This paper concerns a class of quasi-linear parabolic and elliptic partial differential equations in a bounded domain with nonlocal boundary conditions. The equations under consideration are non-degenerate depending on the property of the diffusion coefficient. By using the upper and lower solutions and monotone iteration, the aim of the paper is to show existence and uniqueness of solutions for the time-dependent problem, existence of maximal and minimal steady-state solutions of the elliptic problem, and the asymptotic behavior of the time-dependent solutions in relation to the steady-state solutions.