Stability result of a viscoelastic plate equation with past history and a logarithmic nonlinearity

• Main Author: Adel M. Al-Mahdi
• Format: Article
• Language: English
• Published: SpringerOpen 2020-05-01
• Series: Boundary Value Problems
• Subjects: Stability; Logarithmic Sobolev inequalities; Infinite memory; Plate equation; Convexity
• Online Access: http://link.springer.com/article/10.1186/s13661-020-01382-9

Abstract In this paper, we are concerned with the decay rate of the solution of a viscoelastic plate equation with infinite memory and logarithmic nonlinearity. We establish an explicit and general decay rate results with imposing a minimal condition on the relaxation function. In fact, we assume that the relaxation function h satisfies h ′ ( t ) ≤ − ξ ( t ) H ( h ( t ) ) , t ≥ 0 , $$ h^{\prime}(t)\le-\xi(t) H\bigl(h(t)\bigr),\quad t\geq0, $$ where the functions ξ and H satisfy some conditions. Our proof is based on the multiplier method, convex properties, logarithmic inequalities, and some properties of integro-differential equations. Moreover, we drop the boundedness assumption on the history data, usually made in the literature. In fact, our results generalize, extend, and improve earlier results in the literature.