On the ultimate energy bound of solutions to some forced second-order evolution equations with a general nonlinear damping operator

Abstract

Under suitable growth and coercivity conditions on the nonlinear damping operator $g$ which ensure nonresonance, we estimate the ultimate bound of the energy of the general solution to the equation $ü\left(t\right)+Au\left(t\right)+g\left(\dot{u}\left(t\right)\right)=h\left(t\right)$, $t\in {ℝ}^{+}$, 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 $C\left(1+\parallel h{\parallel }^4\right)$, where $\parallel h\parallel$ stands for the $L^{\infty }$ norm of $h$ with values in $H$ and the growth of $g$ does not seem to play any role. If $g$ behaves like a power for large values of the velocity, the ultimate bound has quadratic growth with respect to $\parallel h\parallel$ and this result is optimal. If $h$ is antiperiodic, we obtain a much lower growth bound and again the result is shown to be optimal even for scalar ODEs.