Convergence to steady states of solutions to nonlinear integral evolution equations

We prove that any global bounded solution of the nonlinear evolutionary integral equation $$dot{u}(t) + intlimits_0^t a(t-s)mathcal{E}'(u(s))ds =f(t), quad t >0 $$ tends to a single equilibrium state for long time (i.e., ${mathcal{E}'(vartheta)=0}$ where ${vartheta= lim_{t rightarrow infty} u(t)}$ on a real Hilbert space), where ${mathcal{E}'}$ is the Fréchet derivative of a functional ${mathcal{E}}$ , which satisfies the ?ojasiewicz?CSimon inequality near ${vartheta}$ . The vector-valued function f and the scalar kernel a satisfy suitable conditions.