Global Well-Posedness and Polynomial Decay for a Nonlinear Viscoelastic Equation with Variable Density and Memory

Aims: This paper investigates a nonlinear viscoelastic equation with variable density and memory. Study Design: Polynomial decay. Place and Duration of Study: This paper was completed at the School of Mathematics and Statistics at Southwest University during from May 2024 to February 2025. Methodology: Using Faedo-Gal¨erkin method and Energy method. Results: We study the global well-posedness and show the polynomial decay results with more general and weaker assumptions (compared with the previous studies) on the memory kernel. Conclusion: We study the well-posedness and polynomial stability, which is different from others, where the multiplying method and energy method were used to study the polynomial stability. This result substantially improves earlier results in the literature.