Stabilization and Control for the Subcritical Semilinear Wave Equation in a Bounded Domain With a Cauchy-Ventcel Boundary Conditions
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摘要: 分析RN的有界域中半线性波方程解的指数衰减特性，有界域具有Cauchy-Ventcel型边界条件，并且球体外部作用着阻尼项．在对非线性作出适当又自然的假设后，倘若非线性在无穷大处为亚临界时，有限能量解的指数衰减性满足局部一致性．粗略地说，亚临界性意味着，在无穷大处非线性增长率次数不大于5．B.Dehman、G.Lebeau和E.Zuazua得到了R3和RN中的经典能量（用于估计局限于球体外部以能量形式表示的解的总能量）不等式和Strichartz估计的结果，使得研究RN有界域（域内及其边界上是亚临界非线性，边界为Cauchy-Ventcel型连续）中半线性波方程的稳定性与可控性成为可能．Abstract: The exponential decay property of solutions of the semilinear wave equation in bounded domain of RN(N is equals or greater than 1) with a damping term which is effective on the exterior of a ball and with boundary conditions of Cauchy-Ventcel type was analyzed. Under suitable and natural assumptions on the nonlinearity, it was proved that the exponential decay holds locally uniformly for finite energy solutions that provided the nonlinearity is subcritical at infinity. Subcriticality means, roughly speaking, that the nonlinearity grows at infinity atmost as a power is less than 5. The results obtained in R3 and RN (N equals to or greater than 1) by B. Dehman, G. Le beau and E. Zuazua on the inequalities of the classical energy (which estimate the total energy of solutions in terms of the energy localized in the exterior of a ball) and on Strichartz's estimates, allow us to give an application to the stabilization contro llability of the semilinear wave equation in abounded domain of RN (N equals to orgreater than 1) with a subcritical nonlinearity on the domain and its boundary and with conditions on the boundary of Cauchy-Ventcel type.
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Key words:
- stabilization /
- exact controllability /
- limit pro blems /
- semilinear /
- subcritical /
- partial differential equations /
- Cauchy-Ventcel
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