On the explicit construction of an ODE which has the same dynamics as a scalar parabolic PDE
Assuming f satisfies conditions ensuring the infinite dimensional dynamical system defined by the scalar parabolic equation *) \(u_ t=(au_ x)_ x+f(x,u)\), \(x\in (0,1)\), admits a global attractor \({\mathcal A}\), we consider the problem of describing the dynamics of *) restricted to \({\mathcal A}\) by a system of ordinary differential equations. We show that, provided the conductivity function a(x) is near to a limit singular function, such a system of differential equations can be explicitly constructed. Our proof is based on the reduction to a finite dimensional invariant manifold which contains \({\mathcal A}\).
scientific article; zbMATH DE number 4032210
scientific article