SIAM J. MATH. ANAL. c  2006 Society for Industrial and Applied Mathematics Vol. 38, No. 3, pp. 741–758 APPROXIMATE TRAVELING WAVES IN LINEAR REACTION-HYPERBOLIC EQUATIONS * AVNER FRIEDMAN † AND GHEORGHE CRACIUN ‡ Abstract. Linear reaction-hyperbolic equations arise in the transport of neurofilaments and membrane-bound organelles in axons. The profile of the solution was shown by simulations to be approximately that of a traveling wave; this was also suggested by formal calculations [M. C. Reed, S. Venakides, and J. J. Blum, SIAM J. Appl. Math., 50 (1990), pp. 167–180]. In this paper we prove such a result rigorously. Key words. axonal transport, hyperbolic equations, asymptotic approximations, traveling waves AMS subject classifications. 35L45, 92C20, 92C40 DOI. 10.1137/050637947 1. Introduction. This paper is concerned with the mathematical analysis of reaction-hyperbolic equations which describe transport of materials along a straight ray l 0 = {x :0 <x< ∞}. The model is motivated from biology; it describes the transport of proteins and other molecules along the axon of a neuron. The proteins are formed near the nucleus of the cell, that is, at x = 0, and are transported to various locations along the axon, moving towards the synaptic end. Some material is also transported back, in retrograde motion. The transported materials include, for example, vesicles, membrane-bound organelles, and neurofilaments. Motor pro- teins attached to a vesicle (or a neurofilament) carry this cargo as they pace along a microtubule, step by step, energized by adenosine triphosphate (ATP) molecules. While some of the motors may be moving along a microtubule, others may be “rest- ing” on-track, or even off-track, for a while. Thus the model has to deal with several populations of vesicles, depending in what state of motion they are. Earlier models of axonal transport were developed by Reed and Blum [10, 1, 2]. Using mass reaction laws and conservation of mass, they derived a system of hyperbolic equations and studied (mostly numerically) the particle concentration profile along the axon. The numerical results show that the transport of the particle concentrations has the pro- file of “approximate traveling waves”; experimentally, they arise from radiolabeling proteins in the soma and then observing the progress of the wave of label as it goes down the axon. The wave goes at constant velocity, but the front spreads so it is only approximately a traveling wave. Reed, Venakides, and Blum [11] considered a mathe- matical problem derived from such a transport model, in the biologically relevant case when the transition between the various populations is fast relative to the transport. Recent experimental results and computational models [4, 6, 8, 12, 13] also describe the dynamics of such transport. * Received by the editors August 10, 2005; accepted for publication (in revised form) March 6, 2006; published electronically July 31, 2006. This work was supported by the National Science Foundation under agreement 0112050. http://www.siam.org/journals/sima/38-3/63794.html † Mathematical Biosciences Institute, Ohio State University, Columbus, OH 43210 (afriedman@ mbi.ohio-state.edu). ‡ Department of Mathematics and Department of Biomolecular Chemistry, University of Wiscon- sin, Madison, WI 53706 (craciun@math.wisc.edu). 741