The aim of this work is to demonstrate the effectiveness of the extension theory of symmetric operators in the investigation of the stability of standing waves for the nonlinear Schrödinger equations with two types of non-linearities (power and logarithmic) and two types of point interactions ($\delta$- and $\delta'$-) on a star graph. Our approach allows us to overcome the use of variational techniques in the investigation of the Morse index for self-adjoint operators with non-standard boundary conditions which appear in the stability study. We also demonstrate how our method simplifies the proof of the stability results known for the NLS equation with point interactions on the line.

In this paper, we study the long time behavior of energy solutions for a class of wave equation with time-dependent potential and speed of propagation. We introduce a classification of the potential term, which clarifies whether the solution behaves like the solution to the wave equation or Klein-Gordon equation. Moreover, the derived linear estimates are applied to obtain global (in time) small data energy solutions for the Cauchy problem to semilinear Klein-Gordon models with power nonlinearity.

In the paper, we derive a fractional version of the Du Bois-Reymond lemma for a generalized Riemann-Liouville derivative (derivative in the Hilfer sense). It is a generalization of well known results of such a type for the Riemann-Liouville and Caputo derivatives. Next, we use this lemma to investigate critical points of a some Lagrange functional (we derive the Euler-Lagrange equation for this functional).