When I reading papers concerning energy decay rates for second order evolution equations, authors use higher-order energy method to deal with. For example, Timoshenko system $$\rho_{1}\varphi_{tt}-b_{1}(\varphi_{x}+\psi)_{x}=0,$$ $$\rho_{2}\psi_{tt}-b_{2}\psi_{xx}+\int_{0}^{t}g(t-s)\psi_{xx}(x,s)ds+b_{1}(\varphi_{x}+\psi)=0.$$ For giving a energy decay rate of the system, in some papers authors differentiate the above system twice and combine multiplier method to get a energy decay rate, for instance, $E(t)\leq C/t$, where $C=C(E(0),E_{1}(0),E_{2}(0))$.
But for nonlinear damping system, I can not find out any reference concerning this kind of theme: Energy decay rates. For example, to deal with $$\rho_{1}\varphi_{tt}+f(\varphi_{t})-b_{1}(\varphi_{x}+\psi)_{x}=0,$$ $$\rho_{2}\psi_{tt}-b_{2}\psi_{xx}+\int_{0}^{t}g(t-s)\psi_{xx}(x,s)ds+b_{1}(\varphi_{x}+\psi)=0.$$ When we differentiate this system twice, we should differentiate $f(\varphi_{t})$ twice, that is, $f'(\varphi_{t})\varphi_{tt}$ and $f''(\varphi_{t})\varphi_{tt}^{2}+f'(\varphi_{t})\varphi_{ttt}$. How to do next? Any reference can refer? Any help are appreciated.Thanks!
How about some analytical mechanics? Define the Lagrangian density $L = K-V$ with the kinetic and potential energies $$ K = \tfrac12\rho_1\varphi_t^2 + \tfrac12\rho_2 \psi_t^2 , $$ $$ V = \tfrac12 b_1 (\varphi_x +\psi)^2 + \tfrac12 b_2 \psi_x^2 , $$ see Wikipedia. Calculus of variations leads to the following Euler-Lagrange equations governing the free motion $$ \rho_1 \varphi_{tt} - b_1 (\varphi_x + \psi)_x = 0 , $$ $$ \rho_2\psi_{tt} - b_2\psi_{xx} + b_1 (\varphi_x + \psi) = 0 . $$ (I don't know the origin of the term $g *_t \psi_{xx}$ in OP; it must be some sort of memory effect [1]). To add dissipation to the system, one might introduce a Rayleigh dissipation function $D = \frac12 \eta \varphi_t^2$ such that the first PDE above becomes $$ \rho_1 \varphi_{tt} - b_1 (\varphi_x + \psi)_x = -\eta \varphi_t . $$ Using these assumptions, one might be able to prove the decay of Hamiltonian energy $\mathcal{E} = \int \left(K + V\right)\text dx$ over time (see this post), which is sometimes required out of thermodynamical considerations, or for the sake of well-posedness. Nevertheless, other forms of the dissipation might lead to energy decay as well. For instance, one might want to attempt the derivation of a dissipative theory from the nonlinear dissipation function $D = \frac14\eta \varphi_t^4$, see relevant literature below and references therein.
[1] Jin-Han Park, and Jum-Ran Kang. "Energy decay of solutions for Timoshenko beam with a weak non-linear dissipation." IMA Journal of Applied Mathematics 76.2 (2011): 340-350. doi:10.1093/imamat/hxq040