Global existence and asymptotic stability of smooth solutions to a fluid dynamics model of biofilms in one space dimension

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• 作者：Roberta Bianchini^a ; ^{bianchin@uniroma2.it" class="auth_mail" title="E-mail the corresponding author} ; Roberto Natalini^b ; ^{roberto.natalini@cnr.it" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Fluid dynamics models ; Dissipative hyperbolic equations ; Biofilms ; Global existence ; Asymptotic stability
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2016
• 出版时间：15 February 2016
• 年：2016
• 卷：434
• 期：2
• 页码：1909-1923
• 全文大小：363 K

文摘

In this paper, we present an analytical study, in the one space dimensional case, of the fluid dynamics system proposed in [3] to model the formation of biofilms. After showing the hyperbolicity of the system, we show that, in an open neighborhood of the physical parameters, the system is totally dissipative near its unique non-vanishing equilibrium point. Using this property, we are able to prove existence and uniqueness of global smooth solutions to the Cauchy problem on the whole line for small perturbations of this equilibrium point and the solutions are shown to converge exponentially in time at the equilibrium state.