On global instability of solutions to hyperbolic equations with non-Lipschitz nonlinearity

In a bounded domain $\Omega \subset \mathbb{R}^n$, we consider the following hyperbolic equation
$$\begin{equation*} \begin{cases} v_{tt} = \Delta_p v+\lambda |v|^{p-2}v-|v|^{\alpha-2}v,& x\in \Omega, \\ v\bigr{|}_{\partial \Omega}=0. \end{cases} \end{equation*}$$
We assume that $1<\alpha<p<+\infty$; this implies that the nonlinearity in the right hand side of the equation is of a non-Lipschitz type. As a rule, this type of nonlinearity prevent us from applying standard methods from the theory of nonlinear differential equations. An additional difficulty arises due to the presence of the $p$-Laplacian $\Delta_p (\cdot):=\text{div}(|\nabla(\cdot)|^{p-2}\nabla(\cdot))$ in the equation. In the first result, the theorem on the existence of the so-called stationary ground state of the equation is proved. The proof of this result is based on the Nehari manifold method. In the main result of the paper we state that each stationary ground state is unstable globally in time. The proof is based on the development of an approach by Payne and Sattinger introduced for studying the stability of solutions to hyperbolic equations.