New general decay results for a Moore–Gibson–Thompson equation with memory

Abstract

In this paper, we consider the Moore–Gibson–Thompson equation with a memory term in the subcritical case, which arises in high-frequency ultrasound applications accounting for thermal flux and molecular relaxation times. For a class of relaxation functions satisfying $g^{\prime }\left(t\right)\le -\xi \left(t\right)H\left(g\right(t\left)\right)$ for H to be increasing and convex function near the origin and $\xi \left(t\right)$ to be a nonincreasing function, we establish the optimal explicit and general energy decay result, from which we can recover the earlier exponential rate in the special case.

COMMUNICATED BY:

• Iren Lasiecka

Keywords:

• General stability
• Moore–Gibson–Thompson equation
• decay rate
• energy method

2000 Mathematics Subject Classifications:

• 35B35
• 35Q70
• 35G05
• 45D05
• 74D99

Acknowledgments

The authors are grateful to the anonymous referees and the editor for their useful remarks and comments.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by the National Natural Science Foundation of China [grant number 11771216], the Natural Science Foundation of Jiangsu Province [grant number BK20151523], the Six Talent Peaks Project in Jiangsu Province [grant number 2015-XCL-020] and the Qing Lan Project of Jiangsu Province.