Remarks on contractions of reaction-diffusion PDE's on weighted L^2 norms

where ∆ denotes a diffusion operator, is central to many biological applications, in fields ranging from pattern formation in development to ecology. One of the central topics of research in this context is the question of how the stability of solutions of the PDE relates to stability of solutions of the underlying ordinary differential equation (ODE) dx dt (t) = F (x(t), t). This paper shows that when solutions of this ODE have a certain contraction property, namely μ2,P (JF (u, t)) < 0 uniformly on u and t, where μ2,P is a logarithmic norm (matrix measure) associated to a P -weighted L2 norm, the associated PDE, subject to no-flux (Neumann) boundary conditions, enjoys a similar property, if P 2D +DP 2 > 0. This result complements a similar result shown in [1] which, while allowing norms L with p not necessarily equal to 2, had the restriction that it only applied to diagonal matrices P . Here, P is allowed to be an arbitrary positive definite symmetric matrix. The paper also discusses an example of biological interest, as well as examples that illustrate when the results in [1] apply but the current result does not.