Stability and gradient dynamical systems (Q1884573)

This paper deals with infinite-dimensional dynamical systems. The author considers as a phase space metric ones. In the case, when \(X\) is a compact manifold (or even locally compact), there is an extensive qualitative theory of dynamical systems associated with stability and bifurcation of the orbit structure. If \(X\) is not locally compact, the theory is far from complete. The author presents recent developments in this direction. Moreover, he presents an application of the theory to functional-differential equations, as well as quasilinear parabolic equations and linearly damped nonlinear wave equations.