Global existence of solutions to a coupled parabolic-hyperbolic system with moving boundary

Authors:
Y. S. Choi and Craig Miller

Journal:
Proc. Amer. Math. Soc. **139** (2011), 3257-3270

MSC (2010):
Primary 35K10, 35K59, 35L04

DOI:
https://doi.org/10.1090/S0002-9939-2011-10801-3

Published electronically:
January 24, 2011

MathSciNet review:
2811281

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A cell motility study leads to a moving boundary problem governed by a system of parabolic-hyperbolic equations. Establishing the parabolicity of one of the governing equations requires a priori bound analysis. Such bounds also exclude the formation of shock in the hyperbolic equation. Speeds of the moving boundaries can then be controlled, which eventually leads to the global existence of solutions.

**1.**Fabiana Cardetti and Y.S. Choi, A parabolic-hyperbolic system modeling a moving cell, Electronic J. of Differential Equations 2009 (2009), No. 95, 1-11. MR**2530134 (2010h:35046)****2.**Y.S. Choi, P. Groulx, and R. Lui, Moving boundary problem for a one-dimensional crawling nematode sperm cell model, Nonlinear Analysis: Real World Applications 6 (2005), 874-898. MR**2165218 (2006f:35283)****3.**Y.S. Choi, J. Lee, and R. Lui, Traveling wave solutions for a one-dimensional crawling nematode sperm cell model, J. Math. Biol. 49 (2004), 310-328. MR**2102761 (2005g:92011)****4.**Y.S. Choi and R. Lui, Linearized stability of traveling cell solutions arising from a moving boundary problem. Proc. Amer. Math. Soc. 135 (2007), 743-753. MR**2262870 (2007h:37119)****5.**Y.S. Choi and R. Lui, Existence of traveling domain solutions for a two-dimensional moving boundary problem, Trans. Amer. Math. Soc. 361 (2009), 4027-4044. MR**2500877 (2010j:35623)****6.**O.A. Ladyžhenskaya, V.A. Solonnik and N.N. Ural'ceva,*Linear and Quasilinear Equations of Parabolic Type*, Translations of Mathematical Monographs, 23, Amer. Math. Soc., 1967. MR**0241822 (39:3159b)****7.**A. Mogilner, Mathematics of cell motility: have we got its number?, J. Math. Biol. 58 (2009), 105-134. MR**2448425 (2009k:92018)****8.**A. Mogilner and D.W.Verzi, A simple 1-D physical model for the crawling nematode sperm cell, J. Stat. Phys. 110 (2003), 1169-1189.**9.**T.D. Pollard and G.G. Borisy, Cellular motility driven by assembly and disassembly of actin filaments, Cell, 112 (2003), 453-465.**10.**M. Protter and H.F. Weinberger, Maximum Principles in Differential Equations, Prentice-Hall, Englewood Cliffs, NJ, 1967. MR**0219861 (36:2935)****11.**S.M. Rafelski and J.A. Theriot, Crawling toward a unified model of cell mobility: Spatial and temporal regulation of actin dynamics, Annual Rev. Biochem. 73 (2004), 209-39.

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Additional Information

**Y. S. Choi**

Affiliation:
Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269-3009

Email:
choi@math.uconn.edu

**Craig Miller**

Affiliation:
Department of Mathematics, University of New Haven, 300 Boston Post Road, West Haven, Connecticut 06516

Email:
cmiller@newhaven.edu

DOI:
https://doi.org/10.1090/S0002-9939-2011-10801-3

Keywords:
Global existence,
coupled parabolic-hyperbolic equations,
method of characteristics

Received by editor(s):
March 11, 2010

Received by editor(s) in revised form:
March 15, 2010, and August 9, 2010

Published electronically:
January 24, 2011

Communicated by:
Matthew J. Gursky

Article copyright:
© Copyright 2011
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.