Study of semi-linear $ \sigma $-evolution equations with frictional and visco-elastic damping

Tuan Anh Dao; Hironori Michihisa

doi:10.3934/cpaa.2020079

Article Contents

2020, Volume 19, Issue 3: 1581-1608. Doi: 10.3934/cpaa.2020079

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Study of semi-linear $ \sigma $-evolution equations with frictional and visco-elastic damping

Tuan Anh Dao^1,2, , and
Hironori Michihisa^3,

1.
School of Applied Mathematics and Informatics, Hanoi University of Science and Technology, No.1 Dai Co Viet road, Hanoi, Vietnam
2.
Faculty for Mathematics and Computer Science, TU Bergakademie Freiberg, Prüferstr. 9, 09596, Freiberg, Germany
3.
Department of Mathematics, Graduate School of Science, Hiroshima University, Higashi-Hiroshima 739-8526, Japan

^* Corresponding author
^* Corresponding author

Received: July 2019

Revised: September 2019

Published: March 2020

The first author is supported by Vietnamese Government's Scholarship (Grant number: 2015/911)

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Abstract

In this article, we study semi-linear $ \sigma $-evolution equations with double damping including frictional and visco-elastic damping for any $ \sigma\ge 1 $. We are interested in investigating not only higher order asymptotic expansions of solutions but also diffusion phenomenon in the $ L^p-L^q $ framework, with $ 1\le p\le q\le \infty $, to the corresponding linear equations. By assuming additional $ L^{m} $ regularity on the initial data, with $ m\in [1, 2) $, we prove the global (in time) existence of small data energy solutions and indicate the large time behavior of global obtained solutions as well to semi-linear equations. Moreover, we also determine the so-called critical exponent when $ \sigma $ is integers.

Keywords:

$ \sigma $-evolution equations,
frictional damping,
visco-elastic damping,
asymptotic profile; diffusion phenomenon,
global existence,
critical exponent.

Mathematics Subject Classification: Primary: 35G25, 35B40; Secondary: 35B33, 35C20.

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References

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