As we know in civil engineering, it is the first task for constructing banks or building structures to predict the subsidence of the underlying ground. When ground of clay saturated with pore water is compressed, the water flows out and the total volume decreases. Consequently, the ground subsides and the strength of the clay increases. We call this phenomena consolidation. For the simplicity of analysis, we usually assume that the thickness of the clay stratum will not vary during consolidation and neglect the effect by its decrease. Alternatively in this paper, we formulate a mathematical model of one dimensional consolidation problem, taking the decrease of the thickness of the clay stratum into consideration. And we show the existence and uniqueness of the exact solution of the problem under the assumption of small initial data
