TY - JOUR

T1 - Global existence and optimal decay rates for the Timoshenko system

T2 - The case of equal wave speeds

AU - Mori, Naofumi

AU - Xu, Jiang

AU - Kawashima, Shuichi

N1 - Funding Information:
The authors would like to thank the referee for various comments which improve the presentation of the manuscript. J. Xu is partially supported by the National Natural Science Foundation of China ( 11471158 ), the Program for New Century Excellent Talents in University ( NCET-13-0857 ) and the NUAA Fundamental Research Funds ( NS2013076 ). The work is also partially supported by Grant-in-Aid for Scientific Researches (S) 25220702 and (A) 22244009 . Appendix A For convenience of reader, in this section, we review the Littlewood–Paley decomposition and definitions for Besov spaces and Chemin–Lerner spaces in R n ( n ≥ 1 ), see [4] for more details. Let ( φ , χ ) is a couple of smooth functions valued in [0,1] such that φ is supported in the shell C ( 0 , 3 4 , 8 3 ) = { ξ ∈ R n | 3 4 ≤ | ξ | ≤ 8 3 } , χ is supported in the ball B ( 0 , 4 3 ) = { ξ ∈ R n | | ξ | ≤ 4 3 } satisfying χ ( ξ ) + ∑ q ∈ N φ ( 2 − q ξ ) = 1 , q ∈ N , ξ ∈ R n and ∑ k ∈ Z φ ( 2 − k ξ ) = 1 , k ∈ Z , ξ ∈ R n ∖ { 0 } . For f ∈ S ′ (the set of temperate distributions which is the dual of the Schwarz class S ), define Δ − 1 f : = χ ( D ) f = F − 1 ( χ ( ξ ) F f ) , Δ q f : = 0 for q ≤ − 2 ; Δ q f : = φ ( 2 − q D ) f = F − 1 ( φ ( 2 − q | ξ | ) F f ) for q ≥ 0 ; Δ ˙ q f : = φ ( 2 − q D ) f = F − 1 ( φ ( 2 − q | ξ | ) F f ) for q ∈ Z , where F f , F − 1 f represent the Fourier transform and the inverse Fourier transform on f , respectively. Observe that the operator Δ ˙ q coincides with Δ q for q ≥ 0 . Denote by S 0 ′ : = S ′ / P the tempered distributions modulo polynomials P . We first give the definition of homogeneous Besov spaces. Definition A.1 For s ∈ R and 1 ≤ p , r ≤ ∞ , the homogeneous Besov spaces B ˙ p , r s is defined by B ˙ p , r s = { f ∈ S 0 ′ : ‖ f ‖ B ˙ p , r s < ∞ } , where ‖ f ‖ B ˙ p , r s = { ( ∑ q ∈ Z ( 2 q s ‖ Δ ˙ q f ‖ L p ) r ) 1 / r , r < ∞ , sup q ∈ Z 2 q s ‖ Δ ˙ q f ‖ L p , r = ∞ . Similarly, the definition of inhomogeneous Besov spaces is stated as follows. Definition A.2 For s ∈ R and 1 ≤ p , r ≤ ∞ , the inhomogeneous Besov spaces B p , r s is defined by B p , r s = { f ∈ S ′ : ‖ f ‖ B p , r s < ∞ } , where ‖ f ‖ B p , r s = { ( ∑ q = − 1 ∞ ( 2 q s ‖ Δ q f ‖ L p ) r ) 1 / r , r < ∞ , sup q ≥ − 1 2 q s ‖ Δ q f ‖ L p , r = ∞ . On the other hand, we also present the definition of Chemin–Lerner spaces first initialed by J.-Y. Chemin and N. Lerner [5] , which are the refinement of the space–time mixed spaces L T θ ( B ˙ p , r s ) or L T θ ( B p , r s ) . Definition A.3 For T > 0 , s ∈ R , 1 ≤ r , θ ≤ ∞ , the homogeneous mixed Chemin–Lerner spaces L ˜ T θ ( B ˙ p , r s ) is defined by L ˜ T θ ( B ˙ p , r s ) : = { f ∈ L θ ( 0 , T ; S 0 ′ ) : ‖ f ‖ L ˜ T θ ( B ˙ p , r s ) < + ∞ } , where ‖ f ‖ L ˜ T θ ( B ˙ p , r s ) : = ( ∑ q ∈ Z ( 2 q s ‖ Δ ˙ q f ‖ L T θ ( L p ) ) r ) 1 r with the usual convention if r = ∞ . Definition A.4 For T > 0 , s ∈ R , 1 ≤ r , θ ≤ ∞ , the inhomogeneous Chemin–Lerner spaces L ˜ T θ ( B p , r s ) is defined by L ˜ T θ ( B p , r s ) : = { f ∈ L θ ( 0 , T ; S ′ ) : ‖ f ‖ L ˜ T θ ( B p , r s ) < + ∞ } , where ‖ f ‖ L ˜ T θ ( B p , r s ) : = ( ∑ q ≥ − 1 ( 2 q s ‖ Δ q f ‖ L T θ ( L p ) ) r ) 1 r with the usual convention if r = ∞ . We further define C ˜ T ( B p , r s ) : = L ˜ T ∞ ( B p , r s ) ∩ C ( [ 0 , T ] , B p , r s ) and C ˜ T 1 ( B p , r s ) : = { f ∈ C 1 ( [ 0 , T ] , B p , r s ) | ∂ t f ∈ L ˜ T ∞ ( B p , r s ) } , where the index T will be omitted when T = + ∞ . By Minkowski's inequality, L ˜ T θ ( B p , r s ) may be linked with the usual space–time mixed spaces L T θ ( B p , r s ) . Remark A.1 It holds that ‖ f ‖ L ˜ T θ ( B p , r s ) ≤ ‖ f ‖ L T θ ( B p , r s ) if r ≥ θ ; ‖ f ‖ L ˜ T θ ( B p , r s ) ≥ ‖ f ‖ L T θ ( B p , r s ) if r ≤ θ .
Publisher Copyright:
© 2014 Elsevier Inc.

PY - 2015/3/5

Y1 - 2015/3/5

N2 - We first show the global existence and optimal decay rates of solutions to the classical Timoshenko system in the framework of Besov spaces. Due to the non-symmetric dissipation, the general theory for dissipative hyperbolic systems (see [31]) cannot be applied to the Timoshenko system directly. In the case of equal wave speeds, we construct global solutions to the Cauchy problem pertaining to data in the spatially Besov spaces. Furthermore, the dissipative structure enables us to give a new decay framework which pays less attention on the traditional spectral analysis. Consequently, the optimal decay estimates of solution and its derivatives of fractional order are shown by time-weighted energy approaches in terms of low-frequency and high-frequency decompositions. As a by-product, the usual decay estimate of L1(R)-L2(R) type is also shown.

AB - We first show the global existence and optimal decay rates of solutions to the classical Timoshenko system in the framework of Besov spaces. Due to the non-symmetric dissipation, the general theory for dissipative hyperbolic systems (see [31]) cannot be applied to the Timoshenko system directly. In the case of equal wave speeds, we construct global solutions to the Cauchy problem pertaining to data in the spatially Besov spaces. Furthermore, the dissipative structure enables us to give a new decay framework which pays less attention on the traditional spectral analysis. Consequently, the optimal decay estimates of solution and its derivatives of fractional order are shown by time-weighted energy approaches in terms of low-frequency and high-frequency decompositions. As a by-product, the usual decay estimate of L1(R)-L2(R) type is also shown.

KW - Critical Besov spaces

KW - Global existence

KW - Optimal decay estimates

KW - Timoshenko system

UR - http://www.scopus.com/inward/record.url?scp=84920684659&partnerID=8YFLogxK

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U2 - 10.1016/j.jde.2014.11.003

DO - 10.1016/j.jde.2014.11.003

M3 - Article

AN - SCOPUS:84920684659

VL - 258

SP - 1494

EP - 1518

JO - Journal of Differential Equations

JF - Journal of Differential Equations

SN - 0022-0396

IS - 5

ER -