SOME $L^Q (\mathbb{R})$-NORM DECAY ESTIMATES ($Q\in[1,+\infty]$) FOR TWO CAUCHY SYSTEMS OF TYPE RAO-NAKRA SANDWICH BEAM WITH A FRICTIONAL DAMPING OR AN INFINITE MEMORY

Aissa Guesmia

doi:10.11948/20220055

2022 Volume 12 Issue 6

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Aissa Guesmia. SOME $L^Q (\mathbb{R})$-NORM DECAY ESTIMATES ($Q\in[1,+\infty]$) FOR TWO CAUCHY SYSTEMS OF TYPE RAO-NAKRA SANDWICH BEAM WITH A FRICTIONAL DAMPING OR AN INFINITE MEMORY[J]. Journal of Applied Analysis & Computation, 2022, 12(6): 2511-2540. doi: 10.11948/20220055

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Aissa Guesmia. SOME $L^Q (\mathbb{R})$-NORM DECAY ESTIMATES ($Q\in[1,+\infty]$) FOR TWO CAUCHY SYSTEMS OF TYPE RAO-NAKRA SANDWICH BEAM WITH A FRICTIONAL DAMPING OR AN INFINITE MEMORY[J]. Journal of Applied Analysis & Computation, 2022, 12(6): 2511-2540. doi: 10.11948/20220055

SOME $L^Q (\mathbb{R})$-NORM DECAY ESTIMATES ($Q\in[1,+\infty]$) FOR TWO CAUCHY SYSTEMS OF TYPE RAO-NAKRA SANDWICH BEAM WITH A FRICTIONAL DAMPING OR AN INFINITE MEMORY

Aissa Guesmia

Institut Elie Cartan de Lorraine, UMR 7502, Université de Lorraine, 3 Rue Augustin Fresnel, BP 45112, 57073 Metz Cedex 03, France

Author Bio: Aissa Guesmia, Email address: aissa.guesmia@univ-lorraine.fr

Abstract

Abstract

In this paper, we consider two systems of type Rao-Nakra sandwich beam in the whole line $ \mathbb{R} $ with a frictional damping or an infinite memory acting on the Euler-Bernoulli equation. When the speeds of propagation of the two wave equations are equal, we show that the solutions do not converge to zero when time goes to infinity. In the reverse situation, we prove some $ L^2 (\mathbb{R}) $-norm and $ L^1 (\mathbb{R}) $-norm decay estimates of solutions and theirs higher order derivatives with respect to the space variable. Thanks to interpolation inequalities and Carlson inequality, these $ L^2 (\mathbb{R}) $-norm and $ L^1 (\mathbb{R}) $-norm decay estimates lead to similar ones in the $ L^q (\mathbb{R}) $-norm, for any $ q\in [1,+\infty] $. In our both $ L^2 (\mathbb{R}) $-norm and $ L^1 (\mathbb{R}) $-norm decay estimates, we specify the decay rates in terms of the regularity of the initial data and the nature of the control.
- Rao-Nakra sandwich beam /
- frictional damping /
- infinite memory /
- unbounded domain /
- asymptotic behavior /
- $L^2 (\mathbb{R})$-norm and $L^1 (\mathbb{R})$-norm decay estimates /
- energy method /
- Fourier analysis
MSC: 34B05, 34D05, 34H05, 35B40, 35L45, 74H40, 93D20, 93D15

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References

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