\({\mathfrak L}^ 2\)-perturbations of periodic equilibria of reaction diffusion systems
The paper considers reaction diffusion systems \((u_ j)_ t= \tau_ j \Delta u_ j P_ j\) \((u_ 1, \dots, u_ n)\), where \(P_ j\) is a polynomial and \(u_ j= u_ j (x_ 1, \dots, x_ m)\) are functions on \(\mathbb{R}^ m\), \(m\leq 3\). Under the assumption that a smooth periodic equilibrium solution \(v= (v_ 1, \dots, v_ n)\), which is \(L_ j\)- periodic with respect to \(x_ j\) \((j=1, \dots,m)\) is given, the stability of this equilibrium under smooth perturbations \(\varphi\in L^ 2 (\mathbb{R}^ m)\) is investigated. To this end the spectrum of \((*)\) \(D\Delta d_ u F(v)\) as an unbounded operator on \(L^ 2 (\mathbb{R}^ m)^ n\) is studied, where \(D= (\Delta_{ij} \tau_ j)\) \((i,j= 1,\dots, n)\) and \(F(u)= (P_ 1 u,\dots, P_ n u)\) \((u= (u_ 1,\dots,u_ n))\) and \(v\) a smooth \(L_ 1, \dots, L_ m\)-periodic equilibrium solution of the elliptic system \(D\Delta u F(u)=0\). This spectrum is described as the union of the spectra of a family of boundary value problems for \((*)\) subject to a \(\theta\)-periodic boundary condition, \(\theta\in [0,2 \pi]^ m\). The author assumes the principle of linearized stability and instability, the proof that it applies to the evolution equations considered here is announced to appear in a subsequent paper. As an application it is shown that the passage from \(L\)-periodic perturbations to smooth \(L^ 2\) perturbations can only decrease or eventually leave unchanged stability status. As an example the scalar case is considered, showing that with the exception where the rightmost points of the corresponding spectra are 0, periodic and \(L^ 2\)-stability (resp., instability) coincide. Then a system is constructed which is periodically stable but Lyapunov unstable with respect to smooth \(L^ 2\)- perturbations.
scientific article; zbMATH DE number 736837