# 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 & Continuous Dynamical Systems - A, 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, (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.   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), (2013), 185.  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.  doi: 10.1007/s10955-006-9022-1.  Google Scholar [5] J. K. Hale, Ordinary Differential Equations,, Dover Publications, (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, (2003), 183.  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.  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.  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.  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.  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, (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.  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.  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.   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.  doi: 10.1016/j.cam.2005.11.010.  Google Scholar

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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, (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.   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), (2013), 185.  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.  doi: 10.1007/s10955-006-9022-1.  Google Scholar [5] J. K. Hale, Ordinary Differential Equations,, Dover Publications, (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, (2003), 183.  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.  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.  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.  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.  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, (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.  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.  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.   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.  doi: 10.1016/j.cam.2005.11.010.  Google Scholar
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