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In this paper the finite speed of propagation of solutions and the continuous dependence on the nonlinearity of a degenerate parabolic partial differential equation are discussed. Our objective is to derive an explicit expression for the speed of propagation and the large time behavior of the solution and to show that the solution continuously depends on the nonlinearity of the equation.

This work is concerned with the inverse problem of determining initial value of the Cauchy problem for a nonlinear diffusion process with an additional condition on free boundary. Considering the flow of water through a homogeneous isotropic rigid porous medium, we have such desire: for every given positive constants $K$ and $T_0$, to decide the initial value $u_0$ such that the solution $u(x,t)$ satisfies ${sup}_{x\in H_u\left(T_0\right)}\left|x\right|\ge K$, where $H_u\left(T_0\right)=\{x\in {ℝ}^N:u(x,T_0)>0\}$. In this paper, we first establish a priori estimate $u_t\ge C\left(t\right)u$ and a more precise Poincaré type inequality...