文章基本信息

标题：Global existence, energy decay and blow-up of solutions for wave equations with time delay and logarithmic source
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作者：Sun-Hye Park
期刊名称：Advances in Difference Equations
印刷版ISSN：1687-1839
电子版ISSN：1687-1847
出版年度：2020
卷号：2020
期号：1
页码：1-17
DOI：10.1186/s13662-020-03037-6
出版社：Hindawi Publishing Corporation
摘要：In this paper, we study the wave equation with frictional damping, time delay in the velocity, and logarithmic source of the form $$ u_{tt}(x,t) - \Delta u (x,t) \alpha u_{t} (x,t) \beta u_{t} (x, t- \tau ) = u(x,t) \ln \bigl\vert u(x,t) \bigr\vert ^{\gamma } . $$ There is much literature on wave equations with a polynomial nonlinear source, but not much on the equations with logarithmic source. We show the local and global existence of solutions using Faedo–Galerkin’s method and the logarithmic Sobolev inequality. And then we investigate the decay rates and infinite time blow-up for the solutions through the potential well and perturbed energy methods.
关键词：Wave equation; Logarithmic source; Time delay; Energy decay; Blow-up