Inertial manifold theory for a class of reaction-diffusion equations on thin tubular domains
The author is concerned with the dynamical system defined by the following reaction diffusion system \[ \begin{cases} u_ t = \Delta u + f(u,v,x,y) & \text{in \((0,\infty) \times \Omega_ \varepsilon\)}\\ v_ t = \Delta v + g(u,v,x,y) & \text{in \((0,\infty) \times \Omega_ \varepsilon\)}\\ \text{Neumann boundary condition},\end{cases}\tag{1} \] where \(\Omega_ \varepsilon\) is a thin tubular domain defined by \(\Omega_ \varepsilon = \{(x,y) \in \mathbb{R}^ 1 \times \mathbb{R}^ N \mid 0 \leq x \leq L,\quad y/\varepsilon \in D_ x\}\), where \(D_ x = r(x)D \subset \mathbb{R}^ N\) and \(D\) is an open set with \(0\in D\). The author considers the limiting situation of \(\varepsilon \to 0\) in (1) and proves (1) approaches the system of equations \[ \left\{\begin{aligned} \overline{u}_ t = \Delta\overline{u} + f(\overline{u}, \overline{v}, x,0) & \text{ in }(0,\infty) \times (0,L)\\ \overline{v}_ t = \Delta \overline{v} + g(\overline{u}, \overline{v}, x,0)& \text{ in }(0,\infty) \times (0,L)\end{aligned}\right. \] from the dynamical system point of view. This work can be regarded as an extension of the work of Hale and Raugel who dealt with the scalar equation in a similar situation. The inertial manifold method is used.
scientific article; zbMATH DE number 558857