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.

Keywords:

• Abstract evolution equation
• blow up
• infinite memory
• local existence
• time-varying delay

AMS Subject Classifications:

• 35L90
• 35B44
• 35A01

Acknowledgements

The authors are highly grateful to the anonymous referee for his/her valuable comments and suggestions for the improvement of the paper. This research work is supported by the General Direction of Scientific Research and Technological Development (DGRSDT), Algeria.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Correction Statement

This article has been republished with minor changes. These changes do not impact the academic content of the article.

Additional information

Funding

This research work is supported by the General Direction of Scientific Research and Technological Development (DGRSDT), Algeria.