Global existence and blow-up of solutions for infinitely degenerate semilinear hyperbolic equations with logarithmic nonlinearity

Abstract

In this paper, we study the initial-boundary value problem for a class of infinitely degenerate semilinear hyperbolic equations with logarithmic nonlinearity $$ u_{tt}-\triangle_{X} u=u\log | u | , $$ where $X= (X_1,X_2,...,X_m)$ is an infinitely degenerate system of vector fields, and $$ {\triangle_X} = \sum\limits_{j = 1}^m {X_j^2} $$ is an infinitely degenerate elliptic operator. By using the logarithmic Sobolev inequality and a family of potential wells, we first prove the invariance of some sets. Then, by the Galerkin method, we obtain the global existence and blow-up in finite time of solutions with low initial energy or critical initial energy.