On the principle of linearized stability for variational inequalities
We study the Lyapunov stability of stationary solutions of parabolic variational inequalities of the form \[ (du(t)/dt+F(u(t)),v-u(t))\geq 0,\quad v\in K,\quad u(t)\in K,\quad u(0)=u_ 0, \] where K is a closed convex subset of a real Hilbert space. We derive conditions on \(F(u_ 0)\), \(F'(u_ 0)\) and K, which are sufficient for the stability or the instability of a stationary solution \(u_ 0\). In some special cases, these conditions can be written in terms of eigenvalues of the corresponding variational inequality \(0\neq u\in K_ 0:\) \((F'(u_ 0)u- \lambda u,v-u)\geq 0,v\in K_ 0,\) where \(K_ 0\) is the closure of the set \(\cup _{\alpha >0}\alpha (K-u_ 0)\), and then they are analogous to those occurring in the principle of linearized stability for semilinear parabolic equations. An application with F being a semilinear elliptic differential operator is given.
scientific article; zbMATH DE number 4046495