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November/December 2019 Global existence and blow-up of solutions for infinitely degenerate semilinear hyperbolic equations with logarithmic nonlinearity
Hua Chen, Jing Wang, Huiyang Xu
Differential Integral Equations 32(11/12): 639-658 (November/December 2019).

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.

Citation

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Hua Chen. Jing Wang. Huiyang Xu. "Global existence and blow-up of solutions for infinitely degenerate semilinear hyperbolic equations with logarithmic nonlinearity." Differential Integral Equations 32 (11/12) 639 - 658, November/December 2019.

Information

Published: November/December 2019
First available in Project Euclid: 22 October 2019

zbMATH: 07144907
MathSciNet: MR4021257

Subjects:
Primary: 35L70, 35L80

Rights: Copyright © 2019 Khayyam Publishing, Inc.

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Vol.32 • No. 11/12 • November/December 2019
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