Convergence of wave equation with friction

Question

I’m studying the following 1-D wave equation with friction $$\begin{cases} u_{tt}+2\epsilon u_t-u_{xx}=0,\,x\in(0,\pi),\,t>0,\\ u_x(0,t)=0=u_x(\pi,t)w,\\ u(x,0)=u_0(x),\,u_t(x,0)=0,\end{cases}$$ for $0<\epsilon<1$. I’ve been able to prove the uniqueness of the solution by energy techniques, and by separation of variables I ended up getting that the solution is $$u(x,t)=\sum e^{-\epsilon t}\left( A_n\cos (\sqrt{n^2-\epsilon^2}t)+B_n\sin (\sqrt{n^2-\epsilon^2}t)\right)\cos (nx).$$ Now $$u_0(x)=u(x,0)=\sum A_n\cos (nx)\implies A_n=\frac{2}{\pi}\int_0^\pi u_0(x)\cos(nx)dx.$$ The other boundary tells us that $$0=u_t(x,0)=-\epsilon u(x,0)+\sum e^{-\epsilon t}\sqrt{n^2-\epsilon^2}\left(-A_n\sin (\sqrt{n^2-\epsilon^2}0)+B_n\cos (\sqrt{n^2-\epsilon^2}0)\right)\cos (nx)\implies u_0(x)=\sum\frac{\sqrt{n^2-\epsilon^2}}{\epsilon} B_n\cos(nx)\implies B_n=\frac{2\epsilon}{\sqrt{n^2-\epsilon^2}}\int_0^\pi u_0(x)\cos (nx)dx.$$ Now I want to prove that $$u(x,t)\to \frac{1}{\pi}\int_0^\pi u_0(x)dx,$$ when $t\to\infty$, and if it converges uniformly. But I don’t know how to proceed, I tried using Bessel’s equality for the coefficients of the Fourier series of $u_0(x)$, but I didn’t end up with anything. Any help will be very appreciated.