Manuscript submitted to Website: http://AIMsciences.org AIMS’ Journals Volume 00, Number 0, Xxxx XXXX pp. 000–000 CHARACTERIZATION OF TURING DIFFUSION-DRIVEN INSTABILITY ON EVOLVING DOMAINS Georg Hetzer Department of Mathematics and Statistics Auburn University Auburn University, AL 36849, USA Anotida Madzvamuse Department of Mathematics University of Sussex Mantell Build. Brighton, BN19RF, UK and Wenxian Shen Department of Mathematics and Statistics Auburn University Auburn University, AL 36849, USA Abstract. In this paper we establish a general theoretical framework for Tur- ing diffusion-driven instability for reaction-diffusion systems on time-dependent evolving domains. The main result is that Turing diffusion-driven instability for reaction-diffusion systems on evolving domains is characterised by Lya- punov exponents of the evolution family associated with the linearised system (obtained by linearising the original system along a spatially independent so- lution). This framework allows for the inclusion of the analysis of the long- time behavior of the solutions of reaction-diffusion systems. Applications to two special types of evolving domains are considered: (i) time-dependent do- mains which evolve to a final limiting fixed domain and (ii) time-dependent domains which are eventually time periodic. Reaction-diffusion systems have been widely proposed as plausible mechanisms for pattern formation in mor- phogenesis. 1. Introduction. Reaction-diffusion equations (RDEs) have been widely proposed as plausible models of pattern forming processes in developmental biology [27]. On fixed domains, Turing [38] derived the conditions under which a linearised reaction- diffusion system admits a linearly stable spatially homogeneous steady state in the absence of diffusion and yet, becomes unstable under appropriate conditions in the presence of diffusion to yield a spatially varying inhomogeneous steady state. This process is now well-known as diffusively-driven instability and is of particular in- terest in developmental biological pattern formation as a means of initiating self 2000 Mathematics Subject Classification. Primary: 35K55, 35K57; Secondary: 37B55, 37C60, 37C75. Key words and phrases. Reaction-diffusion systems, evolving domains, growing domains, Tur- ing diffusion-driven instability, Lyapunov exponents, nonautonomous systems, evolution semigroup theory, exponential dichotomy.. 1