In this talk, we will consider the of porous media equations when the initial data are continuous and compactly supported. The major feature is the parabolicity degenerate at the moving boundary. By introducing the proper weighted Sobolev space, which captures the degenerative at the moving boundary, the well-posedness of the local solution is established at first. Then, the global solution closed to the Barenblatt solution is constructed, and converges to the Barenblatt solution as time goes to infinity. Our results particularly give a positive answer of the open problem proposed by Lee and Vazquez on convexity.

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