Non-autonomous set-valued dynamical processes: Asymptotics and applications

• Applicant: Professor Dr. Peter E. Kloeden
• Subject Area: Mathematics
• Term: from 2010 to 2014
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 169166715

• Final Report Year: 2014

Final Report Abstract

The main thrust of this project concerned the long-term dynamical behaviour of non-autonomous setvalued processes on non-linear metric state spaces with different topologies, in particular when considered as the domain and the image space. This provides an abstract framework for diverse types of differential equations that arise in a variety of interesting applications, such as the evolution of sets and shapes which require metric rather than Banach spaces. Of particular interest was to verify the existence on non-autonomous attractors in such systems, which often first requires that existence and other properties of solutions are established. A broad range of examples was considered, both deterministic and stochastic. A new class of stochastic differential equations was introduced in which particular sample paths could depend on other sample paths, a simple example of which are SDEs with expectations in their coefficient functions. A further type of existence proof of solutions, both with and without uniqueness, was used to circumvent to lack of compactness criteria for spaces of mean-square random variables. Another new development was the introduction of mean-square random dynamical systems as deterministic non-autonomous dynamical systems (two-parameter processes) between spaces of mean-square random variables with different sigma-algebras resulting from the non-anticipative nature of the driving noise. The existence of mean-square random attractors, essentially deterministic pullback attractors was established in some representative examples involving different methods to overcome the lack of a compactness criterion. This work extends the mean-square approach which is common in physics and the engineering sciences. In addition, J.-P. Aubin’s idea of morphological equations was extended to stochastic morphological evolution equations to describe evolutions of non-convex random closed sets. Deterministic systems were also investigated. The existence of pullback and pathwise random attractors were verified for single-valued non-autonomous systems generated by various semi-linear and quasi-linear parabolic equations. The proofs involved asymptotic compactness criteria and norm-toweak topologies on the state spaces. Biological applications motivated two major investigations of deterministic systems, one of which resulted in a set-valued differential equation with an infinite delay. This was to model the switching-off (and maintaining this state) of reaction terms in say a tumour or an environmental pollution problem. This led to a mathematically very challenging problem, for which the existence of solutions problem was resolved. The second application concerns a class of multiscale models for tumour cell migration involving chemotaxis, haptotaxis and subcelluluar dynamics. As a spin-off from the set-valued methods developed in the project, it was possible to establish the existence (with or without uniqueness) results for fuzzy set differential equations under very general conditions such as lack of fuzzy convexity, compact level sets and normality. Hüllermeier’s formulation of fuzzy differential equations is considered as a starting point – rather than the classically used Hukuhara derivative (with its restrictive and thus inconvenient consequences).

Publications

• Stochastic morphological evolution equations. J. Differential Equations 251, No.10 (2011), pp. 2950 – 2979
  Kloeden, P.E. and Lorenz, Th.
  
• Mean-square random dynamical systems. J. Differential Equations 253, No.5 (2012), pp. 1422 – 1438
  Kloeden, P.E. and Lorenz, Th.
  
• Pullback attractors for non-autonomous quasilinear parabolic equations with dynamical boundary conditions. Discrete Contin. Dyn. Syst. Ser. B 17 (2012), no.7, pp. 2635 - 2651
  Yang, Lu, Yang, Meihua and Kloeden, P.E.
  
• A Peano-like theorem for stochastic differential equations with non-local sample dependence. Stoch. Anal. Appl. 31, No.1 (2013), pp.19 – 30
  Kloeden, P.E. and Lorenz, Th.
  
• Fuzzy differential equations without fuzzy convexity. Fuzzy Sets and Systems 230 (2013), pp. 65 – 81
  Kloeden, P.E. and Lorenz, Th.