Exponential decay for semilinear distributed systems with damping
The author considers the abstract system \[ y''(t)+Ay(t)+Dy'(t)+F(y(t),y'(t))By(t)=0, \] \[ y(0)=y_0,\quad y'(0)=y_1, \] which is a model for vibrating processes where damping is accounted for by the term \(Dy'(t)\); \(B\) and \(D\) are linear bounded operators from \(H\) to \(H\), where \(H\) is real Hilbert space. The operator \(D\) is assumed to be selfadjoint and to satisfy: \((Dv,v)\geq 0\), \(|Dv|^2\leq \text{const} (Dv,v)\) for every \(v\in H\). The operator \(A\) is the unique linear operator from \(V\subset H\) to \(V'\) which is characterized by \((Av,w)_{V'V}=(v,w)_{V}\) for every \(v,w\in V\). Two different nonlinearities are considered: \(F(y(t),y'(t))=f((By(t),y'(t)))\) and \(F(y(t),y'(t))=f((By(t),y(t)))\). The stability of the given equation is studied. The main goal is to characterize the classes of damping operators \(D\) such that under additional assumptions the energy specified by \(E(t)=(|y(t)|^2+|y'(t)|^2)/2\) decays exponentially to zero when \(t\to\infty\). Local, nonuniform and uniform exponential stability is investigated. Applications to the wave and Euler-Bernoulli equations are given.
scientific article; zbMATH DE number 1884298