## Differential and Integral Equations

### 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.

#### Article information

Source
Differential Integral Equations, Volume 32, Number 11/12 (2019), 639-658.

Dates
First available in Project Euclid: 22 October 2019