Differential and Integral Equations

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

Hua Chen, Jing Wang, and Huiyang Xu

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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

Permanent link to this document
https://projecteuclid.org/euclid.die/1571731513

Mathematical Reviews number (MathSciNet)
MR4021257

Zentralblatt MATH identifier
07144907

Subjects
Primary: 35L70: Nonlinear second-order hyperbolic equations 35L80: Degenerate hyperbolic equations

Citation

Chen, Hua; Wang, Jing; Xu, Huiyang. Global existence and blow-up of solutions for infinitely degenerate semilinear hyperbolic equations with logarithmic nonlinearity. Differential Integral Equations 32 (2019), no. 11/12, 639--658. https://projecteuclid.org/euclid.die/1571731513


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