On the study of the hyperbolic $ p(.) $-biharmonic equation with no flux boundary condition

Roumaissa Khalfallaoui; Abderrazak Chaoui; Khaled Zennir; Safa M. Mirgani; Roumaissa Khalfallaoui; Abderrazak Chaoui; Khaled Zennir; Safa M. Mirgani

doi:10.3934/math.20251064

AIMS Mathematics

2025, Volume 10, Issue 10: 23943-23957. doi: 10.3934/math.20251064

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Research article

On the study of the hyperbolic $ p(.) $-biharmonic equation with no flux boundary condition

1.
Applied Mathematics and Modeling Laboratory, Department of Mathematics, University of 8 May 1945, Guelma, Algeria; khalfallaoui.roumaissa@univ-guelma.dz (R. K.), chaoui.abderezak@univ-guelma.dz (A. C.)
2.
Department of Mathematics, College of Science, Qassim University, Saudi Arabia; k.Zennir@qu.edu.sa (K. Z)
3.
Department of Mathematics and Statistics, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 13318, Saudi Arabia; smmmohamed@imamu.edu.sa (S. M. M.)

Received: 18 June 2025 Revised: 26 August 2025 Accepted: 10 October 2025 Published: 21 October 2025
MSC : 35A01, 35B40, 35L35

In the present paper, we studied a hyperbolic $ p(x) $-biharmonic problem with known no-flux values on the smooth part of the boundary in variable exponent Sobolev spaces. The global existence of a weak solution, using the Galerkin method, was proved. The blow-up of the solution with negative initial functional energy in finite time was discussed using the concavity method. The interaction between the variable exponent in the main equation and the boundary condition is crucial in determining the presence/absence of solutions.
- weak solution,
- p(x)-biharmonic equation,
- variable exponent,
- nonlinear equations,
- negative initial energy,
- energy and industry,
- global existence,
- blow-up of the solution
Citation: Roumaissa Khalfallaoui, Abderrazak Chaoui, Khaled Zennir, Safa M. Mirgani. On the study of the hyperbolic $ p(.) $-biharmonic equation with no flux boundary condition[J]. AIMS Mathematics, 2025, 10(10): 23943-23957. doi: 10.3934/math.20251064

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Abstract

In the present paper, we studied a hyperbolic $ p(x) $-biharmonic problem with known no-flux values on the smooth part of the boundary in variable exponent Sobolev spaces. The global existence of a weak solution, using the Galerkin method, was proved. The blow-up of the solution with negative initial functional energy in finite time was discussed using the concavity method. The interaction between the variable exponent in the main equation and the boundary condition is crucial in determining the presence/absence of solutions.

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