# 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:

show all references

##### References:
 [1] Türker Özsarı, Nermin Yolcu. The initial-boundary value problem for the biharmonic Schrödinger equation on the half-line. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3285-3316. doi: 10.3934/cpaa.2019148 [2] Virginie Bonnaillie-Noël. Harmonic oscillators with Neumann condition on the half-line. Communications on Pure & Applied Analysis, 2012, 11 (6) : 2221-2237. doi: 10.3934/cpaa.2012.11.2221 [3] António J.G. Bento, Nicolae Lupa, Mihail Megan, César M. Silva. Integral conditions for nonuniform $μ$-dichotomy on the half-line. Discrete & Continuous Dynamical Systems - B, 2017, 22 (8) : 3063-3077. doi: 10.3934/dcdsb.2017163 [4] Lena-Susanne Hartmann, Ilya Pavlyukevich. Advection-diffusion equation on a half-line with boundary Lévy noise. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 637-655. doi: 10.3934/dcdsb.2018200 [5] Feliz Minhós, Hugo Carrasco. Solvability of higher-order BVPs in the half-line with unbounded nonlinearities. Conference Publications, 2015, 2015 (special) : 841-850. doi: 10.3934/proc.2015.0841 [6] Bogdan Sasu, Adina Luminiţa Sasu. On the dichotomic behavior of discrete dynamical systems on the half-line. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 3057-3084. doi: 10.3934/dcds.2013.33.3057 [7] Nicola Abatangelo, Serena Dipierro, Mouhamed Moustapha Fall, Sven Jarohs, Alberto Saldaña. Positive powers of the Laplacian in the half-space under Dirichlet boundary conditions. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1205-1235. doi: 10.3934/dcds.2019052 [8] Minoo Kamrani. Numerical solution of partial differential equations with stochastic Neumann boundary conditions. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5337-5354. doi: 10.3934/dcdsb.2019061 [9] Chérif Amrouche, Yves Raudin. Singular boundary conditions and regularity for the biharmonic problem in the half-space. Communications on Pure & Applied Analysis, 2007, 6 (4) : 957-982. doi: 10.3934/cpaa.2007.6.957 [10] Ademir Fernando Pazoto, Lionel Rosier. Uniform stabilization in weighted Sobolev spaces for the KdV equation posed on the half-line. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1511-1535. doi: 10.3934/dcdsb.2010.14.1511 [11] Guangsheng Wei, Hong-Kun Xu. On the missing bound state data of inverse spectral-scattering problems on the half-line. Inverse Problems & Imaging, 2015, 9 (1) : 239-255. doi: 10.3934/ipi.2015.9.239 [12] Nguyen Thieu Huy, Pham Van Bang. Invariant stable manifolds for partial neutral functional differential equations in admissible spaces on a half-line. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 2993-3011. doi: 10.3934/dcdsb.2015.20.2993 [13] Márcio Cavalcante, Chulkwang Kwak. Local well-posedness of the fifth-order KdV-type equations on the half-line. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2607-2661. doi: 10.3934/cpaa.2019117 [14] Piotr Kowalski. The existence of a solution for Dirichlet boundary value problem for a Duffing type differential inclusion. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2569-2580. doi: 10.3934/dcdsb.2014.19.2569 [15] Shaoyong Lai, Yong Hong Wu, Xu Yang. The global solution of an initial boundary value problem for the damped Boussinesq equation. Communications on Pure & Applied Analysis, 2004, 3 (2) : 319-328. doi: 10.3934/cpaa.2004.3.319 [16] Pablo Amster, Man Kam Kwong, Colin Rogers. A Neumann Boundary Value Problem in Two-Ion Electro-Diffusion with Unequal Valencies. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2299-2311. doi: 10.3934/dcdsb.2012.17.2299 [17] Julien Dambrine, Nicolas Meunier, Bertrand Maury, Aude Roudneff-Chupin. A congestion model for cell migration. Communications on Pure & Applied Analysis, 2012, 11 (1) : 243-260. doi: 10.3934/cpaa.2012.11.243 [18] Grégoire Allaire, Yves Capdeboscq, Marjolaine Puel. Homogenization of a one-dimensional spectral problem for a singularly perturbed elliptic operator with Neumann boundary conditions. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 1-31. doi: 10.3934/dcdsb.2012.17.1 [19] Igor Kossowski, Katarzyna Szymańska-Dębowska. Solutions to resonant boundary value problem with boundary conditions involving Riemann-Stieltjes integrals. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 275-281. doi: 10.3934/dcdsb.2018019 [20] T. J. Christiansen. Resonances and balls in obstacle scattering with Neumann boundary conditions. Inverse Problems & Imaging, 2008, 2 (3) : 335-340. doi: 10.3934/ipi.2008.2.335

2018 Impact Factor: 1.143