I am working on a fluid-structure-interaction evolution problem: let us suppose that the motion of the rectangular body $B$, with boundary $\partial B=S$, immersed in an unbounded 2D channel $\mathbb{R}\times (-L,L)$, is governed by a nonlinear oscillator equation with restoring force $f=f(h)$, and forced by the fluid lift exerted on $B$: \begin{equation}\label{eq:ODE} \ddot{h}+f(h)=-\hat{e}_2\cdot{\langle \mathbf{T}\cdot\hat{n},1\rangle}_S \qquad \text{in}\,\,(0,T),\qquad (1) \end{equation}where $\mathbf{T}= \mathbf{T}(u,p)=-p\mathbf{I}+\mu[\nabla u+(\nabla u)^T]$.
We suppose that $u\in L^2(0,T; H^1(\Omega))$, velocity field of the fluid, is $\textit{known}$ and thus \begin{equation} \mathbf{T}(u,p)\cdot\hat{n}\in L^2(0,T;W^{-\tfrac{2}{3},\tfrac{3}{2}}(S)), \end{equation} In (1), ${\langle \cdot,\cdot\rangle}_S$ precisely labels the duality between $W^{-\tfrac{2}{3},\tfrac{3}{2}}(S)$ and its dual space $W^{\tfrac{2}{3},3}(S)$.
We assume that $f(h)$ satisifes the following hypotheses: $f\in C^1(-L+1,L-1)$ and \begin{equation} f'(h)>0\, \forall h\in (-L+1,L-1), \quad \lim_{|h|\to L-1}\frac{|f(h)|}{(|L-1|-|h|)^{3/2}}=+\infty \end{equation} The ODE (1) enjoys existence and uniqueness of solutions $h\in H^2(0,T)$.
How do I prove that this solution is also $\textit{bounded}$?