Reaction-diffusion equation on thin domains
Reaction-diffusion equations of the form \[ u_t- \Delta u- f(u)- h(x, y)= 0,\quad (x, y)= (x_1,\dots, x_n, y)\in Q_\varepsilon,\tag{1} \] under Neumann (or other) boundary conditions are considered. The domain \(Q_\varepsilon\) is assumed to be thin in one direction: \(Q_\varepsilon= \{(x,y)\in \mathbb{R}^n\times \mathbb{R}: x\in \Omega,\;0< y< g(x, \varepsilon)\}\). Here, \(\Omega\) is a polygonal domain in \(\mathbb{R}^n\) and \(g\) is a \(C^3\) function satisfying \[ g(x, 0)= 0\quad\text{and} \quad g_0(x):= {\partial g\over \partial\varepsilon} (x, 0)> 0. \] The main goal of the paper is to compare the dynamics of (1) with the ``limit'' equation \[ u_t- {1\over g_0} \sum (g_0 u_{x_i})_{x_i}- f(u)- h(x)=0, \quad x\in \Omega,\tag{2} \] (with boundary conditions corresponding to the original ones). Under appropriate assumptions (including dissipativity), the authors prove that the family of attractors \(A_\varepsilon\), \(\varepsilon\geq 0\), of (1) (for \(\varepsilon> 0\)) and (2) (for \(\varepsilon= 0\)) is upper semicontinuous at \(\varepsilon= 0\). Moreover, if \(n= 1\) and the equilibria of (2) are hyperbolic then the attractors are continuous at \(\varepsilon= 0\) and the flows on them are all equivalent if \(0\leq \varepsilon\leq \varepsilon_0\) with a sufficient small \(\varepsilon_0\). The latter result involves analysis of the inertial manifolds of (1), (2).
scientific article; zbMATH DE number 810286