Global Existence, Exponential Decay and Blow-up in Finite Time for a Class of Finitely Degenerate Semilinear Parabolic Equations

Abstract

In this paper, we study the initial-boundary value problem for the semilinear parabolic equations ut − ∆Xu = |u|p−1u, where X = (X1, X2, ⋯ , Xm) is a system of real smooth vector fields which satisfy the Hormander’s condition, and $$\Delta_X\;=\;\sum\limits_{j = 1}^m\;{X_j^2}$$ finitely degenerate elliptic operator. Using potential well method, we first prove the invariance of some sets and vacuum isolating of solutions. Finally, by the Galerkin method and the concavity method we show the global existence and blow-up in finite time of solutions with low initial energy or critical initial energy, and also we discuss the asymptotic behavior of the global solutions.