This note is somehow a continuation of the previous survey [107] dedicated to the memory of Prof. Vasilii V. Zhikov. Here we present further results contained in [105] concerning the Kirchhoff wave initial boundary value problems in the so-called degenerate case, that is, when the involved Kirchhoff function M can be zero at zero, and involving the fractional Laplacian, nonlinear damping and source terms in \(\Omega \times \mathbb {R}^{+}_0\) , where \(\Omega \subset \mathbb {R}^n\) is a bounded domain with Lipschitz boundary \(\partial \Omega \) . The first part of the survey briefly presents the main results of [105], concerning global existence, vacuum isolating, asymptotic behavior and blow-up of solutions. The second part is devoted to the the blow-up phenomenon of the initial boundary value problem for the periodic rotation-two-component Camassa–Holm (R2CH) system given in [79], as well as a precise blow-up criterion, two new blow-up statements and the exact blow-up rate for strong solutions of the R2CH system in question.