Generic hyperbolicity for reaction diffusion equations on symmetric domains (Q1090839)

Let B be a ball in \(R^ n\). A solution \(u(x)\in H^ 2(B)\cap H^ 1_ 0(B)\) of the problem \(\Delta u+f(u)=0\), \(u|_{\partial B}=0\) (f\(\in {\mathcal F}\), the set of \(C^ 2\)-functions with \(C^ 2\) Whitney topology) is called hyperbolic if there is no nontrivial solution of the linearized problem \(\Delta v+f'(u(x))v=0\) in B, \(v|_{\partial B}=0\). It is proved that for f's from a residual subset of \({\mathcal F}\) all radially symmetric solutions are hyperbolic. On the other hand it is shown that for \(n=2\) there exists a nonempty open subset \({\mathcal H}\) of \({\mathcal F}\) such that for each \(f\in {\mathcal H}\) the problem has a (nonsymmetric) nonhyperbolic solution.