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 lim t → + ∞ 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.