Well-posedness and long time behavior for p-Laplacian equation with nonlinear boundary condition

In this paper, we study the homogeneous nonlinear boundary value problem for the p-Laplacian equation u t − △ p u + a ( x , t ) | u | σ − 2 u − b ( x , t ) | u | ν − 2 u = h ( x , t ) . We prove the existence of weak solutions which is global or local in time in dependence on the relation between the exponent of nonlinear part in boundary value and p. Boundedness of weak solution is proved. We established conditions of uniqueness. We prove also the properties of extinction in a finite time, finite speed propagation and waiting time. Lastly, by using the energy method, we obtain sufficient conditions that the solutions of this problem with non-positive initial energy blow up in finite time.