Analysis of evolution equation with variable-exponent memory modeling multiscale viscoelasticity
Yiqun Li, Xiangcheng Zheng
TL;DR
The paper addresses an evolution equation with non-positive type variable-exponent memory that models multiscale viscoelasticity. It introduces a perturbation-based kernel splitting $k(t)=\beta_{1-\alpha_0}(t)+\tilde{g}(t)$ to reformulate the problem into a tractable modal system and proves well-posedness in $H^1(L^2)$ with a priori bounds, along with a higher-regularity estimate under enhanced data, using spectral decomposition and Mittag-Leffler representations. The results provide a rigorous theoretical foundation for numerical analysis and simulations of memory-laden viscoelastic materials, with the initial exponent value $\alpha(0)=\alpha_0$ governing the observed singular behavior. The methodology offers a framework for extending to error estimates for discretizations and for exploring coupling with the Laplacian in variable-exponent kernels.
Abstract
We investigate the well-posedness and solution regularity of an evolution equation with non-positive type variable-exponent memory, which describes multiscale viscoelasticity in materials with memory. The perturbation method is applied for model transformation, based on which the well-posedness is proved. Then the weighted solution regularity is derived, where the initial singularity is characterized by the initial value of variable exponent.