Resolvent estimates and quantification of nonlinear stability
The authors investigate the role played by resolvent estimates for the stability of ordinary and parabolic equations. In particular they consider the following equation in the Euclidean space \(u_t=Au + f(u)\) where \(A\) is a linear operator an Euclidean space (\(n\times n\)-matrix) and \(f\) is a nonlinear term. It is assumed that \(\text{Re }\sigma (A)<0\) where \(\sigma (A)\) is the spectrum of \(A\). Under various assumptions on \(f\), stability conditions are derived. Note that the authors established some resolvent estimates, which follow from Theorem 17.3.1 of the book [\textit{M. I. Gil'}, Stability of finite and infinite dimensional systems, Kluwer, Dordrecht (1998; Zbl 0916.93002)]. In addition, the equations of the type \[ u_t=u_{xx}+u_x+uu_x \] on a finite segment are investigated.
scientific article; zbMATH DE number 1471192