Asymptotic behavior of solutions for the Chipot-Weissler equation with free boundary

In this talk, we consider the free boundary problem for the Chipot-Weissler equation. It is well known that global existence or blowup of solutions of nonlinear parabolic equations depends on which one dominating the model, the source or absorption, and on the absorption coefficient for the balance case of them. The aim of our investigation was to study the influence of exponents of source and absorption, initial data and free boundary on the asymptotic behavior of solutions. At first, the ecological meaning of this model is explained by deriving the equation and the free boundary condition. Then, local existence and uniqueness are discussed, and the continuous dependence on initial data and comparison principle are proved. Furthermore, the finite time blow-up and global solution are given by constructing sub- and super-solutions. In the different ranges of exponents and initial conditions, finite time blow-up solutions, global fast solutions and global slow solutions are classified. Finally, the problem with double free boundaries will be discussed too.