# American Institute of Mathematical Sciences

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 and 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. [2] P. Amster and A. Deboli, A Neumann boundary-value problem on an unbounded interval, Electronic J. of Differential Equations, 2008 (2008), 1-5. [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. [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. [5] J. K. Hale, Ordinary Differential Equations, Dover Publications, Mineola, NY, 2009. [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. [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. [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. [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. [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. [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. [12] T. L. Stepien, Collective Cell Migration in Single and Dual Cell Layers, Ph.D. thesis. University of Pittsburgh, 2013. [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. [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. [15] K. Szymańska, Resonant problem for some second-order differential equation on the half-line, Electron. J. Differential Equations, (2007), 1-9. [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.

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##### 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. [2] P. Amster and A. Deboli, A Neumann boundary-value problem on an unbounded interval, Electronic J. of Differential Equations, 2008 (2008), 1-5. [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. [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. [5] J. K. Hale, Ordinary Differential Equations, Dover Publications, Mineola, NY, 2009. [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. [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. [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. [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. [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. [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. [12] T. L. Stepien, Collective Cell Migration in Single and Dual Cell Layers, Ph.D. thesis. University of Pittsburgh, 2013. [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. [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. [15] K. Szymańska, Resonant problem for some second-order differential equation on the half-line, Electron. J. Differential Equations, (2007), 1-9. [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.
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