Reaction-diffusion systems governed by non-autonomous forms

We consider a semilinear problemu′(t)+A(t)u(t)=f(u(t)),u(0)=u0, where A(t) is associated with a non-autonomous form a(t,⋅,⋅). Using an invariance principle for closed, convex sets in the underlying Hilbert space we find conditions for global solutions. This can be applied to reaction diffusion systems on L2(Ω)N. Our point is that the forms a(t,⋅,⋅) need only to be submarkovian to carry over invariance of the (scalar) reaction equation to the reaction-diffusion system.