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**Existence of global solutions for a weakly coupled system of semilinear viscoelastic damped \(\sigma\)-evolution equations.**
*(English)*
Zbl 1441.35159

Summary: We prove the global (in time) existence of small data solutions from energy spaces based on \(L^q\) spaces, \(q \in (1,\infty)\), to the Cauchy problem for a weakly coupled system of semilinear viscoelastic damped \(\sigma \)-evolution equations, where we consider nonlinearity terms with powers \(p_1, p_2 > 1\) and any \(\sigma_1, \sigma_2 \geq 1\) 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 \(q \in (1,\infty)\) and \(m \in [1,q)\), we apply \((L^m \cap L^q)-L^q\) 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 \(\sigma_1, \sigma_2, m\) and \(q\) bring some benefits to relax the restrictions on the admissible exponents \(p_1, p_2\).

### MSC:

35L56 | Initial value problems for higher-order hyperbolic systems |

35L30 | Initial value problems for higher-order hyperbolic equations |

35L71 | Second-order semilinear hyperbolic equations |

35R11 | Fractional partial differential equations |

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\textit{T. A. Dao}, Rocky Mt. J. Math. 50, No. 2, 527--542 (2020; Zbl 1441.35159)

### References:

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