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This is a Quantitative Study study.

October 17, 2025Open Access

On the Existence and Behavior of Solution for p(x)‐Biharmonic‐Type Hyperbolic Equation With Nonlinear Damping and Variable Exponent Source

Key Points

Global existence of solutions established via potential well theory and continuous principles.
Stability properties of the solution identified using the Komornik lemma, demonstrating qualitative insights.
Investigation into the blow-up of solutions under conditions with negative initial energy, revealing critical parameter influences.
The study contributes to existing knowledge on the behavior of fourth-order equations with nonlinear features.

Abstract

A fourth‐order p ( x )‐biharmonic‐type hyperbolic equation with variable‐exponent nonlinearities is considered. The global existence of solutions has been obtained by potential well theory and the continuous principle. Qualitative properties related to the stability of the solution of this equation are obtained using the method of the well‐known Komornik lemma. The blow‐up of solutions with negative initial energy under certain conditions on the parameters of the equation is investigated.

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