On the fractional order hyperbolic equation with random coefficients

Abstract

Fractional order partial differential equations are very important mathematical models in describing lots of anomalous dynamic behaviors in multiple disciplines. This paper focuses upon the fractional order hyperbolic equation, which is influenced by different types of time-dependent oscillating coefficients on the principal operator part. We apply the essential techniques from harmonic analysis and probability analysis to explore the upper bound of loss of regularity. Furthermore, in order to demonstrate the optimality of the estimates, an appropriate counter-example with periodic coefficients is constructed to show the lower bound of loss of regularity by the application of instability arguments.

Keywords:

• Factional order hyperbolic equation
• Random oscillating coefficients
• Microlocal analysis
• Loss of regularity
• Instability argument

2010 Mathematics Subject Classifications:

• 35L10
• 35L90
• 35S10
• 81V10

Disclosure statement

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

Additional information

Funding

This project is supported by Natural Science Foundation of Jiangsu Province [BK 20191257]