Global attractor for a class of doubly nonlinear abstract evolution equations
Global attractor for a class of doubly nonlinear abstract evolution equations
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一类双非线性抽象演化方程的全局吸引子
DOI：
10.3934/dcds.2006.14.801
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发表时间：
2006
影响因子：
1.1
通讯作者：
A. Segatti
中科院分区：
文献类型：
--
作者：
A. Segatti
本文考虑Hilbert空间$\H$中抽象非线性演化方程的Cauchy问题
$\A(u'(t))+ \B(u(t))-\lambda u(t)$$f \mbox{in } \H \mbox{ for a.e. }t\in (0,+\infty)$$u(0)=u_{0},$
哪里$\A$是一个最大的（可能多值）单调
从Hilbert空间$\H$到自身的算子，而$\B$是一个在$\H$满足合适相容条件的紧子层的固有凸下半连续函数φ: $\H\rightarrow (-\infty,+\infty]$的次微分。最后，$\lambda$是一个正常数。利用逼近-先验估计-极限过程证明了解的存在性。本文的主要结果是所有解的集合在由势φ定义域给出的相空间中产生了John M. Ball[8]意义上的广义半流。该过程具有点耗散性和渐近紧性；此外，构造了全局吸引子，该吸引子可以吸引系统的所有轨迹，这些轨迹相对于一个与未知约束严格相关的度量。给出了一些涉及偏微分方程的问题的应用。
In this paper we consider the Cauchy problem for the abstract nonlinear evolution equation in a Hilbert space $\H$
$\A(u'(t))+ \B(u(t))-\lambda u(t)$ ∋ $f \mbox{in } \H \mbox{ for a.e. }t\in (0,+\infty)$ $u(0)=u_{0},$
where $\A$ is a maximal (possibly multivalued) monotone
operator from the Hilbert space $\H$ to itself, while $\B$ is the subdifferential of a proper, convex and lower semicontinuous function φ:$\H\rightarrow (-\infty,+\infty]$ with compact sublevels in $\H$ satisfying a suitable compatibility condition. Finally, $\lambda$ is a positive constant. The existence of solutions is proved by using an approximation-a priori estimates-passage to the limit procedure. The main result of this paper is that the set of all the solutions generates a Generalized Semiflow in the sense of John M. Ball [8] in the phase space given by the domain of the potential φ. This process is shown to be point dissipative and asymptotically compact; moreover the global attractor, which attracts all the trajectories of the system with respect to a metric strictly linked to the constraint imposed on the unknown, is constructed. Applications to some problems involving PDEs are given.