Propagation profile of support for evolution p-Laplacian with convection in half space

Consider the Cauchy-Dirichlet problem in half space for a one-dimensional evolution p-Laplacian with convection for p>2, and pay attention to the interface ξ(t)=sup{x;u(x,t)>0}. It is well known that limt→+∞ξ(t)=+∞ in the absence of the convection, while the inclusion of the first-order term may change the property of finite (or infinite) speed of propagation. In this paper, it will be shown that the nonlinear convection plays a very important role to the evolution of ξ(t). For the convection with promoting diffusion, the fast propagation phenomenon occurs (i.e. u(x,t)>0 whenever t>0) if the convection is strong enough, otherwise, ξ(t) remains finite and non-localized. While under the convection with counteracting diffusion, if the convection is strong enough, localization (even shrinking and extinction) appears, otherwise, ξ(t) keeps non-localized. In addition, it is found that the time-related boundary data are significant also to the behavior of solutions: the decay or incremental rates of the boundary data affect not only the contraction or expansion of the supports, but also the propagation speed of the interface.