This work covers a range of topics in mathematical analysis and partial differential equations. It begins with remarks on bifurcation from the essential spectrum and explores constructions using analytic semigroups, along with abstract exponential decay results for eigenfunctions. The discussion includes Wiener regularity and heat semigroups on spaces of continuous functions, as well as the existence of infinitely many solutions for nonlinear elliptic systems. Hölder regularity for linear fractional evolution equations and non-gradient perturbations of Ornstein-Uhlenbeck semigroups are examined. The text also addresses jumping nonlinearities and local singular variations of domains in semilinear elliptic equations, alongside singular semilinear elliptic equations in L1. Further topics include moving surfaces and abstract parabolic evolution equations, evolution equations with memory-type dissipation, and the large-time behavior of solutions to semilinear parabolic equations with nonlinear dynamical boundary conditions. The properties of the Navier-Stokes operator and bifurcations of stationary solutions in planar exterior domains are analyzed, along with discussions on the loss of convexity in simple closed curves under surface diffusion. The work also delves into fiber spinning equations in nonisothermal viscous flow, radial solutions of quasilinear boundary value problems, and operator-valued Fourier multipliers. Other
