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摘要
In this paper, we study a free boundary problem modeling grain hydration. The grain is submersed in water, and there are two free boundaries occurring in this problem. The outer free boundary r = R(t) is the boundary separating the grain region {r < R(t)} from water region {r > R(t)}. The inner boundary r = S(t) is the boundary separating the wet region {r > S(t)} from the dry region {r < S(t)}. Water penetrates from water region into grain region and also from wet region into dry region-causing both free boundaries to move. We show that this problem globally in time is well posed, and admits a unique solution with two stages. We prove that there exists a T* < infinity such that the inner free boundary reaches the origin at T*: S(T* - 0) = 0. The problem on the time interval [0,T*] constitutes stage I and on the time interval [T*, infinity) constitutes stage coproduct. We establish that lim(t ->infinity) R(t) = R-infinity < infinity, and that lim(t ->infinity) u(r, t) = u(infinity) (the saturated moisture content). The solution is singular at time t = 0; we give an explicit characteristics of singularity at t = 0. The solution is also singular near point (0,T*); we shall use approximation to study the behavior of the solution.
