Energy decay of nonlocal viscoelastic equations with nonlinear damping and polynomial nonlinearity
Qingqing Peng, Yikan Liu
TL;DR
The paper addresses energy decay for a variable-coefficient nonlocal viscoelastic wave equation with nonlinear damping and polynomial nonlinearity. It develops a Lyapunov-based framework to prove polynomial energy decay under modest memory-kernel assumptions and further enhances decay rates when the kernel satisfies a convexity condition, with explicit rates tied to auxiliary convex functions. The main contributions include establishing global well-posedness, a polynomial decay result, and a refined decay bound under (A5), plus illustrative examples of faster exponential-type decays. This work advances understanding of memory effects in nonlinear viscoelastic systems and informs design of damping mechanisms in materials with temporal nonlocality.
Abstract
This paper is concerned with the energy decay of a viscoelastic variable coefficient wave equation with nonlocality in time as well as nonlinear damping and polynomial nonlinear terms. Using the Lyapunov method, we establish a polynomial energy decay for the solution under relatively weak assumptions regarding the kernel of the nonlocal term. More specifically, we improve the decay rate of the energy by additionally imposing a certain convexity assumption on the kernel. Several examples are provided to confirm such improvements to faster polynomial or even exponential decays.