In this paper we study a convection-reaction-diffusion equation of the form $u_t=\epsilon (h(u)u_x{)}_x-f(u{)}_x+f^{\prime }(u),t>0,$ with a nonlinear diffusion in a bounded interval of the real line. In particular, we first focus our attention on the existence of stationary solutions with at most one zero inside the interval, studying their behavior with respect to the viscosity coefficient $\epsilon >0$ and their stability/instability properties. Then, we investigate the large time behavior of the solutions for finite times and the asymptotic regime. We also show numerically that, for a particular class of initial data, the so-called metastable behavior occurs, meaning that the time-dependent solution persists for an exponentially long (with respect to $\epsilon$) time in a transition non-stable phase, before converging to a stable configuration.