Abstract
In this paper, we consider the initial-boundary problem for semilinear wave equation with a new condition on the source term $ f(u) $ such as \begin{document}$ \begin{equation*} \alpha \int_0^{u } f(s)ds \leq uf(u) + \beta u^2 +\alpha \sigma, \end{equation*} $\end{document} for some positive constants $ \alpha $, $ \beta $, and $ \sigma $, where $ \alpha >2 $ and $ \beta < \frac{\lambda_1(\alpha -2)}{2} $ with $ \lambda_1 $ being a first eigenvalue of Dirichlet-Laplacian. The condition on $ f(u) $ with $ \beta = 0 $ and $ \sigma = 0 $ was studied in [13] and [3]. By introducing a family of potential wells, we are able to establish the invariant sets, vacuum isolation of solutions, and a threshold result of global existence and nonexistence of solutions with the new condition on a source term. Moreover, we discuss the global existence and nonexistence of solutions for problem with initial condition $ E(0)