Pullback attractors of reaction-diffusion inclusions with space-dependent delay
Peter E. Kloeden Thomas Lorenz

Inspired by biological phenomena with effects of switching off (maybe just for a while), we investigate non-autonomous reaction-diffusion inclusions whose multi-valued reaction term may depend on the essential supremum over a time interval in the recent past (but) pointwise in space. The focus is on sufficient conditions for the existence of pullback attractors. If the multi-valued reaction term satisfies a form of inclusion principle standard tools for non-autonomous dynamical systems in metric spaces can be applied and provide new results (even) for infinite time intervals of delay. More challenging is the case without assuming such a monotonicity assumption. Then we consider the parabolic differential inclusion with the time interval of delay depending on space and extend the approaches of norm-to-weak semigroups to a purely metric setting. This provides completely new tools for proving pullback attractors of non-autonomous dynamical systems in metric spaces.

keywords: Reaction-diffusion equation parabolic differential inclusion with space-dependent delay dead core retarded functional evolution inclusion existence of solutions non-autonomous pullback attractors set-valued dynamical system with norm-to-weak closed graph
Mutational inclusions: Differential inclusions in metric spaces
Thomas Lorenz
The focus of interest is how to extend ordinary differential inclusions beyond the traditional border of vector spaces. We aim at an existence theorem for solutions whose values are in a given metric space.
   In the nineties, Aubin suggested how to formulate ordinary differential equations and their main existence theorems in metric spaces: mutational equations (which are quite similar to the quasidifferential equations of Panasyuk). Now the well-known Antosiewicz-Cellina Theorem is extended to so-called mutational inclusions. It provides new results about nonlocal set evolutions in R N .
keywords: morphological equation. Mutational equation (semilinear) evolution equation differential inclusion
On the modeling of moving populations through set evolution equations
Rinaldo M. Colombo Thomas Lorenz Nikolay I. Pogodaev
We introduce a class of set evolution equations that can be used to describe population's movements as well as various instances of individual-population interactions. Optimal control/management problems can be formalized and tackled in this framework. A rigorous analytical structure is established and the basic well posedness results are obtained. Several examples show possible applications and their numerical integrations display possible qualitative behaviors of solutions.
keywords: evolution of measures Set evolution equations confinement problems differential inclusions agents-population interactions.
Reaction-diffusion equations with a switched--off reaction zone
Peter E. Kloeden Thomas Lorenz Meihua Yang
Reaction-diffusion equations are considered on a bounded domain $\Omega$ in $\mathbb{R}^d$ with a reaction term that is switched off at a point in space when the solution first exceeds a specified threshold and thereafter remains switched off at that point, which leads to a discontinuous reaction term with delay. This problem is formulated as a parabolic partial differential inclusion with delay. The reaction-free region forms what could be called dead core in a biological sense rather than that used elsewhere in the literature for parabolic PDEs. The existence of solutions in $L^2(\Omega)$ is established firstly for initial data in $L^{\infty}(\Omega)$ and in $W_0^{1,2}(\Omega)$ by different methods, with $d$ $=$ $2$ or $3$ in the first case and $d$ $\geq$ $2$ in the second. Solutions here are interpreted in the sense of integral or strong solutions of nonhomogeneous linear parabolic equations in $L^2(\Omega)$ that are generalised to selectors of the corresponding nonhomogeneous linear parabolic differential inclusions and are shown to be equivalent under the assumptions used in the paper.
keywords: Reaction-diffusion equation dead core existence of solutions. memory inclusion equations discontinuous right-hand sides

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