STABILITY IMPLIES CONSTANCY FOR FULLY AUTONOMOUS REACTION-DIFFUSION-EQUATIONS ON FINITE METRIC GRAPHS

We show that there are no stable stationary nonconstant solutions of the evolution problem (1) for fully autonomous reaction-diffusion-equations on the edges of a finite metric graph G under continuity and Kirchhoff flow transition conditions at the vertices. (1) {u is an element of C(G X [0,infinity)) boolean AND C-K(2,1)(G X (0,infinity)), partial derivative(t)u(j )= partial derivative(2)(j)u(j) + f(u(j)) on the edges k(j), (K) Sigma(N)(j=1)d(ij)c(ij)partial derivative(j)u(j) (v(i), t) = 0 at the vertices v(i).( )