Prove that a function is locally Lipschitz

Question

I am studying the paper "F. D. Araruna, P. Braz E Silva, E. Zuazua, Asymptotic limits and stabilization for the 1D nonlinear Mindlin-Timoshenko system"[http://link.springer.com/article/10.1007/s11424-010-0137-8] and I have a doubt in the end of page 419. The authors proved that the function $$f=\left[\psi_x\left(\eta_x+\frac{1}{2}\psi_x^2\right)-\widetilde{\psi}_x\left(\widetilde{\eta}_x+\frac{1}{2}\widetilde{\psi}_x^2\right)\right]_x,$$ where $\psi,\widetilde{\psi}\in H_0^2(0,L)$ and $\eta,\widetilde{\eta}\in H^1(0,L)\cap\{v\in L^2(0,L):\int_0^L v(x)dx=0\}$, is locally Lipschitz, but I believe that the proof of it is wrong. I think they disregarded the derivative with respect to $x$, ie, I think they just felt $$f=\psi_x\left(\eta_x+\frac{1}{2}\psi_x^2\right)-\widetilde{\psi}_x\left(\widetilde{\eta}_x+\frac{1}{2}\widetilde{\psi}_x^2\right).$$ The authors really screwed up or am I mistaken? If they have wrong, does anyone know how to prove that $ f $ is locally Lipschitz?