Asymptotic behavior and blow-up of solutions for infinitely degenerate semilinear parabolic equations with logarithmic nonlinearity

In this paper, we study the initial-boundary value problem for infinitely degenerate semilinear parabolic equations with logarithmic nonlinearity u(t) - Delta(X)u = u log vertical bar u vertical bar, where X = (X-1, X-2, ... , X-m) is an infinitely degenerate system of vector fields, and Delta(X) := Sigma(m)(j=1) X-j(2) is an infinitely degenerate elliptic operator. Using potential well method, we first prove the invariance of some sets and vacuum isolating of solutions. Then, by the Galerkin method and the logarithmic Sobolev inequality, we obtain the global existence and blow-up at +infinity of solutions with low initial energy or critical initial energy, and we also discuss the asymptotic behavior of the solutions. (C) 2018 Elsevier Inc. All rights reserved.