Trajectory statistical solutions and Liouville type theorem for nonlinear wave equations with polynomial growth

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.