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

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.

Keywords

fourth-order viscoelastic equation / global existence / general energy decay / nonlinear equations / finite-time blow-up / logarithmic nonlinearity / variable exponent spaces / strong damping

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