On inertial manifolds for reaction-diffusion equations on genuinely high-dimensional thin domains
On inertial manifolds for reaction-diffusion equations on genuinely high-dimensional thin domains
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DOI：
10.4064/sm154-3-6
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发表时间：
2002-07
影响因子：
0.8
通讯作者：
M. Prizzi;K. P. Rybakowski
中科院分区：
文献类型：
--
作者：
M. Prizzi;K. P. Rybakowski
本文研究了薄空间区域上的一类半线性反应扩散方程，该方程靠近一个低维子流形。当厚度趋向于零时，畴塌缩到（的子集）$M$上。正如在以前的论文中所证明的那样（M。普里齐，M. Rinaldi和K. P. Rybakowski，Curved thin domains and parabolic equations，Stud.Math.151），上述族有一个极限方程，它是定义在某个抽象极限相空间上的抽象半线性抛物方程。本文的目标之一是给出更易于处理的极限相空间的特征。在附加的假设条件下，我们也给出了一个简单的描述的极限方程。此外，如果M是一个球面，并且上述方程的非线性是耗散的，我们证明了，如果厚度足够小，相应的方程具有一个惯性流形，即一个包含方程吸引子的不变流形。因此，我们得到的存在性惯性流形的反应扩散方程的某些类薄区域的真正的高维。
In this paper we study a family of semilinear reaction-diffusion equations on thin spatial domains, lying close to a lower dimensional submanifold $M$. As the thickness tends to zero, the domains collapse onto (a subset of) $M$. As it was proved in a previous paper (M. Prizzi, M. Rinaldi and K. P. Rybakowski, Curved thin domains and parabolic equations, Stud. Math. 151), the above family has a limit equation, which is an abstract semilinear parabolic equation defined on a certain abstract limit phase space. One of the objectives of this paper is to give more manageable characterizations of the limit phase space. Under additional hypotheses, we also give a simple description of the limit equation. If, in addition, $M$ is a sphere and the nonlinearity of the above equations is dissipative, we prove that, if the thickness is small enough, the corresponding equation possesses an inertial manifold, i.e. an invariant manifold containing the attractor of the equation. We thus obtain the existence of inertial manifolds for reaction-diffusion equations on certain classes of thin domains of genuinely high dimension.