− p−1

=C

 p−1  χ(l−r≥0) 2rs p  r (u) L p¯ L q¯



2( j −l)(s p −s0 ) χ( j −l≤0)2ls p  l (u) p¯ q¯ L L

·

∞ 

x

t

l∈Z






x

t

r=−∞



x

s

2

0 ) (l−r)(− p−1

 p−1  χ(l−r≥0) 2rs p  r (u) L p¯ L q¯ t

r=−∞

 p ≤ C sup 2 j s p   j (u) L p¯ L q¯ , t

j

x



x

(3.6)

where χ( j −l≤0) = 1 for l ≥ j , and χ( j −l≤0) = 0 for l < j . The last inequality follows from the fact that s p > s0 and from the Young inequality. Similarly, we also have  p 2 j (s p −s0 ) II L β L β ∗ ≤ C sup 2 j s p   j (u) L p¯ L q¯ . t

This proves (3.1).

x

t

j

x




4. Proofs of the main results From (2.9), we see that the problem (1.1) and (1.2) is equivalent to the following integral equation:  t u(x, t) = W˙ (t)u 0 (x) + W (t)u 1 (x) ± W (t − τ )u p (x, τ )dτ. (4.1) 0

A. Guo, S. Cui / Nonlinear Analysis 65 (2006) 802–824

811

For a given pair u 0 and u 1 in suitable function spaces, let S be the mapping u → Su defined by  t (Su)(x, t) = W˙ (t)u 0 (x) + W (t)u 1 (x) ± W (t − τ )u p (x, τ )dτ 0

≡ uL + uNL,

(4.2)

t

where u L = W˙ (t)u 0 (x) + W (t)u 1 (x) and u N L = ± 0 W (t − τ )u p (x, τ )dτ . Then clearly a solution of (4.1) is a fixed point of the mapping S and vice versa. Thus, to prove Theorems 1.1 and 1.2, we only need to prove that the mapping S has fixed points in suitable function spaces. First we recall a lemma given by Planchon [15]: s,q Lemma 4.1. Let s > 0, E be a Banach space, q ∈ [1, ∞], and define B˙ E by the following condition: s,q iff {2 j s   j f  E }l q < ∞. f ∈ B˙ E  j js ˆ ˙ s,q Then, if f = j f j , where f j ∈ B(0, 2 ) and {2  f j  E }l q < ∞, we have f ∈ B E with appropriate norm control.

Using this lemma, we can prove the following preliminary result: Lemma 4.2. Let f k ∈ G ∞,θ for 1 ≤ k ≤ p, then the following inequality holds:   p p      j (s p −s0 )  sup 2 fk  ≤C  fk G ∞,θ .  j   β β∗ j 1 1

(4.3)

Lt Lx

Proof. The proof of this Lemma isessentially routine modifications of Lemma 3.1, therefore we only sketch the proof. Since f k = ∞ j =−∞  j f k , by Lemma 4.1 the key point here is to estimate   p     2 j (s p −s0 )   j f k  .  1  β β∗ Lt Lx

Applying the H¨older inequality and the Bernstein inequality, we have   p p     j (s p −s0 )  j (s p −s0 )   f ≤ 2  f    j f k  β p¯ ( p−1) β ∗ q¯ ( p−1) 2  j k j 1 L p¯ L qx¯ ∗ t  β β∗  1 p−β ¯ q−β ¯ 2 L L Lt Lx

t

≤ 2 j (s p −s0 )   j f1  L p¯ L q¯ x

t

= 2 j s p   j f 1  L p¯ L q¯ t

x

= 2 j s p   j f 1  L p¯ L q¯ t

≤C

p 

x

p 

p  2 p 

2

q−β ¯ ∗) j ( qn¯ − βn(∗ q( ) ¯ p−1)

Lt ∗

−s

2

  j fk 

2 ¯ ) 0 + n − n(q−β j ( p−1 ) q¯ β ∗ q( ¯ p−1)

  j f k  L p¯ L q¯ t

2 j s p   j f k  L p¯ L q¯ t

2

 fk G ∞,θ .

1

This concludes the proof of this lemma.

β p¯ ( p−1) q¯ p−β ¯ Lx




x

x

x

812

A. Guo, S. Cui / Nonlinear Analysis 65 (2006) 802–824

Similarly, we can also prove the following lemma: Lemma 4.3. Let f 1 ∈ G q,θ (1 ≤ q < ∞) and f k ∈ G ∞,θ for 2 ≤ k ≤ p. Then the following inequality holds:   q  q1   p p       ∞   2 j (s p−s0 )   fk  ≤  f1 G q,θ  fk G ∞,θ . (4.4)  j   β β∗   j =−∞ 1 2 Lt Lx

Proof. By similar calculations as in the proofs of Lemmas 3.1 and 4.2, we have   q  q1   p ∞       2 j (s p −s0 )     j fk    β β∗   j =−∞ 1

 ≤

∞ 

Lt Lx




j =−∞

≤  f 1 G q,θ

2 j s p   j f 1  L p¯ L q¯ t

p 

p 

x

q  q1 2 j s p   j f k  L p¯ L q¯ t

2

x

 fk G ∞,θ .

2




This ends the proof of Lemma 4.3.

s ,B ˙ s−2 Lemma 4.4. Let (u 0 , u 1 ) ∈ (B2,q 2,q ν > 0 such that



H˙ −2) and s > s p . Then there exist s p < s < s and

u L  Fq,T ≤ C T ν (u 0  B s + (−∆)−1 u 1  B s ). 2,q

(4.5)

2,q

s1 0 ⊂ Lα, ⊂ B˙ α,2 Proof. Since 2 < α, for any 2 < µ < α satisfying 0 < α − µ << 1 we have B˙ µ,2

where s1 = µn − αn . Let γ = γ (µ), namely γ1 = n2 ( 12 − µ1 ); (γ (µ), µ) is an admissible pair. It is clear that γ > α. Thus, from the H¨oder inequality and Lemma 2.2, we get 1  q q  jsp u L  Fq,T = 2   j u L  L αT L αx  ≤

j ∈Z

q  1 −1 2 j s p T α γ   j u L L γ L α T

j ∈Z



≤C

 j ∈Z

≤ CT

T



 ≤ CT

q

x

q  q1 2

jsp

  j u L  L γ B˙ s1 T

1 1 α−γ

1 1 α−γ

1 1 α−γ

1

µ,2

q  2 j s p (2 j s1   j u 0  L 2 + 2 j (s1−2)   j u 1  L 2 )

1

j ∈Z

q  2 j (s p +s1 )   j u 0  L 2 + 2 j (s p −2+s1 )   j u 1  L 2 j ∈Z

q

1 q

A. Guo, S. Cui / Nonlinear Analysis 65 (2006) 802–824

813

≤ C T ν (u 0  B˙ s + u 1  B˙ s −2 ), 2,q

2,q

ν

≤ C T (u 0  B s + (−∆)−1 u 1  B s ), 2,q

where ν =

−

1 α

2,q

and s = s p + s1 = s p + n( µ1 − α1 ). Hence the desired assertion holds.

1 γ




s −2

p p Lemma 4.5. Let p ∈ N, (u 0 , u 1 ) ∈ ( B˙ 2,q , B˙ 2,q ) and u ∈ G q,θ,T , 1 ≤ q < ∞. Let S be the

s

4 or n ≥ 4 and p > 1 + n8 , nonlinear operator defined by (4.2). If either n = 3 and p > 1 + n−2 then there exists a corresponding ν > 0 such that the following inequality holds:

Su L p L xp ≤ C T ν (u 0  B˙ s p + u 1  ˙ s p −2 + uG q,θ,T ). p

T

(4.6)

B2,q

2,q

 Proof. Since Su = ∞ j =−∞  j Su, from the Minkowski inequality we see that (4.6) follows if we prove that  p   j Su L p L xp ≤ C T ν (u 0  B˙ s p + u 1  ˙ s p −2 + uG q,θ,T ). (4.7) T

j ∈Z

B2,q

2,q

We first consider the low frequency terms in the left-hand side of (4.7). Note that the pair (∞, 2) is admissible. Then, from s1 ⊂ B˙ 0p,2 ⊂ L p , B˙ 2,2

where s1 = ( n2 − np ), from Lemma 2.2 and Lemma 6.2.1 in [1], for any s2 ≥ s1 we have   j u L  L ∞ L xp ≤ C  j u L  L ∞ B˙ s1 T

T

2,2

≤ C(  j W˙ (t)u 0  L ∞ B s2 +   j W (t)u 1  L ∞ B s2 ) T

≤ C(2

j s2

T

2,2

  j u 0L 2 + 2

j (s2 −2)

2,2

  j u 1  L 2 ).

Therefore 1 1   q q q q   j (s p −s2 ) j (s −s ) 2 2 p 2   j u L  L ∞ B˙ s1   j u L  L ∞ L xp ≤C j ∈Z

≤C

T





2

jsp

j ∈Z

T

j ∈Z

  j u 0L 2 + 2

j (s p −2)

  j u 1L 2

q

1 q

≤ C(u 0  B˙ s p + u 1  ˙ s p −2 ).

(4.8)

B2,q

2,q

Similarly, from Lemma 2.2 and Lemma 6.2.1 in [1] we get   j u N L  L ∞ L xp ≤ C  j u N L  L ∞ B s2 ≤ C  j u p  T

T

≤ C  j u p  ≤ C2

j (s2 −s0 )

2,2

β

2,2

s −2

L T Hβ2

β

≤ C  j u p  L β H s2 −s0 T

  j u L β L β∗ . p

Thus, from the proof of Lemma 3.1, we have

s −2

2 L T Bβ,2

T

x

β∗

814

A. Guo, S. Cui / Nonlinear Analysis 65 (2006) 802–824



q  2 j (s p −s2 )   j u N L  L ∞ L xp

1



q  ≤C 2 j (s p −s0 )   j u p  L β L β ∗

q

T

j ∈Z

T

j ∈Z

p−1

1 q

x

p

≤ CuG ∞,θ,T uG q,θ,T ≤ CuG q,θ,T . This, combined with (4.8) and (4.9), yields 1  q q  p j (s p −s2 ) 2   j (Su) L ∞ L xp ≤ C(u 0  B˙ s p + u 1  ˙ s p −2 + uG q,θ,T ). T

j ∈Z

B2,q

2,q

(4.9)

(4.10)

Now let s2 = s1 + n2 . Clearly, s p − s2 = −

n n 4 − + < 0. p−1 2 p

Thus, from (4.10) and the H¨oder inequality, we have −1  j =−∞

  j Su L ∞ L xp = T

−1  j =−∞



2− j (s p −s2 ) 2 j (s p −s2 )   j Su L ∞ L xp T

−1  

≤

2

j (s p −s2 )

−1  

×

 q1

T

j =−∞



  j Su L ∞ L xp

q

2

− j (s p −s2 )

q

 1 q

j =−∞ p

≤ C(u 0  B˙ s p + u 1  ˙ s p −2 + uG q,θ,T ). B2,q

2,q

This implies, again from the H¨older inequality, that −1  j =−∞

1

p

  j Su L p L xp ≤ C T p (u 0  B˙ s p + u 1  ˙ s p −2 + uG q,θ,T ). T

2,q

B2,q

(4.11)

Next we consider the high frequency terms in the left-hand side of (4.7). Since 4 < p<∞ for n = 3, and n−2 8 1+ < p <∞ for n > 4, n 1+

4 n 2 n n 4 n p−1 − p < min( p , 2 − p ). Take a real number a0 such that a0 ≥ 0 and p−1 − p < 2 n n 2 2n 1 n 1 1 2n a0 ≤ min( p , 2 − p ). Let l = a0 and p1 = n−2a0 . Clearly, l = 2 ( 2 − p1 ) and 2 ≤ p1 ≤ n−2 , so that (l, p1 ) is an admissible pair. Note that, by taking s1 = pn1 − np = n2 − np − a0 (> 0), we have

it follows that

B˙ sp11 ,2 ⊂ B˙ 0p,2 ⊂ L p . Thus, from Lemma 2.2 and Lemma 6.2.1 in [1], we get   j u L Ll

T

p

Lx

≤ C  j u L  L l

T

s B˙ p1 ,2 1

≤ C(2 j s1   j u 0  L 2 + 2 j (s1−2)   j u 1  L 2 ).

A. Guo, S. Cui / Nonlinear Analysis 65 (2006) 802–824

Therefore   2 j (s p −s1 )   j u L  L l j ∈Z

≤C

T



 j ∈Z

q

1 q

p

Lx

815

q  q1



 2 j (s p −s1 )   j u L  L l ≤C

T

j ∈Z

2 j s p   j u 0  L 2 + 2 j (s p −2)   j u 1  L 2

q

s B˙ p1 ,2 1

1 q

≤ C(u 0  B˙ s p + u 1  ˙ s p −2 ).

(4.12)

B2,q

2,q

Similarly, from Lemma 2.2 and Lemma 6.2.1 in [1], we get   j u N L Ll

T

p

Lx

≤ C  j u N L  L l

T

≤ C  j u p 

β

s B˙ p1 ,2

≤ C  j u p 

1

s −2

L T H˙ β1

s −2

β

1 L T B˙ β,2

≤ C2 j (s1−s0 )   j u p  L β L β ∗ . T

x

Thus, from the proof of Lemma 3.1, we have 1 1   q q q q   j (s p −s1 ) j (s −s ) p p 0   j u N L  L l L xp ≤C   j u L β L β∗ 2 2 T

j ∈Z

T

j ∈Z

p−1

x

p

≤ CuG ∞,θ,T uG q,θ,T ≤ CuG q,θ,T . This, combined with (4.12) and (4.13), gives  1 q q  p 2 j (s p −s1 )   j (Su) L l L xp ≤ C(u 0  B˙ s p + u 1  ˙ s p −2 + uG q,θ,T ). T

j ∈Z

B2,q

2,q

(4.13)

(4.14)

4 Next, since l > p and s p −s1 = a0 + np − p−1 > 0, from (4.14) and the H¨oder inequality we have +∞  j =0

  j Su L l

p T Lx

=

+∞  j =0

≤

2− j (s p −s1 ) 2 j (s p −s1 )   j Su L l

 +∞  

2

j (s p −s1 )

j =0

×

p

T

 +∞  

2

  j Su L l

q

Lx

1 q

p T Lx

− j (s p −s1 )

q

 1 q

j =0 p

≤ C(u 0  B˙ s p + u 1  ˙ s p −2 + uG q,θ,T ). B2,q

2,q

Using the H¨older inequality again we obtain +∞  j =0

  j Su L p L xp ≤ C T T

( 1p − 1l )

p

(u 0  B˙ s p + u 1  ˙ s p −2 + uG q,θ,T ). 2,q

B2,q

Summing up (4.14) and (4.15), we get (4.6). This completes the proof. We are now ready to prove Theorems 1.1 and 1.2




(4.15)

816

A. Guo, S. Cui / Nonlinear Analysis 65 (2006) 802–824

sp s p −2 Proof of Theorem 1.2. (1) For a given (u 0 , u 1 ) ∈ ( B˙ 2,∞ , B˙ 2,∞ ) and a positive number a to be specified later, we define a metric space (Ya , d) as follows: ! "  Ya = w ∈ E ∞ F∞ : w E∞ + w F∞ ≤ a ,

d(u, v) = u − v E∞ + u − v F∞ . It is clear that (Ya , d) is a nonempty complete metric space. Let S be the mapping defined by (4.2). We prove that, for s suitably chosen a, S maps Ya into itself, and is a contraction mapping when restricted on Ya . By localizing in frequency and using (2.18) we get ˙ (t)  j u 0 + W (t)  j u 1  L α L α W˙ (t)  j u 0 + W (t)  j u 1  L ∞ 2 + W t x t Lx ≤ C(  j u 0  L 2 + 2−2 j   j u 1  L 2 ). Hence jsp 2 j s p   j u L L ∞   j u L  L αt L αx ≤ C(2 j s p   j u 0  L 2 + 2 j (s p −2)   j u 1  L 2 ). 2 +2 t Lx

It follows that u L  E∞ + u L  F∞ ≤ C(u 0  B˙ s p + u 1  ˙ s p −2 ).

(4.16)

B2,∞

2,∞

Similarly, by localizing in frequency and using (2.19) and Lemma 6.2.1 in [1] we get   t   t          j  j w(t − τ ) f (x, τ )dτ + w(t − τ ) f (x, τ )dτ  ∞ 2   α α  0

≤ C2

−s0 j

0

Lt Lx

Lt Lx

  j ( f (x, t)) L β L β ∗ . t

(4.17)

x

Hence jsp 2 j s p   j u N L L ∞   j u N L  L αt L αx ≤ C2 j (s p −s0 )   j (u p ) L β L β ∗ . 2 +2 t Lx t

x

It follows, from Lemma 3.1, that u N L  E∞ + u N L  F∞ ≤ C sup 2 j (s p −s0 )   j (u p ) L β L β ∗ ≤

p CuG ∞,θ

≤

j p  Cu F∞ E∞

t

≤ Ca

p

for u ∈

x

YaT .

(4.18)

Combing (4.16) and (4.18) we see that, for any u ∈ Ya , Su E∞ + Su F∞ ≤ C(u 0  B˙ s p + u 1  ˙ s p −2 ) + Ca p ≤ C(δ + a p ). 2,∞

(4.19)

B2,∞

−

p

We now take an a > 0 sufficiently small such that Ca p−1 ≤ 14 , and δ ≤ (4C) p−1 . Then we get Su E∞ + Su F∞ ≤ a. Hence S maps Y a into itself. Next we prove that S is a contraction mapping when restricted on Ya . Let u i ∈ YTa , i = 1, 2. From Lemma 4.2 we have Su 1 − Su 2  E∞ + Su 1 − Su 2  F∞ ≤ C2 j (s p −s0 )   j (u 1 − u 2 ) L β L β ∗ t x    p−1      p−k−1 = C2 j (s p −s0 )  j (u 1 − u 2 ) u k1 u 2    k=0 p

p

β

β∗

Lt Lx

A. Guo, S. Cui / Nonlinear Analysis 65 (2006) 802–824 p−1

817

p−1

≤ C(u 1 G ∞,θ + u 2 G ∞,θ )u 1 − u 2 G ∞,θ ≤

1 (u 1 − u 2  E∞ + u 1 − u 2  F∞ ), 2

and the desired assertion follows. From the Banach fixed point theorem, the assertion (1) immediately follows. sp s p −2 (2) For a given (u 0 , u 1 ) ∈ ( B˙ 2,q , B˙ 2,q ), and the constant C in (4.19), we define a metric space (Y, d) as follows: #  Y = w ∈ Eq Fq : w Eq + w Fq ≤ 2C(u 0  B˙ s p + u 1  ˙ s p −2 ), B2,q 2,q $ − 1 w E∞ + u − v F∞ ≤ (4C) p−1 , d(u, v) = u − v Eq + u − v Fq . From Theorem 1.2 (1) we see that Su E∞ + Su F∞ ≤ (4C)

1 − p−1

u E∞ + u F∞ ≤ (4C)

if

1 − p−1

.

Similarly as in the proofs of inequalities (4.16) and (4.18), we have u L  Eq + u L  Fq ≤ C(u 0  B˙ s p + u 1  ˙ s p −2 ), 2,q

(4.20)

B2,q

p−1

u N L  Eq + u N L  Fq ≤ CuG ∞,θ uG q,θ .

(4.21)

Combining (4.20) and (4.21), we obtain p−1

Su Eq + Su Fq ≤ C(u 0  B˙ s p + u 1  ˙ s p −2 + uG ∞,θ uG q,θ ). 2,q

(4.22)

B2,q

From Theorem 1.2(1) we see that, for any u ∈ Y , Su Eq + Su Fq ≤ 2C(u 0  B˙ s p + u 1  ˙ s p −2 ). 2,q

B2,q

Hence S maps Y into itself. Moreover, for u i ∈ Y (i = 1, 2), Lemma 4.3 shows that  1 ∞  q q  p p 2 j (s p−s0 )   j (u 1 − u 2 ) L β L β ∗ Su 1 − Su 2  Eq + Su 1 − Su 2  Fq ≤ C t

j =−∞

     p−1   ∞      p−k−1 2 j (s p −s0 )  j (u 1 − u 2 ) =C u k1 u 2   j =−∞   k=0 p−1

β

x

q  q1    β∗

Lt Lx

p−1

≤ C(u 1 G ∞,θ + u 2 G ∞,θ )u 1 − u 2 G q,θ ≤

1 (u 1 − u 2  Eq + u 1 − u 2  Fq ). 2

Thus S is a contraction mapping on Y , and the second assertion of Theorem 1.2 is proved. The third assertion of Theorem 1.2 will be proved later.

818

A. Guo, S. Cui / Nonlinear Analysis 65 (2006) 802–824

Proof of Theorem 1.1. (1) We replace the time interval [0, ∞) by [0, T ] in all estimates. Since u L  Eq,T + u L  Fq,T ≤ C(u 0  B˙ s p + u 1  ˙ s p −2 , ) B2,q

2,q

we have, for ( p1 , q1 ) = (∞, 2) and ( p2 , q2 ) = (α, α), 1  ∞  q q  2 j s p   j u L  L pi L qi < +∞, i = 1, 2. x

T

j =−∞

This implies that there exists jn ∈ N such that, for | j | ≥ jn ,  1 q q   1 jsp 2   j u L  L pi L qi < δ, T x 2 | j |≥ j

(4.23)

n

where δ is a constant that can be small enough. For this jn we have  1 q q   jsp 2   j u L  L pi L qi lim = 0. T →0

x

T

| j |< jn

Therefore, there exists T0 > 0 such that, for 0 < T < T0 ,  1 q q   1 jsp 2   j u L  L pi L qi < δ. T x 2 | j |< j

(4.24)

(4.25)

n

From this inequality and (4.23) and (4.25) we get, for 0 < T < T0 , u L  Eq,T + u L  Fq,T < δ. Now define a metric space (YaT , d) as follows: ! "  YaT = w ∈ E q,T Fq,T : w Eq,T + w Fq,T < a , d(u, v) = u − v Eq,T + u − v Fq,T . Then, by using a similar argument to that used in the proof of Theorem 1.2(1), we get the result of Theorem 1.1(1). (2) If u 0 = u 1 = 0, then it is clear that u = 0 is a solution of (4.1). In what follows we assume that |u 0 | + |u 1 | = 0. For two positive numbers a, T to be specified later, we redefine the metric space (YTa , d) as follows: !  p p  s YTa LT Lx : = w ∈ C([0, T ]; B2,q ) Fq,T " s + wG s w Fq,T + wG q,θ,T + w L p L xp + λ(w L ∞ ) < a , B T 2,q q,θ,T T

d(u, v) = u − v Fq,T + u − vG q,θ,T + u − v L p L xp T

s + u − vG s + λ(u − v L ∞ ), q,θ,T T B2,q

where λ =

W (t )u 0 +W (t )u 1  Fq,T

u 0  B s +D −2 u 1  B s 2,q

, and

2,q

wG sq,θ,T = {2 j s   j u L p¯ L q¯ }l q T

x

A. Guo, S. Cui / Nonlinear Analysis 65 (2006) 802–824

with

1 p¯

=

=

1−θ 1 α , q¯

θ 2

+

1−θ α ,

and θ = 1 −

n+4 np .

819

u 0  ˙ s +u 1  ˙ s −2

Let s and ν be as in Lemma 4.4. Then λ ≤ C T ν u

B 2,q

0  B s +D 2,q

B 2,q

−2 u

1  Bs 2,q

. We shall prove that, for

suitably chosen a and T , S maps YTa into itself and is a contraction mapping. We first assume that 0 < T ≤ 1. First, from Corollary 2.3, Lemma 4.4 and the proof of Lemma 3.1 we see that, for any u ∈ YTa s−2 s and (u 0 , u 1 ) ∈ (B2,q , B2,q ), there hold −1 {2 j s p   j u L  L ∞ u 1  B s ), 2 }l q ≤ C(u 0  s + (−∆) B t Lx 2,q

2,q

u L  Fq,T ≤ C T ν (u 0  B s + (−∆)−1 u 1  B s ), 2,q

2,q

p

{2 j s p   j u N L  L ∞ , 2 }l q ≤ Cu G q,θ,T t Lx p

u N L  Fq,T ≤ CuG q,θ,T . Thus, by complex interpolation with θ , we get u L G q,θ,T ≤ C T (1−θ)ν (u 0  B s + (−∆)−1 u 1  B s ), 2,q

u N L G q,θ,T ≤

2,q

p CuG q,θ,T .

It follows that Su Fq,T + SuG q,θ,T ≤ C(T (1−θ)ν (u 0  B s + (−∆)−1 u 1  B s ) + a p ) for u ∈ YTa . 2,q

2,q

(4.26) s , it is clear that Secondly, from the definition of the Besov space B2,q s ≤ S0 (Su) L ∞ L 2x + Su L ∞ B˙ s Su L ∞ T B2,q T

T

2,q

≤ S0 (u L ) L ∞ L 2x + S0 (u N L ) L ∞ L 2x + u L  L ∞ B˙ s + u N L  L ∞ B˙ 2,q . T

T

T

2,q

T

Since S0 (u L ) L ∞ L 2x T

S0 (u N L ) L ∞ L 2x T

≤ C(S0 (u 0 ) L 2 + S0 (u 1 ) H˙ −2 ) ≤ C(u 0  L 2 + u 1  H˙ −2 ),   t    p  = W (t − τ )u (τ )dτ S   0 L ∞ L 2x

0

T  t      p  ≤ sup  W (t − τ )(S0 (u (τ ))dτ )  t ∈[0,T ]  

0

L 2x

t ≤ sup

t ∈[0,T ]

(G 1 (·, t − τ ) ∗ ψ0 ) ∗ (u p (τ )) L 2 dτ 0

t ≤ sup

t ∈[0,T ]

(G 1 (·, t − τ ) ∗ ψ0 ) L 2 (u p (τ )) L 1 dτ 0 p

≤ C(T + 1)u L p L p , T

x

820

A. Guo, S. Cui / Nonlinear Analysis 65 (2006) 802–824

we have p

S0 (Su) L ∞ L 2x ≤ C(u 0  L 2 + u 1  H˙ −2 + (T + 1)u L p L p ). T

T

(4.27)

x

Using Corollary 2.3, and following the proof of Lemma 3.1, we deduce that  u L  L ∞ B˙ s T

= sup

2,q

t ∈[0,T ]

 ≤  ≤

q  2 j s   j u L L 2

1 q

j ∈Z

q  2 j s (W˙ (t)  j u 0  L 2 + W (t)  j u 1  L 2 ) j ∈Z

q  2 j s   j u 0  L 2 + 2 j (s−2)  j u 1  L 2

1 q

1 q

j ∈Z

≤ C(u 0  B˙ s + u 1  B˙ s−2 ),  u N L  L ∞ B˙ s = sup T

2,q

t ∈[0,T ]

2,q

2,q

q  2 j s   j u N L L 2

1 q

j ∈Z



q  2 j (s−s0)   j (u p ) L β L β ∗ ≤C t

j ∈Z

1 q

x

p−1

≤ CuG sq,θ,T uG ∞,θ,T . Thus  Su L ∞ B˙ s ≤ T

2,q

 j ∈Z

1 q

(2   j Su L ∞ L 2x ) js

q

T

p−1

≤ C(u 0  B˙ s + u 1  B˙ s−2 + uG sq,θ,T uG ∞,θ,T ). 2,q

We now estimate 

%

(4.28)

2,q

 q & q1 j s   Su α α 2 . From Corollary 2.3, we see that L L j j ∈Z T x

q  2 j s   j u L  L αT L αx

1



q  ≤ 2 j s   j u 0  L 2 + 2 j (s−2)  j u 1  L 2

q

j ∈Z

j ∈Z

≤ C(u 0  B˙ s + u 1  B˙ s−2 ), 

 j ∈Z

2 j s   j u N L  L αT L αx

q



1 q

2,q



2,q

q  2 j (s−s0)   j (u p ) L β L β ∗ ≤C t

j ∈Z

p−1

≤ CuG sq,θ,T uG ∞,θ,T .

x

1 q

1 q

A. Guo, S. Cui / Nonlinear Analysis 65 (2006) 802–824

821

Therefore, 1  q q  p−1 js 2   j Su L αT L αx ≤ C(u 0  B˙ s + u 1  B˙ s−2 + uG sq,θ,T uG ∞,θ,T ). (4.29) 2,q

j ∈Z

2,q

By complex interpolation, from (4.28) and (4.29) we obtain p−1

SuG sq,θ,T ≤ C(u 0  B˙ s + u 1  B˙ s−2 + uG sq,θ,T uG ∞,θ,T ) 2,q

2,q

−1 p s + (−∆) s +a ) ≤ C(u 0  B2,q u 1  B2,q

Hence, using the fact that λ ≤

CT ν

u 0  ˙ s +u 1  ˙ s −2 B 2,q

B 2,q

u 0  B s +D −2 u 1  B s 2,q

≤ 1), we obtain

for u ∈ YTa .

(4.30)

and, particularly, λ ≤ C (for 0 < T

2,q

−1 p s + SuG s s + (−∆) s ) + λa ) ) ≤ C(λ(u 0  B2,q u 1  B2,q λ(Su L ∞ T B2,q q,θ,T

≤ C(T ν (u 0  B s + (−∆)−1 u 1  B s ) + a p ). 2,q

(4.31)

2,q

Finally, from Lemma 4.5 we have, for u ∈ YTa and T ∈ (0, 1] Su L p L xp ≤ C(T ν (u 0  B s + (−∆)−1 u 1  B s ) + a p ). T

2,q

(4.32)

2,q

Combining (4.26), (4.31) and (4.32), we obtain SuYTa ≤ C(T ν (u 0  B s + u 1  B s −2 ) + a p ), 2,q

for u ∈ YTa .

2,q

We now take a > 0 sufficiently small such that Ca k ≤ 14 , and then take T > 0 sufficiently small a a a such that T ν ≤ 4C(u  +D −2 u  ) . Then we have SuYT ≤ a for u ∈ YT . Hence, S maps 0

YTa

Bs 2,q

1

Bs 2,q

into itself. Next, for u 1 , u 2 ∈ YTa , using Lemma 4.3 and a similar argument as before, we can easily deduce that, by taking T = T (u 0  B s + (−∆)−1 u 1  B s ) smaller as necessary, we have 2,q

2,q

Su 1 − Su 2  Fq,T + Su 1 − Su 2 G q,θ,T ≤

p−1 C(u 1 G q,θ,T

p−1

+ u 2 G q,θ,T )u 1 − u 2 G q,θ,T , p−1

p−1

T

T

s + Su 1 − Su 2 G s λ(Su 1 − Su 2  L ∞ ) ≤ C(u 1 Y a + u 2 Y a )u 1 − u 2 YTa , T B2,q q,θ,T

Su 1 − Su 2  L p L xp ≤ T

p−1 C(u 1 G q,θ,T

p−1 + u 2 G q,θ,T )u 1

− u 2 G q,θ,T .

Summing the above inequalities and using the fact that Ca p−1 ≤ 14 , we get 1 u 1 − u 2 YTa . 2 Thus S is a contraction mapping on YTa , and the assertion (2) follows. The proof of Theorem 1.1 is complete.
p−1

p−1

T

T

Su 1 − Su 2 YTa ≤ C(u 1 Y a + u 2 Y a )u 1 − u 2 YTa ≤

We now prove Theorem 1.2 (3). From Theorem 1.1 (2) we know that there exists a unique s ), where T = T (u  + u  ). Hence, to prove local solution u ∈ C([0, T ]; B2,q 0 Bs 1 B s −2 2,q

2,q

assertion (3) of Theorem 1.2, we only need to prove that, for any given T > 0, this solution u can be extended to the time interval [0, T ]. Following line by line the proof of the nonlinear

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A. Guo, S. Cui / Nonlinear Analysis 65 (2006) 802–824

estimate established in Lemma 3.1, we can deduce that, for any µ ∈ [0, s] and T > 0, there holds u L ∞ B˙ µ + uG µ T

2,q

q,θ,T

p−1

≤ C(u 0  B˙ µ + u 1  B˙ µ−2 + uG∞,θ uG µ 2,q

q,θ,T

2,q

).

From the proof of Theorem 1.2 (1) we see that, if u 0  B˙ s p + u 1  ˙ s p −2 < δ, then 2,∞

B2,∞

uG ∞,θ < (4C)−1 . p−1

Thus we have u L ∞ B˙ µ + uG µ 2,q

T

q,θ,T

−1 s + (−∆) s ). ≤ 2C(u 0  B˙ µ + u 1  B˙ µ−2 ) ≤ 2C(u 0  B2,q u 1  B2,q 2,q

2,q

Taking µ = 0 and µ = s, respectively, and then summing the results, we obtain −1 s s + (−∆) s ). u L ∞ ≤ 2C(u 0  B2,q u 1  B2,q T B2,q s This proves that u can be extended to all time intervals [0, +∞), and u ∈ L ∞ ([0, +∞); B2,q ). This proves assertion (3) of Theorem 1.2.


Proof of Theorem 1.3. From the proofs of (3.1) and Theorem 1.2, we see that  1 ∞  q q  p−1 2 j (s p −s0 )   j (u p ) L β L β ∗ ≤ CuG ∞,θ uG q,θ < +∞. j =−∞

t

x

Thus there exists jn ∈ N such that, for | j | ≥ jn , 1  q q   1 2 j (s p −s0 )   j (u p ) L β L β ∗ < , t x n | j |≥ j

(4.33)

n

and there exists an increasing sequence of time {Tn } for which  q  q1   1 j (s p −s0 ) p 2   j (u ) L β < . β∗ n [Tn ,∞) L x | j |< j

(4.34)

n

Let u L ,n be the solution of the free beam equation with the same data as u at t = Tn . Then u − u L ,n has zero data at T = Tn and satisfies ∂t2 (u − u L ,n ) + 2 (u − u L ,n ) = ±u p ,

t > Tn .

Hence (4.17), (4.33) and (4.34) give   1 ∞  q q   jsp 2   j (u − u L ,n ) L 2x
j =−∞

q  q1

β∗ [Tn ,∞) L x

1
(4.35)

A. Guo, S. Cui / Nonlinear Analysis 65 (2006) 802–824

823

Next, notice that for m > n, since u(Tm ) = u L ,m (Tm ), we have 1 ∂t (u L ,m (Tm ) − u L ,n (Tm )) ˙ s p −2 < C , B2,q n 1 u L ,m (Tm ) − u L ,n (Tm ) B˙ s p < C . 2,q n From the equality

(4.36) (4.37)

  j v(·, t)2H˙ 2 +   j ∂t v(·, t)2L 2 =   j f 2H˙ 2 +   j g2L 2 , x

x

where v(x, t) is the solution of the free beam equation with initial data v(x, 0) = f (x), vt (x, 0) = g(x), and (4.36) and (4.37), we have 1 u L ,m (0) − u L ,n (0) B˙ s p + ∂t (u L ,m (0) − u L ,n (0)) ˙ s p −2 < C . B2,q 2,q n sp s p −2 + Thus {(u L ,n (0), ∂t u L ,n (0))} is a Cauchy sequence in B˙ 2,q × B˙ 2,q . If we let (u + 0 , u 1 ) be the limit, then from (4.35) we conclude that s lim u(t) − u + L (t) B˙ p = 0.

t →+∞

2,q

u+ L (t)

+ is the solution of the free beam equation with initial data (u + The assertion that 0 , u 1 ) is immediate. This proves Theorem 1.3.


Acknowledgement This work is supported by the China National Natural Science Foundation under the grant number 10471157. References [1] J. Bergh, J. L¨ofstr¨om, Interpolation Spaces. An Introduction, in: Grundlehern Math. Wiss., No.223, Springer-Verlag, Berlin, 1976. [2] G. Chen, Z. Yang, Existence and non-existence of global solutions for a class of nonlinear wave equations, Math. Methods Appl. Sci. 23 (2000) 615–631. [3] T. Cazenave, Semilinear Schr¨odinger Equations, Amer. Math. Soc., Providence, 2003. [4] C.E. Kenig, G. Ponce, L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. pure Appl. Math. 46 (1993) 527–620. [5] C.E. Kenig, G. Ponce, L. Vega, Higher-order nonlinear dispersive equations, Proc. Amer. Math. Soc. 122 (1994) 157–166. [6] C.E. Kenig, G. Ponce, L. Vega, Quadratic forms for the 1-D semilinear Schr¨odinger equation, Trans. Amer. Math. Soc. 348 (8) (1996) 3323–3353. [7] C.E. Kenig, G. Ponce, L. Vega, On the ill-posedness of some canonical dispersive equations, Duke Math. J. 106 (3) (2001) 617–633. [8] M. Keel, C.D. Sogge, On existence and scattering with minimal regularity for semilinear wave equations, J. Funct. Anal. 130 (2) (1995) 357–426. [9] J. Liu, Free vibrations for an asymmetric beam equation, Nonlinear Anal. 51 (2002) 487–497. [10] J. Liu, Free vibrations for an asymmetric beam equation II, Nonlinear Anal. 56 (2004) 415–432. [11] S.P. Levandosky, Decay estimates for fourth order wave equations, J. Differential Equations. 143 (1998) 360–413. [12] J. Marsden, Lecture on geometric methods in mathematical physics, in: CBMS, vol. 37, SIAM, Philadelphia, 1981. [13] L. Molinet, F. Ribaud, On the Cauchy problem for the generalized Korteweg-de Vries equation, Comm. Partial Differential Equations 28 (2003) 2065–2091. [14] L. Molinet, F. Ribaud, Well-posedness results for the generalized Benjamin–Ono equation with small initial data, J. Math. Pures Appl. 83 (2004) 277–311.

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