论文标题
非自治非线性进化方程的不变设定分叉理论
An Invariant Set Bifurcation Theory for Nonautonomous Nonlinear Evolution Equations
论文作者
论文摘要
在本文中，我们建立了一个非自主动力系统$（\ va_ \ lam，\ 0）_ {x，\ ch} $的不变集合理论，由进化方程\ be \ be \ be be \ label {e0} \ ch = \ ch [f（\。，u）] \ ee在希尔伯特太空$ x $上，其中$ a $是部门运算符，$ \ lam $是双面参数，$ f（\。，u）：\ r \ r \ ra x $ is compact，$ f（t，0）compact，$ f（t，0） $ f（\。，u）$。用$ \ va_ \ lam表示：= \ va_ \ lam（t，p）u $ cocycle semiflow由等式生成。根据$ f $的其他一些假设，我们表明，作为参数$ \ lam $跨越$ a $ a $的特征$ \ lam_0 \ $ 0 $ $ $ $ $ $ b_ \ lam（\。）$ $ 0 $ $ \ lam_0 $。此外，$$ \ lim _ {\ lam \ ra \ laM_0}下面定义的$ x^α$（$ \ a \ geq0 $）是与$ a $相关的分数功率空间。 我们的结果是基于本地中央不变歧管上的回拔吸引子分叉的，$ \ cm^\ lam_ {loc}（\。）$。
In this paper we establish an invariant set bifurcation theory for the nonautonomous dynamical system $(\va_\lam,\0)_{X,\cH}$ generated by the evolution equation \be\label{e0}u_t+Au=\lam u+p(t,u),\hs p\in \cH=\cH[f(\.,u)]\ee on a Hilbert space $X$, where $A$ is a sectorial operator, $\lam$ is the bifurcation parameter, $f(\.,u):\R\ra X$ is translation compact, $f(t,0)\equiv0$ and $\cH[f]$ is the hull of $f(\.,u)$. Denote by $\va_\lam:=\va_\lam(t,p)u$ the cocycle semiflow generated by the equation. Under some other assumptions on $f$, we show that as the parameter $\lam$ crosses an eigenvalue $\lam_0\in\R$ of $A$, the system bifurcates from $0$ to a nonautonomous invariant set $B_\lam(\.)$ on one-sided neighborhood of $\lam_0$. Moreover, $$\lim_{\lam\ra\lam_0}H_{X^\a}\(B_\lam(p),0\)=0,\hs p\in P,$$ where $H_{X^\a}(\.,\.)$ denotes the Hausdorff semidistance in $X^\a$ (here $X^α$ ($\a\geq0$) defined below is the fractional power spaces associated with $A$). Our result is based on the pullback attractor bifurcation on the local central invariant manifolds $\cM^\lam_{loc}(\.)$.
