TY - JOUR
T1 - Global existence and decay property of the Timoshenko system in thermoelasticity with second sound
AU - Racke, Reinhard
AU - Said-Houari, Belkacem
N1 - KAUST Repository Item: Exported on 2020-10-01
PY - 2012/9
Y1 - 2012/9
N2 - Our main focus in the present paper is to study the asymptotic behavior of a nonlinear version of the Timoshenko system in thermoelasticity with second sound. As it has been already proved in Said-Houari and Kasimov (2012) [29], the linear version of this system is of regularity-loss type. It is well known (Hosono and Kawashima (2006) [34], Ide and Kawashima (2008) [27], Kubo and Kawashima (2009) [41]) that the regularity-loss property of the linear problem creates difficulties when dealing with the nonlinear problem. In fact, the dissipative property of the problem becomes very weak in the high frequency region and as a result the classical energy method fails. To overcome this difficulty and following Ide and Kawashima (2008) [27] and Ikehata (2002) [30], we use an energy method with negative weights to create an artificial damping which allows us to control the nonlinearity. We prove that for 0≤k≤[s2]-2 with s<8, the solution of our problem is global in time and decays as
AB - Our main focus in the present paper is to study the asymptotic behavior of a nonlinear version of the Timoshenko system in thermoelasticity with second sound. As it has been already proved in Said-Houari and Kasimov (2012) [29], the linear version of this system is of regularity-loss type. It is well known (Hosono and Kawashima (2006) [34], Ide and Kawashima (2008) [27], Kubo and Kawashima (2009) [41]) that the regularity-loss property of the linear problem creates difficulties when dealing with the nonlinear problem. In fact, the dissipative property of the problem becomes very weak in the high frequency region and as a result the classical energy method fails. To overcome this difficulty and following Ide and Kawashima (2008) [27] and Ikehata (2002) [30], we use an energy method with negative weights to create an artificial damping which allows us to control the nonlinearity. We prove that for 0≤k≤[s2]-2 with s<8, the solution of our problem is global in time and decays as
UR - http://hdl.handle.net/10754/562288
UR - https://linkinghub.elsevier.com/retrieve/pii/S0362546X1200140X
UR - http://www.scopus.com/inward/record.url?scp=84862143662&partnerID=8YFLogxK
U2 - 10.1016/j.na.2012.04.011
DO - 10.1016/j.na.2012.04.011
M3 - Article
SN - 0362-546X
VL - 75
SP - 4957
EP - 4973
JO - Nonlinear Analysis: Theory, Methods & Applications
JF - Nonlinear Analysis: Theory, Methods & Applications
IS - 13
ER -