林国广, 朱昌清. 一类非线性高阶Kirchhoff型方程解的渐进性态[J]. 云南大学学报(自然科学版), 2019, 41(5): 867-875. doi: 10.7540/j.ynu.20180777
引用本文: 林国广, 朱昌清. 一类非线性高阶Kirchhoff型方程解的渐进性态[J]. 云南大学学报(自然科学版), 2019, 41(5): 867-875. doi: 10.7540/j.ynu.20180777
LIN Guo-guang, ZHU Chang-qing. Asymptotic behavior of solutions for a class of nonlinear higher-order Kirchhoff-type equations[J]. Journal of Yunnan University: Natural Sciences Edition, 2019, 41(5): 867-875. DOI: 10.7540/j.ynu.20180777
Citation: LIN Guo-guang, ZHU Chang-qing. Asymptotic behavior of solutions for a class of nonlinear higher-order Kirchhoff-type equations[J]. Journal of Yunnan University: Natural Sciences Edition, 2019, 41(5): 867-875. DOI: 10.7540/j.ynu.20180777

一类非线性高阶Kirchhoff型方程解的渐进性态

Asymptotic behavior of solutions for a class of nonlinear higher-order Kirchhoff-type equations

  • 摘要: 研究了一类非线性非局部高阶Kirchhoff型偏微分方程的初边值问题. 首先,利用先验估计和Galerkin方法证明了方程在空间 H_0^m + k(\Omega ) \times H_0^k(\Omega ) 中存在唯一的整体解;然后,采用紧致法证明了该问题生成的解半群 S(t) 存在一个紧的整体吸引子族 A_k;最后,通过线性化方法,证明了算子半群 S(t) 的Frechet可微性以及关于线性化问题体积元的衰减性,从而得到整体吸引子族的Hausdorff维数和Fractal维数估计.

     

    Abstract: The initial-boundary value problems for a class of nonlinear nonlocal higher-order Kirchhoff partial differential equations is studied. Firstly, the existence and uniqueness of the global solution of the equation in space H_0^m + k(\Omega ) \times H_0^k(\Omega ) are proved by prior-estimation and Galerkin method. And then, the compact method is used to prove that the solution semigroup S(t) generated by the problem has a compact global attractor family A_k. Finally, the semigroup of operators is proved by linearization method. The Hausdorff dimension and Fractal dimension estimation of the global attractor family are obtained by using the Frechet differentiability and the attenuation of the volume element for the linearization problem.

     

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