This paper deals with the asymptotic spreading for a class of diffusion equations with degenerate monostable nonlinearity, where the initial value is asymptotically front-like. By the method of squeezing technique and super- and sub-solutions, we prove that the speed of asymptotic spreading may be finite or infinite, which depends on the degeneracy of nonlinearity as well as the decaying rate of the initial value. Different from the non-degenerate case, the traveling wave solution with the critical speed is globally asymptotically stable if the initial value has light tails, while an acceleration propagation may occur with the initial value having heavy tails in the degenerate case. Furthermore, if the initial value has heavy tails but lighter than algebraic, then the speed of the asymptotic spreading is finite, and the balance between finite or infinite speed is established if the initial value has algebraic tails. Our results reflect that the degeneracy may be harmful to the acceleration propagation.

本文讨论了一类具有退化单稳态非线性的扩散方程的渐近扩展，其中初始值是渐近前沿的。通过压缩技术和超子解的方法，我们证明了渐近扩展的速度可以是有限的，也可以是无限的，这取决于非线性的退化程度以及初始值的衰减率。与非退化情况不同，临界速度的行波解在初始值有轻尾时全局渐近稳定，而在退化情况下初始值有重尾时可能发生加速度传播。此外，如果初始值有重尾但比代数轻，那么渐近传播的速度是有限的，如果初始值具有代数尾，则在有限或无限速度之间建立平衡。我们的结果反映了简并性可能对加速传播有害。