\(L^q\)-solvability for an equation of viscoelasticity in power type material

Abstract

In this paper, we are concerned with existence, uniqueness, regularity, and continuous dependence upon the initial data for mild solutions of an equation of viscoelasticity in power type materials in the \(L^q\)-setting. We also analyze the continuation of this solution up a maximal time of existence and a blow-up alternative. Finally, we obtain the global well-posedness for the problem.

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Acknowledgements

This work was partially supported by CNPq under grant 408194/2018-9. Bruno de Andrade is partially supported by CNPq and CAPES/FAPITEC under grants 308931/2017-3, 88881.157450/2017-01 and 88887.157906/2017-00. We are grateful to the reviewers who carefully read our paper and kindly made suggestions to improve the final version of this manuscript.

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Correspondence to Arlúcio Viana.

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de Andrade, B., Silva, C. & Viana, A. \(L^q\)-solvability for an equation of viscoelasticity in power type material. Z. Angew. Math. Phys. 72, 10 (2021). https://doi.org/10.1007/s00033-020-01443-0

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Mathematics Subject Classification

  • 45D05
  • 35Q35
  • 76A10
  • 76A05
  • 76D03
  • 35R11

Keywords

  • Nonlinear Volterra equations
  • PDEs in connection with viscoelasticity
  • Existence and regularity of solutions
  • Power type materials