Well-Posedness and Convergence Analysis of a Nonlocal Model with Singular Matrix Kernel

Abstract

In this paper, we consider a two-dimensional linear nonlocal model involving a singular matrix kernel. For the initial value problem, we first give well-posedness results and energy conservation via Fourier transform. Meanwhile, we also discuss the corresponding Dirichlet-type nonlocal boundary value problems in the cases of both positive and semi-positive definite kernels, where the core is the coercivity of bilinear forms. In addition, in the limit of vanishing nonlocality, the solution of the nonlocal model is seen to converge to a solution of its classical elasticity local model provided that $c_t = 0.$