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March/April 2021 Trajectory statistical solutions and Liouville type theorem for nonlinear wave equations with polynomial growth
Huite Jiang, Caidi Zhao
Adv. Differential Equations 26(3/4): 107-132 (March/April 2021).

Abstract

This article investigates the following family of nonlinear wave equations $$ \partial^2_tu+\gamma(-\Delta)^{\alpha}\partial_t u=\Delta u-f(u)+g(x), \quad x\in \Omega\subset \mathbb{R}^n, \,n\geqslant 1, $$ with $\alpha\in [0, 1/2)$ and $$ |f(u)|\leqslant c_1(1+|u|^{p-1}), $$ where $c_1>0$ is a constant and $p>2$ is arbitrary. We first prove the existence of trajectory attractor and then use the translation semigroup to construct the trajectory statistical solutions for above equations. Further we establish that the constructed trajectory statistical solutions possess an invariance property and satisfy a Liouville type theorem. Moreover, we reveal that the invariance property of the trajectory statistical solutions is a particular situation of the Liouville type theorem.

Citation

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Huite Jiang. Caidi Zhao. "Trajectory statistical solutions and Liouville type theorem for nonlinear wave equations with polynomial growth." Adv. Differential Equations 26 (3/4) 107 - 132, March/April 2021.

Information

Published: March/April 2021
First available in Project Euclid: 5 May 2021

Subjects:
Primary: 34D35 , 35B41 , 76F20

Rights: Copyright © 2021 Khayyam Publishing, Inc.

JOURNAL ARTICLE
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Vol.26 • No. 3/4 • March/April 2021
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