Global attractor of the fractional damping wave equation on ℝ³

ABSTRACT

The paper studies the well-posedness and the longtime behaviors of the fractional damping wave equation on ${\mathbb{R}}^3$ with critical nonlinearity, we firstly prove the well-posedness of weak solutions via an established Strichartz estimate; Secondly, by the uniformly tail estimate and the energy identity, we prove that the corresponding semigroup possesses a global attractor in the energy space $H^1\left({\mathbb{R}}^3\right)\times L^2\left({\mathbb{R}}^3\right)$.

Keywords:

• Wave equation
• dynamic system
• fractional damping
• global attractor

2010 Mathematics Subject Classifications:

• Primary: 35B40
• 35B41
• Secondary: 35B65
• 37L30

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The work is supported by National Natural Science Foundation of China [grant numbers 11471129, 11772291] and Foundation of Henan Educational Committee [grant number 18B110019] and the Doctor Foundation of Henan University of Technology, China [grant number 2019BS041] and Young Teacher Foundation of Xuchang University.