It's a joint work with Dr. Junfeng He. In this talk I shall talk about our recent work on the spatial decay and  stability of multidimensional  cylinder fronts  for the degenerate Fisher type equation. By applying the  generalized center manifold theorem  and asymptotic expansions we obtain the  spatial decay of traveling fronts with all speeds, especially  for the $p$-degree Fisher type equation  we get the precise algebraic decaying rates and the higher order expansion of  traveling fronts with non-critical speeds. By applying spectral analysis and sub-super solution method we prove the exponential stability  of all traveling fronts  in some exponentially weighted spaces and the   Lyapunov stability of traveling fronts with noncritical speeds in some polynomially weighted spaces.   We also investigate the  asymptotic behavior and spreading speed of the solution with more general initial data, which are proved to be solely determined by  the spatial decay   of the initial data at one end.