On the Kneser property for reaction–diffusion systems on unbounded domains

We prove the Kneser property (i.e. the connectedness and compactness of the attainability set at any time) for reaction–diffusion systems on unbounded domains in which we do not know whether the property of uniqueness of the Cauchy problem holds or not.Using this property we obtain that the global attractor of such systems is connected.Finally, these results are applied to the complex Ginzburg–Landau equation.