The scalar reaction-diffusion equation \(u_ t=u_{xx}+f(x,u)+c(x)\alpha (u)\) with Dirichlet boundary conditions \(u(0,t)=u(1,t)=0\) and the associated dynamics are considered in this interesting article. Here f is \(C^{\infty}\) in u, \(c\in C[0,1]\) and \(\alpha (u)=\int^{1}_{0}n(x)u(x)dx\), where \(n\in L^ 2(0,1)\). It is shown that a complicated and very nice dynamics arises for this model if f,c and \(\alpha\) are suitably chosen.
The main result, from which most of the consequences follow, is Theorem 1.1, where it is stated that for generic a(x), any polynomial vector field V on \({\mathbb{R}}^{2m}\) of degree \(N\geq 2\) with \(V(0)=V'(0)=0\), there exist c and g such that for any m-tuple \(w_ 1,...,w_ m\) of reals, \(\pm iw_ j\) are eigenvalues of \(-u_{xx}-a(x)u-c(x)\alpha (u)\) and the reduced vector field for a centre manifold of \(u=0\) with \(f(x,u)=a(x)u+g(x,u)\) coincides with V for orders 2 through N. This implies in particular that trajectories may have \(\omega\)-limit sets of high dimension.
The possibility of prescribing any finite number of purely imaginary eigenvalues of the linearized problem is proven in Section 2 by using the pole assignment theorem from linear control theory. Section 3 is devoted to the centre manifold reduction and there it is shown that the reduced vector field can be prescribed up to any arbitrary finite order under suitable assumptions. That these (algebraic independence of the eigenfunctions) conditions are generically satisfied is proved in Section 4. Finally, a thorough discussion of the relationship with some reaction- diffusion systems and the corresponding shadow systems, the difference between the cases c(x) and n(x) constant or variable and related matters is given, providing an excellent overview of the problem.