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.

中文翻译：

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具有对数非线性的无限退化半线性双曲型方程解的整体存在和爆破

在本文中，我们研究了一类具有对数非线性的无限退化半线性双曲型方程的初边值问题。u u {tt}-\ triangle_ {X} u = u \ log | 你 ，$$，其中$ X =（X_1，X_2，...，X_m）$是向量场的无限简并系统，$$ {\ triangle_X} = \ sum \ limits_ {j = 1} ^ m {X_j ^ 2} $$是无限简并的椭圆算子。通过使用对数的Sobolev不等式和一系列潜在的井，我们首先证明了某些集合的不变性。然后，通过伽勒金方法，获得具有低初始能量或临界初始能量的溶液的整体存在性，并在有限的时间内爆炸。