Orbital stability and instability of periodic wave solutions for $\phi^{4n}$-models

In this work we study the orbital stability/instability in the energy space of a specific family of periodic wave solutions of the general $\phi^{4n}$-model for all $n\in\mathbb{N}$. This family of periodic solutions are orbiting around the origin in the corresponding phase portrait and, in the standing case, are related (in a proper sense) with the aperiodic Kink solution that connect the states $-\tfrac{1}{2}$ with $\tfrac{1}{2}$. In the traveling case, we prove the orbital instability in the whole energy space for all $n\in\mathbb{N}$, while in the standing case we prove that, under some additional parity assumptions, these solutions are orbitally stable for all $n\in\mathbb{N}$.