TIME-DEPENDENT ASYMPTOTIC BEHAVIOR OF THE WAVE EQUATION WITH STRONG DAMPING ON $\mathbb{R}^{N}$

Xudong Luo; Qiaozhen Ma

doi:10.11948/20220264

2023 Volume 13 Issue 5

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Article navigation > Journal of Applied Analysis & Computation > 2023 > 13(5): 2387-2407

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Xudong Luo, Qiaozhen Ma. TIME-DEPENDENT ASYMPTOTIC BEHAVIOR OF THE WAVE EQUATION WITH STRONG DAMPING ON $\mathbb{R}^{N}$[J]. Journal of Applied Analysis & Computation, 2023, 13(5): 2387-2407. doi: 10.11948/20220264

Citation:

Xudong Luo, Qiaozhen Ma. TIME-DEPENDENT ASYMPTOTIC BEHAVIOR OF THE WAVE EQUATION WITH STRONG DAMPING ON $\mathbb{R}^{N}$[J]. Journal of Applied Analysis & Computation, 2023, 13(5): 2387-2407. doi: 10.11948/20220264

TIME-DEPENDENT ASYMPTOTIC BEHAVIOR OF THE WAVE EQUATION WITH STRONG DAMPING ON $\mathbb{R}^{N}$

Xudong Luo^1,,
Qiaozhen Ma^1, ,

1.
College of Mathematics and Statistics, Northwest Normal University, AnningDong Road, Lanzhou 730070, China

Author Bio: Email: luoxudong117@163.com(X. Luo)
Corresponding author: Email: maqzh@nwnu.edu.cn(Q. Ma)
Fund Project: The authors were supported by National Natural Science Foundation of China (No. 11961059)

Abstract

Abstract

We study the longtime dynamics of non-autonomous wave equations with strong damping in the case of critical nonlinearity. First of all, when $ 1\leq p\leq p^{\ast}=\frac{N+2}{(N-2)_{+}} $, we get the well-posedness of strong damped equation with dime-dependent decay coefficient in $ \mathcal{H}_{t}=H^{1}(\mathbb{R}^{N})\times L^{2}(\mathbb{R}^{N}) $, and prove the quasi-stability of weak solution in $ \mathcal{H}_{t, -1}=H^{1}(\mathbb{R}^{N})\times H^{-1}(\mathbb{R}^{N}) $. Then the time-dependent attractor is proved in $ \mathcal{H}_{t} $. Finally, by using the quasi-stability of weak solution, we establish the existence the pullback exponential attractor for non-autonomous dynamical system $ (U(t, \tau), \mathcal{H}_{t}, \mathcal{H}_{t, -1}) $.
- Wave equation /
- critical exponent /
- well-posedness /
- time-dependent attractor /
- pullback exponential attractor
MSC: 35B25, 35B40, 35B41, 37L30

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References

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