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.

In this note we come back to face a problem regarding forward-backward parabolic equations like $(r(x,t) u)_t - u_{xx} = 0$ and $r(x,t) u_t - u_{xx} = 0$ ($r$ is both positive and negative): the continuity of $t \mapsto \int u^2(x,t) |r(x,t)| \, dx$.

This paper is concerned with a Lotka-Volterra competition model with nonlocal diffusion and free boundaries, which describes the evolution of an invasive species with free boundaries in a one-dimensional habitat and a native species in the whole space. We prove the well-posedness of the model and obtain a spreading-vanishing dichotomy for the invasive species. Sharp criteria for spreading and vanishing are also obtained, which reveal significant differences from the local model. Depending on the choice of the kernel function in the nonlocal diffusion operator, it is expected that the nonlocal model here may have accelerated spreading, which would contrast sharply to the local model, where the spreading has finite speed whenever spreading happens.