On the decay rates of solutions for a nonlinearly damped porous system with a delay

Abstract

We consider a nonlinear damped Porous system subjected to a nonlinear delayed damping acting on the volume fraction equation. Namely, we
investigate the following system \begin{equation*} \begin{dcases}
\rho_1 u_{tt}(x,t) - \kappa u_{xx}(x,t)- b \phi_x(x,t) =0 , \\ \rho_2 \phi_{tt}(x,t) - \delta \phi_{xx}(x,t) + b u_{x}(x,t) + \xi \phi(x,t) + \mu_1 g_1( \phi_t(x,t)) + \mu_2 g_2( \phi_t(x,t-\tau ))=0\end{dcases}\end{equation*} together with Dirichlet-Dirichlet boundary conditions in $[0,1] \times [0,+\infty[$. By the classical Faedo-Galerkin procedure, we first
prove the well-posedness of solutions without paying attention on the weights of feedbacks (delayed or not). This improves many earlier
results existing in the literature by removing the usual restrictions imposed on $\mu_1$ and $\mu_2$. Furthermore, by applying the
multiplier method integrated with some properties of convex functions, we establish two general decay estimates with rates that depend on
the speeds of wave propagation and the smoothness of the initial data