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Singular limits in the Cauchy problem for the damped extensible beam equation

Singular limits in the Cauchy problem for the damped extensible beam equation

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RACKE, Reinhard, Shuji YOSHIKAWA, 2014. Singular limits in the Cauchy problem for the damped extensible beam equation

@techreport{Racke2014Singu-30729, series={Konstanzer Schriften in Mathematik}, title={Singular limits in the Cauchy problem for the damped extensible beam equation}, year={2014}, number={334}, author={Racke, Reinhard and Yoshikawa, Shuji} }

Yoshikawa, Shuji 2014 Yoshikawa, Shuji eng Singular limits in the Cauchy problem for the damped extensible beam equation 2015-04-14T08:08:06Z terms-of-use 2015-04-14T08:08:06Z We study the Cauchy problem of the Ball model for an extensible beam: \[\rho \partial_t^2 u + \delta \partial_t u + \kappa \partial_x^4 u + \eta \partial_t \partial_x^4 u = \left(\alpha + \beta \int_{\R} |\partial_x u|^2 dx + \gamma \eta \int_{\R} \partial_t \partial_x u \partial_x u dx \right) \partial_x^2 u.\]. The aim of this paper is to investigate singular limits as $\rho \to 0$ for this problem. In the authors' previous paper \cite{ra-yo} decay estimates of solutions $u_{\rho}$ to the equation in the case $\rho>0$ were shown. With the help of the decay estimates we describe the singular limit in the sense of the following uniform (in time) estimate: \[\| u_{\rho} - u_{0} \|_{L^{\infty}([0,\infty); H^2(\R))} \leq C \rho.\] Racke, Reinhard Racke, Reinhard

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