Séminaire - Bornes ultimes de l’énergie des solutions de certaines équations d’évolution du second ordre avec amortissement non linéaire et terme source borné

Résumé : Under suitable growth and coercivity conditions on the nonlinear damping operator $g$ which insure non-resonance, we estimate the ultimate bound of the energy of the general solution to the equation $\ddotu(t) + g(\dotu(t))+ Au(t)=h(t),\quad t\in\R^+ $ where $A$ is a positive selfadjoint operator on a Hilbert space $H$ and $h$ is a bounded forcing term with values in $H$. In general the bound is of the form $ C(1+ ||h||^4)$ where $||h||$ stands for the $L^\infty$ norm of $h$ with values in $H$ and the actual growth of $g$ does not seem to play any role as long as we are in the non-resonant situation. If $g$ behaves like a power for large values of the velocity, the ultimate bound has a quadratic growth with respect to $||h||$ independently of the power and this result is optimal. If $h$ is anti-periodic, we obtain a much lower growth bound and again the result is shown to be optimal even for scalar ODEs.