Monotonicity and convergence results in order-preserving systems in the presence of symmetry (Q1576806)

The authors deal with the following question: Suppose that a group \(G\) acts on a space \(X\) and that a mapping \(F:X\to X\) is \(G\)-equivariant, that is \(F\circ g=g\circ F\) for every \(g\in G\). Can we say that solutions of the equation \(F(u)= 0\) are \(G\)-invariant? Here the authors establish a general theorem concerning symmetry or monotonicity properties of stable equilibria. The authors apply their theory to travelling waves and pseudo-travelling waves for a certain class of quasilinear diffusion equations. The authors also establish another useful general theorem, which they call the ``convergence theorem'', which roughly states that stability implies asymptotic stability and present various important applications of this result.