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This paper is concerned with the large time behavior of solutions to the onedimensional degenerate p-degree Fisher equation, where the initial data are assumed to be asymptotically front-like and to decay to zero non-exponentially at one end. By applying sub-super solution method we first prove the Lyapunov stability of all the wave fronts with noncritical speeds in some polynomially weighted space. Further, by using semigroup estimates, sub-super solution method and the nonlinear stability of the traveling waves, we obtain the asymptotic estimates of the solutions which indicate that the solution still moves like a wave front and the asymptotic spreading speed of the level set of the solution can be finite or infinite, which is solely determined by the detailed decaying rate of the initial data.

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