In this paper we study the structure of the global attractor for a reactiondiffusion equation in which uniqueness of the Cauchy problem is not guarantied. We prove that the global attractor can be characterized using either the unstable manifold of the set of stationary points or the stable one but considering in this last case only solutions in the set of bounded complete trajectories.

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In this paper we consider reaction-diffusion systems in which the conditions imposed on the nonlinearity provide global existence of solutions of the Cauchy problem, but not uniqueness. We prove first that for the set of all weak solutions the Kneser property holds, that is, that the set of values attained by the solutions at every moment of time is compact and connected. Further, we prove the existence and connectedness of a global attractor in both the autonomous and nonautonomous cases. The obtained results are applied to several models of physical (or chemical) interest: a model of fractional-order chemical autocatalysis with decay, the Fitz-Hugh-Nagumo equation and the Ginzburg-Landau equation.

In this paper we define and study multivalued dynamical processes in Hausdorff topological spaces. Existence theorems for attractors of multivalued processes are proved, their topological properties are studied. The abstract results are applied to a system of phase-field equations without conditions providing uniqueness of solutions and to nonautonomous differential inclusions.

We study the Kneser property (i.e. the compactness and connectedness for the attainability set of solutions) for a reaction-diffusion system including as a particular case the complex Ginzburg-Landau equation and the Lotka-Volterra system with diffusion. Using this property we obtain also that the global attractor of this system in both the autonomous and non-autonomous cases is connected.