On decay and blow-up of solutions for a singular nonlocal viscoelastic problem with a nonlinear source term

TL;DR

This paper investigates the behavior of solutions to a singular nonlocal viscoelastic problem with nonlinear source terms, establishing conditions for global existence, decay, or finite-time blow-up based on initial data.

Contribution

It introduces new criteria for global existence and blow-up in a singular nonlocal viscoelastic model using potential well, convexity, and energy methods.

Findings

Solutions decay to zero under stable initial conditions.

Solutions blow up in finite time under unstable initial conditions.

The decay rate is more general than previously known.

Abstract

In this paper we consider a singular nonlocal viscoelastic problem with a nonlinear source term and a possible damping term. We proved that if the initial data enter into the stable set, the solution exists globally and decays to zero with a more general rate, and if the initial data enter into the unstable set, the solution with non-positive initial energy as well as positive initial energy blows up in finite time. These are achieved by using the potential well theory, the modified convexity method and the perturbed energy method.