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

Decay Rate of Solutions to Timoshenko System with Past History in Unbounded Domains 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 $$\alpha _g$$ α 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 $$\alpha _g=0,$$ α g = 0 , then the $$L^2$$ L 2 -norm of the solution decays with the rate $$(1+t)^{-1/12}$$ ( 1 + t ) - 1 / 12 . The same decay rate has been obtained for $$\alpha _g\ne 0,$$ α g ≠ 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 $$H^s$$ H s -norm of the solution, we need our initial data to be in the space $$H^{s+s_0},\ s_0>1$$ H s + s 0 , s 0 > 1 . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mathematics and Optimization Springer Journals

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

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Publisher
Springer US
Copyright
Copyright © 2016 by Springer Science+Business Media New York
Subject
Mathematics; Calculus of Variations and Optimal Control; Optimization; Systems Theory, Control; Theoretical, Mathematical and Computational Physics; Mathematical Methods in Physics; Numerical and Computational Physics, Simulation
ISSN
0095-4616
eISSN
1432-0606
D.O.I.
10.1007/s00245-016-9336-6
Publisher site
See Article on Publisher Site

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 $$\alpha _g$$ α 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 $$\alpha _g=0,$$ α g = 0 , then the $$L^2$$ L 2 -norm of the solution decays with the rate $$(1+t)^{-1/12}$$ ( 1 + t ) - 1 / 12 . The same decay rate has been obtained for $$\alpha _g\ne 0,$$ α g ≠ 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 $$H^s$$ H s -norm of the solution, we need our initial data to be in the space $$H^{s+s_0},\ s_0>1$$ H s + s 0 , s 0 > 1 .

Journal

Applied Mathematics and OptimizationSpringer Journals

Published: Feb 11, 2016

References

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