Applied Mathematics & Optimization

, Volume 75, Issue 3, pp 403–428 | Cite as

Decay Rate of Solutions to Timoshenko System with Past History in Unbounded Domains

Article

Abstract

We consider the Cauchy problem for the one-dimensional Timoshenko system coupled with the heat conduction, wherein the latter is described by the Gurtin–Pipkin thermal law. We study the decay properties of the system using the energy method in the Fourier space (to build an appropriate Lyapunov functional) accompanied with some integral estimates. We show that the number \(\alpha _g\) (depending on the parameters of the system) found in (Dell’Oro and Pata J Differ Equ 257(2):523–548, 2014), which rules the evolution in bounded domains, also plays a role in an unbounded domain and controls the behavior of the solution. In fact, we prove that if \(\alpha _g=0,\) then the \(L^2\)-norm of the solution decays with the rate \((1+t)^{-1/12}\). The same decay rate has been obtained for \(\alpha _g\ne 0,\) but under some higher regularity assumption. This high regularity requirement is known as regularity loss, which means that in order to get the estimate for the \(H^s\)-norm of the solution, we need our initial data to be in the space \(H^{s+s_0},\ s_0>1\).

Notes

Acknowledgments

The authors wish to thank the referees for their useful remarks and their careful reading of the proofs presented in this paper.

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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Princess Sumaya University for TechnologyAmmanJordan
  2. 2.Alhosn UniversityAbu DhabiUAE

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