Propagation property for anisotropic nonlinear diffusion equation with convection

We consider propagation property for anisotropic diffusion equation with convection in 2 dimension, ∂ t ( u m ) − ∂ x 1 ( | ∂ x 1 u | p 1 − 1 ∂ x 1 u ) − ∂ x 2 ( | ∂ x 2 u | p 2 − 1 ∂ x 2 u ) + u α − 1 ∂ x 1 u = 0 , where p 1 , p 2 , m , α > 0 . Among the results, we show that perturbation for the nonnegative solutions propagates with infinite speed in x 1 -direction and with finite speed in x 2 -direction if 0 < α < m < p 2 . We also show that the anisotropic propagation may appear when the convection term is weak, backward, or even missing, if 0 < p 1 ⩽ m < p 2 , p 1 ⩽ 1 .