On a large time behavior of a solution to a one-dimensional free boundary problem for adsorption phenomena

In this paper we consider a one-dimensional free boundary problem describing adsorption phenomena in a porous media. This problem was already proposed and we established a local existence in time and a uniqueness result in [1] and [4]. Here, through the discussion for the dynamics of the free boundary in wetting and drying processes we suppose a specific form of the growth rate condition for the free boundary. By using this form in our problem we can obtain the global existence of a solution in time and convergence of the free boundary as t→∞t→∞.