The Cauchy Problem for a One Dimensional Nonlinear Peridynamic Model

TL;DR

This paper investigates the mathematical properties of a one-dimensional nonlinear peridynamic model, establishing conditions for existence, smoothness, and finite-time blow-up of solutions in elastic bar dynamics.

Contribution

It provides new proofs of local and global well-posedness and identifies blow-up conditions for nonlinear peridynamic equations.

Findings

Global solutions exist in sublinear and degree ≤3 nonlinear cases.

Finite-time blow-up conditions are characterized.

Solutions are shown to be smooth under certain conditions.

Abstract

This paper studies the Cauchy problem for a one-dimensional nonlinear peridynamic model describing the dynamic response of an infinitely long elastic bar. The issues of local well-posedness and smoothness of the solutions are discussed. The existence of a global solution is proved first in the sublinear case and then for nonlinearities of degree at most three. The conditions for finite-time blow-up of solutions are established.