Complicated asymptotic behavior of solutions for porous medium equation in unbounded space

In this paper, we find that the unbounded spaces Yσ(RN)(0<σ<2m−1) can provide the work spaces where complicated asymptotic behavior appears in the solutions of the Cauchy problem of the porous medium equation. To overcome the difficulties caused by the nonlinearity of the equation and the unbounded solutions, we establish the propagation estimates, the growth estimates and the weighted L1-L∞ estimates for the solutions.