This study investigates the existence and nonexistence of solutions for a nonlinear fourth-order viscoelastic equation with a fractional damping in a bounded domain. The model at hand describes the dynamic behavior of elastic plates, taking into account both memory effects and nonlocal dissipation. In this framework, the viscoelastic term is represented by a convolution with a suitable relaxation kernel, while the damping mechanism is defined through a fractional power of the associated linear operator.

To establish the local existence, we reformulate the governing equation as an abstract evolution system within an appropriate Hilbert space. By demonstrating that the linear operator generates a strongly continuous semigroup of contractions and ensuring the local Lipschitz continuity of the nonlinearities, we apply semigroup theory to prove the local existence and uniqueness of solutions. Furthermore, we provide conditions under which these solutions attain the regularity of weak solutions based on the initial data.

Regarding long-term behavior, we derive energy identities and a priori estimates to show that solutions exist globally for sufficiently small initial data. Here, the interplay between viscoelastic memory and fractional damping enhances the dissipative structure, which is vital for the stabilization of the system. Conversely, we explore the potential for finite-time blow-up when the system is driven by large initial energy. Using a Lyapunov functional approach and the concavity method, we establish that solutions fail to exist globally under specific conditions related to the nonlinear source. Our findings also highlight the role of the fractional damping term, noting that while it may delay the blow-up, it cannot entirely prevent it in the presence of strong nonlinearities. This work extends the existing literature by integrating fractional calculus into the analysis of fourth-order evolution equations, offering new insights into the balance between memory effects and nonlocal dissipation.