arXiv 1409.34482014-09-11math.AP

Boundyshenko梁系统的非线性边界镇定

Nonlinear Boundary Stabilization for Timoshenko Beam System

M. L. Oliveira, A. J. R. Feitosa, M. Milla Miranda

摘要：

本文研究了一类具有非线性边界条件的非线性方程组$$ \left\|\begin{array}{cc} u"-μ(t)Δu+α_1 \displaystyle\sum_{i=1}^{n}\frac{\partial v}{\partial x_{i}}=0,\, \in Ω\times (0, \infty),\\ v"-Δv-α_2 \displaystyle\sum_{i=1}^{n}\frac{\partial u}{\partial x_{i}}=0, \, \in Ω\times (0, \infty), \end{array} \right. $$解的存在性和衰减性，讨论了$$ \left\|\begin{array}{cc} u=v=0 \,\, in \,Γ_{0}\times (0, \infty),\\ \frac{\partial u}{\partial ν} + h_{1}(x,u')=0\, in\,\, Γ_{1}\times (0, \infty),\\ \frac{\partial v}{\partial ν} + h_{2}(x,v')+σ(x)u=0 \, in\, \,Γ_{1}\times (0, \infty), \end{array} \right. $$和在$t=0$上的初始条件.这里，$Ω$是$\mathbb{R}^n$的有界开集，边界$Γ$由两个不相交的部分$Γ_{0}$和$Γ_{1}$组成，$ν(x)$表示$x\in Γ_{1}$的外部单位法向量。函数$h_{i}(x,s),\,\, (i=1,2)$在$s\in \mathbb{R}$中是连续的和强单调的。利用具有特殊基的Galerkin方法、紧性方法和Strauss关于Lipschitz连续函数逼近连续函数的一个结果，得到了上述问题解的存在性.通过使用适当的李雅普诺夫泛函和乘子方法，能量的指数衰减如下。

英文摘要：

This paper is concerned with the existence and decay of solutions of the following Timoshenko system: $$ \left\|\begin{array}{cc} u"-μ(t)Δu+α_1 \displaystyle\sum_{i=1}^{n}\frac{\partial v}{\partial x_{i}}=0,\, \in Ω\times (0, \infty),\\ v"-Δv-α_2 \displaystyle\sum_{i=1}^{n}\frac{\partial u}{\partial x_{i}}=0, \, \in Ω\times (0, \infty), \end{array} \right. $$ subject to the nonlinear boundary conditions, $$ \left\|\begin{array}{cc} u=v=0 \,\, in \,Γ_{0}\times (0, \infty),\\ \frac{\partial u}{\partial ν} + h_{1}(x,u')=0\, in\,\, Γ_{1}\times (0, \infty),\\ \frac{\partial v}{\partial ν} + h_{2}(x,v')+σ(x)u=0 \, in\, \,Γ_{1}\times (0, \infty), \end{array} \right. $$ and the respective initial conditions at $t=0$. Here $Ω$ is a bounded open set of $\mathbb{R}^n$ with boundary $Γ$ constituted by two disjoint parts $Γ_{0}$ and $Γ_{1}$ and $ν(x)$ denotes the exterior unit normal vector at $x\in Γ_{1}$. The functions $h_{i}(x,s),\,\, (i=1,2)$ are continuous and strongly monotone in $s\in \mathbb{R}$. The existence of solutions of the above problem is obtained by applying the Galerkin method with a special basis, the compactness method and a result of approximation of continuous functions by Lipschitz continuous functions due to Strauss. The exponential decay of energy follows by using appropriate Lyapunov functional and the multiplier method.}

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