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

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

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  • 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.

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


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