A class of dissipative nonautonomous nonlocal second-order evolution equations

Abstract

In this paper we consider the following nonlinear and spatially nonlocal second-order evolution equation from nonlocal theory of continuum mechanics

where is a bounded smooth domain in , , , and a is a bounded continuous function. Here, the kernel J is a nonnegative, symmetric bounded function with bounded derivative, satisfying certain growth conditions. We deduce an energy functional associated to these problem, and we study the local and global well posedness, boundedness and asymptotic behavior of its solutions. Additionally we study the stability of the trivial solution associated to these problem.

Keywords:

• Nonlocal equations
• nonautonomous problem
• energy functional
• second-order evolution equation

AMS Subject Classifications:

• 47j35
• 37b55
• 45g15
• 35r09

Acknowledgements

The authors would like to thank the anonymous referees for their constructive comments and suggestions which helped us to improve the original manuscript considerably.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was partially supported by CAPES/CNPq - Procad/Casadinho, Brazil [grant number 552.758/2011-6], [grant number 552.464/2011-2]; FAPESP, Brazil [grant number 2014/03686-3], [grant number 2014/03109-6].