Opuscula Math. 44, no. 1 (2024), 19-47
https://doi.org/10.7494/OpMath.2024.44.1.19

 
Opuscula Mathematica

Local existence for a viscoelastic Kirchhoff type equation with the dispersive term, internal damping, and logarithmic nonlinearity

Sebastião Cordeiro
Carlos Raposo
Jorge Ferreira
Daniel Rocha
Mohammad Shahrouzi

Abstract. This paper concerns a viscoelastic Kirchhoff-type equation with the dispersive term, internal damping, and logarithmic nonlinearity. We prove the local existence of a weak solution via a modified lemma of contraction of the Banach fixed-point theorem. Although the uniqueness of a weak solution is still an open problem, we proved uniqueness locally for specifically suitable exponents. Furthermore, we established a result for local existence without guaranteeing uniqueness, stating a contraction lemma.

Keywords: viscoelastic equation, dispersive term, logarithmic nonlinearity, local existence.

Mathematics Subject Classification: 35A01, 35L20, 35L70.

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  • Sebastião Cordeiro
  • ORCID iD https://orcid.org/0000-0001-5354-2071
  • Federal University of Pará, Faculty of Exact Sciences and Technology, R. Manoel de Abreu, Abaetetuba, 68440-000, Pará, Brazil
  • Carlos Raposo (corresponding author)
  • ORCID iD https://orcid.org/0000-0001-8014-7499
  • Federal University of Bahia, Department of Mathematics, Av. Milton Santos, Salvador, 40170-110, Bahia, Brazil
  • Jorge Ferreira
  • ORCID iD https://orcid.org/0000-0002-3209-7439
  • Federal Fluminense University, Department of Exact Sciences, Avenida dos Trabalhadores, Volta Redonda, 27255-125, Rio de Janeiro, Brazil
  • Communicated by Runzhang Xu.
  • Received: 2023-03-08.
  • Revised: 2023-08-04.
  • Accepted: 2023-08-06.
  • Published online: 2023-10-27.
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Cite this article as:
Sebastião Cordeiro, Carlos Raposo, Jorge Ferreira, Daniel Rocha, Mohammad Shahrouzi, Local existence for a viscoelastic Kirchhoff type equation with the dispersive term, internal damping, and logarithmic nonlinearity, Opuscula Math. 44, no. 1 (2024), 19-47, https://doi.org/10.7494/OpMath.2024.44.1.19

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