On transitions to stationary states in some infinite-dimensional Hamiltonian systems (Q1377948)

The goal of this work is the investigation of transitions to stationary states in nonlinear hyperbolic equations. Generally speaking, such behavior is impossible for finite-dimensional Hamiltonian systems, as well as for linear autonomous systems of equations of Schrödinger type. We consider as a model a system that describes a string that interacts with a nonlinear oscillator. Namely, we establish transitions to stationary states for real solutions of the model system \[ \mu\ddot u(x, t)= Tu''_{xx}(x, t),\quad x\neq 0;\quad m\ddot y(t)= F(y(t))+ T[u_x'(0+, t)- u_x'(0-, t)].\tag{1} \] Here, \(y(t)\equiv u(0, t)\); \(\mu\), \(T>0\), \(m\geq 0\). We assume that the potential of the external field increases at infinity. The main result of this work is the fact that any solution \(u(x, t)\) of the ``nondegenerate'' system (1) that has a finite energy tends to a certain stationary solution as \(t\to+\infty\).