Unboundedness for the Euler–Bernoulli beam equation with a fractional boundary dissipation

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• 作者：Labidi ; Soraya ; Tatar ; Nasser-eddine
• 关键词：Euler&ndash ; Bernoulli beam equation ; Exponential growth ; Fourier transforms ; Fractional derivative ; Hardy&ndash ; Littlewood&ndash ; Sobolev inequality
• 刊名：Applied Mathematics and Computation
• 出版年：2005
• 期刊代码：7_00963003
• 类别：cp
• 出版时间：February 25, 2005
• 卷：161
• 期：3
• 页码：697-706
• 文件大小：214 K

摘要

We consider the Euler–Bernoulli beam problem with some boundary controls involving a fractional derivative. The fractional derivative here represents a fractional dissipation of lower order than one. We prove that the classical energy associated to the system is unbounded in presence of a polynomial nonlinearity. In fact, it will be proved that the energy will grow up as an exponential function as time goes to infinity.