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We focus on the spreading properties of solutions of monostable equationswith non-linear diffusion. We consider both the porous medium diffusion and thefast diffusion regimes. Initial data may have heavy tails, which tends toaccelerate the invasion phenomenon. On the other hand, the nonlinearity mayinvolve a weak Allee effect, which tends to slow down the process. We study thebalance between these three effects (nonlin-ear diffusion, initial tail, KPPnonlinearity/Allee effect), revealing the separation between "no acceleration"and "acceleration". In most of the cases where acceleration occurs, we alsogive an accurate estimate of the position of the level sets.

Interplay of nonlinear diffusion, initial tails and Allee effect on the speed of invasions