Stabilization of a Logarithmic Viscoelastic Wave Equation with the Not Necessarily Decreasing Kernel and Distributed Delay

This paper is concerned with the asymptotic behavior of a viscoelastic wave equation involving distributed delay, logarithmic nonlinearity and dynamic Wentzell boundary conditions. In general, when the memory kernel function g(t) is monotonically decreasing, the system energy’s decay is similar to that of the kernel function. However, this work addresses the case where the kernel function does not necessarily decay; thus, at this point, whether the system energy can still decay, and especially maintain exponential decay, is a very interesting question. Assuming that the kernel function is not necessarily decreasing, which means that it may oscillate, under some proper conditions, utilizing the Lyapunov functional method and constructing auxiliary functions, an exponential decay result is attained. To some extent, the result extends and improves several earlier related results in the literature.