July  2015, 35(7): 3203-3216. doi: 10.3934/dcds.2015.35.3203

Existence and uniqueness of similarity solutions of a generalized heat equation arising in a model of cell migration

1. 

Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, United States

2. 

School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287-1804

Received  June 2014 Revised  December 2014 Published  January 2015

We study similarity solutions of a nonlinear partial differential equation that is a generalization of the heat equation. Substitution of the similarity ansatz reduces the partial differential equation to a nonlinear second-order ordinary differential equation on the half-line with Neumann boundary conditions at both boundaries. The existence and uniqueness of solutions is proven using Ważewski's Principle.
Citation: Tracy L. Stepien, Hal L. Smith. Existence and uniqueness of similarity solutions of a generalized heat equation arising in a model of cell migration. Discrete & Continuous Dynamical Systems, 2015, 35 (7) : 3203-3216. doi: 10.3934/dcds.2015.35.3203
References:
[1]

R. P. Agarwal and D. O'Regan, Infinite Interval Problems For Differential, Difference and Integral Equations, Kluwer Academic Publishers, Boston, MA, 2001. doi: 10.1007/978-94-010-0718-4.  Google Scholar

[2]

P. Amster and A. Deboli, A Neumann boundary-value problem on an unbounded interval, Electronic J. of Differential Equations, 2008 (2008), 1-5.  Google Scholar

[3]

J. C. Arciero and D. Swigon, Equation-based models of wound healing and collective cell migration, in Complex Systems and Computational Biology Approaches to Acute Inflammation (eds. Y. Vodovotz and G. An), Springer, 2013, 185-207. doi: 10.1007/978-1-4614-8008-2_11.  Google Scholar

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T. Callaghan, E. Khain, L. M. Sander and R. M. Ziff, A stochastic model for wound healing, J. Stat. Phys., 122 (2006), 909-924. doi: 10.1007/s10955-006-9022-1.  Google Scholar

[5]

J. K. Hale, Ordinary Differential Equations, Dover Publications, Mineola, NY, 2009. Google Scholar

[6]

M. W. Hirsch and H. L. Smith, Competitive and cooperative systems: A mini-review, in Positive Systems. Proceedings of the First Multidisciplinary Symposium on Positive Systems (POSTA 2003) (eds. Luca Benvenuti, Alberto De Santis and Lorenzo Farina), Lecture Notes on Control and Information Sciences, Vol. 294, Springer-Verlag, Heidelberg, 2003, 183-190. doi: 10.1007/b79667.  Google Scholar

[7]

H. Lian and W. Ge, Solvability for second-order three-point boundary value problems on a half-line, Appl. Math. Lett., 19 (2006), 1000-1006. doi: 10.1016/j.aml.2005.10.018.  Google Scholar

[8]

H. Lian and F. Geng, Multiple unbounded solutions for a boundary value problem on infinite intervals, Bound. Value Probl., 51 (2011), 1-8. doi: 10.1186/1687-2770-2011-51.  Google Scholar

[9]

B. Liu, J. Li and L. Liu, Existence and uniqueness for an m-point boundary problem at resonance on infinite intervals, Comput. Math. Appl., 64 (2012), 1677-1690. doi: 10.1016/j.camwa.2012.01.023.  Google Scholar

[10]

Q. Mi, D. Swigon, B. Rivière, S. Cetin, Y. Vodovotz and D. J. Hackam, One-dimensional elastic continuum model of enterocyte layer migration, Biophys. J., 93 (2007), 3745-3752. doi: 10.1529/biophysj.107.112326.  Google Scholar

[11]

H. L. Smith, Monotone Dynamical Systems: An introduction to the theory of competitive and cooperative systems, American Mathematical Society, Mathematical Surveys and Monographs, 1995.  Google Scholar

[12]

T. L. Stepien, Collective Cell Migration in Single and Dual Cell Layers, Ph.D. thesis. University of Pittsburgh, 2013.  Google Scholar

[13]

T. L. Stepien and D. Swigon, Traveling waves in a one-dimensional elastic continuum model of cell layer migration with stretch-dependent proliferation, SIAM J. Appl. Dyn. Syst., 13 (2014), 1489-1516. doi: 10.1137/130941407.  Google Scholar

[14]

K. Szymańska, On an asymptotic boundary value problem for second order differential equations, J. Appl. Anal., 12 (2006), 109-118. doi: 10.1515/JAA.2006.109.  Google Scholar

[15]

K. Szymańska, Resonant problem for some second-order differential equation on the half-line, Electron. J. Differential Equations, (2007), 1-9.  Google Scholar

[16]

B. Yan, D. O'Regan and R. P. Agarwal, Unbounded solutions for singular boundary value problems on the semi-infinite interval: Upper and lower solutions and multiplicity, J. Comput. Appl. Math., 197 (2006), 365-386. doi: 10.1016/j.cam.2005.11.010.  Google Scholar

show all references

References:
[1]

R. P. Agarwal and D. O'Regan, Infinite Interval Problems For Differential, Difference and Integral Equations, Kluwer Academic Publishers, Boston, MA, 2001. doi: 10.1007/978-94-010-0718-4.  Google Scholar

[2]

P. Amster and A. Deboli, A Neumann boundary-value problem on an unbounded interval, Electronic J. of Differential Equations, 2008 (2008), 1-5.  Google Scholar

[3]

J. C. Arciero and D. Swigon, Equation-based models of wound healing and collective cell migration, in Complex Systems and Computational Biology Approaches to Acute Inflammation (eds. Y. Vodovotz and G. An), Springer, 2013, 185-207. doi: 10.1007/978-1-4614-8008-2_11.  Google Scholar

[4]

T. Callaghan, E. Khain, L. M. Sander and R. M. Ziff, A stochastic model for wound healing, J. Stat. Phys., 122 (2006), 909-924. doi: 10.1007/s10955-006-9022-1.  Google Scholar

[5]

J. K. Hale, Ordinary Differential Equations, Dover Publications, Mineola, NY, 2009. Google Scholar

[6]

M. W. Hirsch and H. L. Smith, Competitive and cooperative systems: A mini-review, in Positive Systems. Proceedings of the First Multidisciplinary Symposium on Positive Systems (POSTA 2003) (eds. Luca Benvenuti, Alberto De Santis and Lorenzo Farina), Lecture Notes on Control and Information Sciences, Vol. 294, Springer-Verlag, Heidelberg, 2003, 183-190. doi: 10.1007/b79667.  Google Scholar

[7]

H. Lian and W. Ge, Solvability for second-order three-point boundary value problems on a half-line, Appl. Math. Lett., 19 (2006), 1000-1006. doi: 10.1016/j.aml.2005.10.018.  Google Scholar

[8]

H. Lian and F. Geng, Multiple unbounded solutions for a boundary value problem on infinite intervals, Bound. Value Probl., 51 (2011), 1-8. doi: 10.1186/1687-2770-2011-51.  Google Scholar

[9]

B. Liu, J. Li and L. Liu, Existence and uniqueness for an m-point boundary problem at resonance on infinite intervals, Comput. Math. Appl., 64 (2012), 1677-1690. doi: 10.1016/j.camwa.2012.01.023.  Google Scholar

[10]

Q. Mi, D. Swigon, B. Rivière, S. Cetin, Y. Vodovotz and D. J. Hackam, One-dimensional elastic continuum model of enterocyte layer migration, Biophys. J., 93 (2007), 3745-3752. doi: 10.1529/biophysj.107.112326.  Google Scholar

[11]

H. L. Smith, Monotone Dynamical Systems: An introduction to the theory of competitive and cooperative systems, American Mathematical Society, Mathematical Surveys and Monographs, 1995.  Google Scholar

[12]

T. L. Stepien, Collective Cell Migration in Single and Dual Cell Layers, Ph.D. thesis. University of Pittsburgh, 2013.  Google Scholar

[13]

T. L. Stepien and D. Swigon, Traveling waves in a one-dimensional elastic continuum model of cell layer migration with stretch-dependent proliferation, SIAM J. Appl. Dyn. Syst., 13 (2014), 1489-1516. doi: 10.1137/130941407.  Google Scholar

[14]

K. Szymańska, On an asymptotic boundary value problem for second order differential equations, J. Appl. Anal., 12 (2006), 109-118. doi: 10.1515/JAA.2006.109.  Google Scholar

[15]

K. Szymańska, Resonant problem for some second-order differential equation on the half-line, Electron. J. Differential Equations, (2007), 1-9.  Google Scholar

[16]

B. Yan, D. O'Regan and R. P. Agarwal, Unbounded solutions for singular boundary value problems on the semi-infinite interval: Upper and lower solutions and multiplicity, J. Comput. Appl. Math., 197 (2006), 365-386. doi: 10.1016/j.cam.2005.11.010.  Google Scholar

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