Moving boundary problem for a one-dimensional crawling nematode sperm cell model

This paper is a continuation of the authors' investigation of the crawling nematode sperm cell model proposed by Mogilner and Verzi in 2003 (J. Stat. Phys. 110 (2003) 1169). In the earlier papers (J. Math. Biol. 49 (2004) 310 and J. Math. 8 (2004) 399), the first and last authors and Juliet Lee proved the existence of traveling wave solutions for the nematode sperm cell model. In this paper, we prove local existence of solutions to the same model and global existence under certain additional assumption on the initial data. Finally, we also provide numerical evidence that the traveling wave solutions found in (J. Math. Biol., accepted for publication) is globally asymptotically stable.