Blow-up result for an abstract evolution problem with infinite memory and time-varying delay

Abstract

In this paper, we consider an abstract second-order viscoelastic equation with infinite memory, time-varying delay and a nonlinear source term. To the best of our knowledge, there is no blow-up result of solutions for the nonlinear viscoelastic equation with time-varying delay. Moreover, our result extends the blow-up result obtained for problems without memory term to those with infinite memory and problems with constant delay to time-varying delay. Under suitable conditions on initial data, the kernel memory function and the involved functionals, we establish the local existence using the semigroup theory and we prove a finite time blow-up result for the solution with positive initial energy. We also give some applications to illustrate our main result.