TY - JOUR
T1 - On an evolution equation in a cell motility model
AU - Mizuhara, Matthew S.
AU - Berlyand, Leonid
AU - Rybalko, Volodymyr
AU - Zhang, Lei
N1 - Funding Information:
The first author was supported by the Department of Defense (DoD) through the National Defense Science & Engineering Graduate Fellowship (NDSEG) Program. He also received partial support from NSF grants DMS-1106666 and DMS-1405769 . The second and third authors were supported by NSF grants DMS-1106666 and DMS-1405769 . The work of fourth author was supported by NSFC grant 11471214 and 1000 Plan for Young Scientists of China .
Funding Information:
The authors would like to extend their gratitude to the referees for their careful reading of the manuscript and insightful comments. The first author was supported by the Department of Defense (DoD) through the National Defense Science and Engineering Graduate Fellowship (NDSEG) Program. He also received partial support from NSF grants DMS-1106666 and DMS-1405769. The second and third authors were supported by NSF grants DMS-1106666 and DMS-1405769. The work of fourth author was supported by NSFC grant 11471214 and 1000 Plan for Young Scientists of China.
Publisher Copyright:
© 2015 Elsevier B.V. All rights reserved.
PY - 2016/4/1
Y1 - 2016/4/1
N2 - This paper deals with the evolution equation of a curve obtained as the sharp interface limit of a non-linear system of two reaction-diffusion PDEs. This system was introduced as a phase-field model of (crawling) motion of eukaryotic cells on a substrate. The key issue is the evolution of the cell membrane (interface curve) which involves shape change and net motion. This issue can be addressed both qualitatively and quantitatively by studying the evolution equation of the sharp interface limit for this system. However, this equation is non-linear and non-local and existence of solutions presents a significant analytical challenge. We establish existence of solutions for a wide class of initial data in the so-called subcritical regime. Existence is proved in a two step procedure. First, for smooth (H2) initial data we use a regularization technique. Second, we consider non-smooth initial data that are more relevant from the application point of view. Here, uniform estimates on the time when solutions exist rely on a maximum principle type argument. We also explore the long time behavior of the model using both analytical and numerical tools. We prove the nonexistence of traveling wave solutions with nonzero velocity. Numerical experiments show that presence of non-linearity and asymmetry of the initial curve results in a net motion which distinguishes it from classical volume preserving curvature motion. This is done by developing an algorithm for efficient numerical resolution of the non-local term in the evolution equation.
AB - This paper deals with the evolution equation of a curve obtained as the sharp interface limit of a non-linear system of two reaction-diffusion PDEs. This system was introduced as a phase-field model of (crawling) motion of eukaryotic cells on a substrate. The key issue is the evolution of the cell membrane (interface curve) which involves shape change and net motion. This issue can be addressed both qualitatively and quantitatively by studying the evolution equation of the sharp interface limit for this system. However, this equation is non-linear and non-local and existence of solutions presents a significant analytical challenge. We establish existence of solutions for a wide class of initial data in the so-called subcritical regime. Existence is proved in a two step procedure. First, for smooth (H2) initial data we use a regularization technique. Second, we consider non-smooth initial data that are more relevant from the application point of view. Here, uniform estimates on the time when solutions exist rely on a maximum principle type argument. We also explore the long time behavior of the model using both analytical and numerical tools. We prove the nonexistence of traveling wave solutions with nonzero velocity. Numerical experiments show that presence of non-linearity and asymmetry of the initial curve results in a net motion which distinguishes it from classical volume preserving curvature motion. This is done by developing an algorithm for efficient numerical resolution of the non-local term in the evolution equation.
UR - http://www.scopus.com/inward/record.url?scp=84958752209&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=84958752209&partnerID=8YFLogxK
U2 - 10.1016/j.physd.2015.10.008
DO - 10.1016/j.physd.2015.10.008
M3 - Article
AN - SCOPUS:84958752209
SN - 0167-2789
VL - 318-319
SP - 12
EP - 25
JO - Physica D: Nonlinear Phenomena
JF - Physica D: Nonlinear Phenomena
ER -