Global existence, general decay, and blow-up of solutions for a fourth-order viscoelastic equation with variable exponents and logarithmic nonlinearities

Mohammad Shahrouzi; Faramarz Tahamtani; Salah Boulaaras; Mohammad Shahrouzi; Faramarz Tahamtani; Salah Boulaaras

doi:10.3934/cam.2026007

Communications in Analysis and Mechanics

2026, Volume 18, Issue 1: 172-207. doi: 10.3934/cam.2026007

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Research article

Global existence, general decay, and blow-up of solutions for a fourth-order viscoelastic equation with variable exponents and logarithmic nonlinearities

1.
Department of Applied Mathematics, Faculty of Mathematical Sciences, Ferdowsi University of Mashhad, P.O.Box 1159, Mashhad 91775, Iran
2.
Department of Mathematics, College of Sciences, Shiraz University, Shiraz, Iran
3.
Department of Mathematics, College of Science, Qassim University, Buraydah, 51452, Saudi Arabia

Received: 05 October 2025 Revised: 12 February 2026 Accepted: 13 February 2026 Published: 10 March 2026
35A01, 35B35, 35B44, 35L75, 74D10

This paper investigates the existence, energy decay, and finite-time blow-up of solutions for a class of fourth-order viscoelastic evolution equations involving variable exponent nonlinearities, logarithmic terms, and strong damping effects. Such models arise naturally in the mathematical description of heterogeneous viscoelastic media and nonlinear plate equations with memory. By applying the Nehari manifold method and suitable energy estimates, we establish the global existence of weak solutions under appropriate assumptions on the initial data and the variable exponents. Moreover, using a perturbed energy method combined with a carefully constructed Lyapunov functional, we prove that the solutions exhibit general energy decay rates depending on the properties of the relaxation kernel and the spatially variable nonlinearities. Finally, we derive sufficient conditions ensuring finite-time blow-up for solutions with negative initial energy, thereby highlighting the competing effects of strong damping, viscoelastic memory, and logarithmic nonlinearities.
- fourth-order viscoelastic equation,
- global existence,
- general energy decay,
- nonlinear equations,
- finite-time blow-up,
- logarithmic nonlinearity,
- variable exponent spaces,
- strong damping
Citation: Mohammad Shahrouzi, Faramarz Tahamtani, Salah Boulaaras. Global existence, general decay, and blow-up of solutions for a fourth-order viscoelastic equation with variable exponents and logarithmic nonlinearities[J]. Communications in Analysis and Mechanics, 2026, 18(1): 172-207. doi: 10.3934/cam.2026007

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Abstract

This paper investigates the existence, energy decay, and finite-time blow-up of solutions for a class of fourth-order viscoelastic evolution equations involving variable exponent nonlinearities, logarithmic terms, and strong damping effects. Such models arise naturally in the mathematical description of heterogeneous viscoelastic media and nonlinear plate equations with memory. By applying the Nehari manifold method and suitable energy estimates, we establish the global existence of weak solutions under appropriate assumptions on the initial data and the variable exponents. Moreover, using a perturbed energy method combined with a carefully constructed Lyapunov functional, we prove that the solutions exhibit general energy decay rates depending on the properties of the relaxation kernel and the spatially variable nonlinearities. Finally, we derive sufficient conditions ensuring finite-time blow-up for solutions with negative initial energy, thereby highlighting the competing effects of strong damping, viscoelastic memory, and logarithmic nonlinearities.

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