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Peer-reviewedCC BY 4.0Axioms

Global Solution, Decay, and Blow-Up of a Viscoelastic Kirchhoff System with Logarithmic Nonlinearity and Acoustic Fractional Delay Boundary Conditions

  • Zahid Ullah1
  • Jianghao Hao1
  • Ahmed Bchatnia2
  • Abdul Aziz1
  1. 1School of Mathematics and Statistics, Shanxi University, Taiyuan 030006, China
  2. 2Department of Mathematics, College of Science, Qassim University, P.O. Box 6644, Buraydah 52571, Saudi Arabia
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Abstract

We examine a nonlinear viscoelastic Kirchhoff model endowed with a logarithmic source. Assuming suitable hypotheses and under the inclusion of nonlinear distributed delay feedback terms, in addition to the acoustic and fractional boundary feedback terms, we obtain global existence, general decay estimates, and finite-time blow-up for negative initial energy, all for a wide range of relaxation kernels. The Kirchhoff term K(∥∇W∥22) significantly changes the structure of the Nehari manifold and requires a new energy functional based on the primitive K^. Our analysis identifies the precise blow-up mechanism and establishes sharp dimension-dependent restrictions on the exponent p. Our outcomes broaden and enhance recent contributions on classical wave equations to the physically meaningful Kirchhoff framework.

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Cite

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BibTeX
@article{ullah2026global,
  title = {{Global Solution, Decay, and Blow-Up of a Viscoelastic Kirchhoff System with Logarithmic Nonlinearity and Acoustic Fractional Delay Boundary Conditions}},
  author = {Zahid Ullah and Jianghao Hao and Ahmed Bchatnia and Abdul Aziz},
  journal = {Axioms},
  year = {2026},
  month = sep,
  doi = {10.3390/axioms15100728},
  url = {https://www.mdpi.com/2075-1680/15/10/728},
  note = {CC BY 4.0},
}
RIS
TY  - JOUR
TI  - Global Solution, Decay, and Blow-Up of a Viscoelastic Kirchhoff System with Logarithmic Nonlinearity and Acoustic Fractional Delay Boundary Conditions
AU  - Zahid Ullah
AU  - Jianghao Hao
AU  - Ahmed Bchatnia
AU  - Abdul Aziz
PY  - 2026
DA  - 2026/09/30
JO  - Axioms
DO  - 10.3390/axioms15100728
UR  - https://www.mdpi.com/2075-1680/15/10/728
AB  - We examine a nonlinear viscoelastic Kirchhoff model endowed with a logarithmic source. Assuming suitable hypotheses and under the inclusion of nonlinear distributed delay feedback terms, in addition to the acoustic and fractional boundary feedback terms, we obtain global existence, general decay estimates, and finite-time blow-up for negative initial energy, all for a wide range of relaxation kernels. The Kirchhoff term K(∥∇W∥22) significantly changes the structure of the Nehari manifold and requires a new energy functional based on the primitive K^. Our analysis identifies the precise blow-up mechanism and establishes sharp dimension-dependent restrictions on the exponent p. Our outcomes broaden and enhance recent contributions on classical wave equations to the physically meaningful Kirchhoff framework.
SN  - 2075-1680
ER  -