Spatial decay and stability of traveling fronts for degenerate Fisher type equations in cylinder

This paper is concerned with the spatial decay and asymptotic stability of multidimensional traveling fronts for the degenerate Fisher type equation in an unbounded cylinder. Firstly, by applying the moving plane argument and the generalized center manifold theorem, we obtain the uniqueness and exponential decay of the traveling front with the critical speed and the non-exponential decay of traveling fronts with non-critical speeds, especially for the p-degree Fisher equation we get the precise algebraic decaying rates and the higher order expansion of traveling fronts with non-critical speeds. Secondly, by applying the spectral analysis and sub-super solution method we prove the nonlinear exponential stability of all traveling fronts in some exponentially weighted spaces and the Lyapunov stability of traveling fronts with non-critical speeds in some polynomially weighted spaces. Finally, by combining the sub-super solution method with the nonlinear stability results, we get the asymptotic behavior and the asymptotic spreading speeds of the solution for more general initial data, which are proved to be determined by the spatial decay of the initial data at one end.