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April 2020 Existence of global solutions for a weakly coupled system of semilinear viscoelastic damped $\sigma$-evolution equations
Tuan Anh Dao
Rocky Mountain J. Math. 50(2): 527-542 (April 2020). DOI: 10.1216/rmj.2020.50.527

Abstract

We prove the global (in time) existence of small data solutions from energy spaces based on L q spaces, q ( 1 , ) , to the Cauchy problem for a weakly coupled system of semilinear viscoelastic damped σ -evolution equations, where we consider nonlinearity terms with powers p 1 , p 2 > 1 and any σ 1 , σ 2 1 in the comparison between two single equations. To do this, by mixing additional L m regularity for the data on the basis of L q - L q estimates, with q ( 1 , ) and m [ 1 , q ) , we apply ( L m L q ) - L q estimates for solutions to the corresponding linear Cauchy problems to treat semilinear problems. In addition, two different strategies allowing no loss of decay and some loss of decay combined with the flexible choice of admissible parameters σ 1 , σ 2 , m and q bring some benefits to relax the restrictions on the admissible exponents p 1 , p 2 .

Citation

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Tuan Anh Dao. "Existence of global solutions for a weakly coupled system of semilinear viscoelastic damped $\sigma$-evolution equations." Rocky Mountain J. Math. 50 (2) 527 - 542, April 2020. https://doi.org/10.1216/rmj.2020.50.527

Information

Received: 26 July 2019; Revised: 4 October 2019; Accepted: 16 October 2019; Published: April 2020
First available in Project Euclid: 29 May 2020

zbMATH: 07210976
MathSciNet: MR4104391
Digital Object Identifier: 10.1216/rmj.2020.50.527

Subjects:
Primary: 35L30 , 35L56 , 35R11

Keywords: $\sigma$-evolution equations , global existence , loss of decay , Viscoelastic damping , weakly coupled system

Rights: Copyright © 2020 Rocky Mountain Mathematics Consortium

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Vol.50 • No. 2 • April 2020
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