Arbitrary decay of solutions for a singular nonlocal viscoelastic problem with a possible damping term

Abstract

In this paper, we consider a singular nonlocal viscoelastic problem with a possible damping term . We use the assumptions on the relaxation function due to Tatar (J. Math. Phys. 2011;52:013502) and establish an arbitrary decay result instead of just exponential or polynomial type no matter or . This decay result improves that of Mesloub and Messaoudi (Acta Appl. Math. 2010;110:705–724) where an exponential or polynomial decay result was given for the case .

Keywords:

• nonlocal viscoelastic problem
• arbitrary decay
• exponential decay
• polynomial decay
• relaxation function

AMS Subject Classifications:

• 35B35
• 35B40
• 35L20

Acknowledgments

This work was sponsored by Qing Lan Project of Jiangsu Province, and was partly supported by the Tianyuan Fund of Mathematics (Grant No. 11026211), the Natural Science Foundation of the Jiangsu Higher Education Institutions (Grant No. 09KJB110005) and the JSPS Innovation Program (Grant No. CXLX12 0490).