Convergence to steady states of solutions to nonlinear integral evolution equations (Q629851)

The author investigates the long-time behavior of bounded solutions of nonlinear integral evolution equation \[ \dot u(t)+\displaystyle\int_0^ta(t-s){\mathcal E}'(u(s))\,ds=f(t),\,\,\,t>0,\leqno(E) \] in a real Hilbert space \(H\), where \({\mathcal E}'\) is the Fréchet derivative of a functional \({\mathcal E}\in C^1(V)\), \(V\subset H\) is a Hilbert space which is densely and continuously included into \(H\), and the vector-valued function \(f\), the scalar kernel \(a\) and the functional \({\mathcal E}\) satisfy some suitable assumptions. By using an appropriate new Lyapunov functional, some compactness properties and the Lojasiewicz-Simon inequality, he proves that any global bounded solution \(u(t)\) of \((E)\) tends for \(t\to\infty\) to a single equilibrium state \(\vartheta\) (i.e. \({\mathcal E}'(\vartheta)=0\)).