# American Institute of Mathematical Sciences

March  2013, 8(1): 23-35. doi: 10.3934/nhm.2013.8.23

## Reaction-diffusion waves with nonlinear boundary conditions

 1 Department of Mathematics, "Gheorghe Asachi" Technical University, Bd. Carol. I, 700506 Iasi, Romania 2 Institut Camille Jordan, UMR 5208 CNRS, University Lyon 1, 69622 Villeurbanne, France

Received  January 2012 Revised  July 2012 Published  April 2013

A reaction-diffusion equation with nonlinear boundary condition is considered in a two-dimensional infinite strip. Existence of waves in the bistable case is proved by the Leray-Schauder method.
Citation: Narcisa Apreutesei, Vitaly Volpert. Reaction-diffusion waves with nonlinear boundary conditions. Networks & Heterogeneous Media, 2013, 8 (1) : 23-35. doi: 10.3934/nhm.2013.8.23
##### References:
 [1] A. Fabiato, Calcium-induced release of calcium from the cardiac sarcoplasmic reticulum,, Am. J. Physiol. Cell. Physiol., 245 (1983), 1. doi: 10.1016/0022-2828(92)90114-F. [2] A. Friedman, "Partial Differential Equations of Parabolic Type,", Prentice-Hall, (1964). [3] N. El Khatib, S. Genieys, B. Kazmierczak and V. Volpert, Reaction-diffusion model of atherosclerosis development,, J. Math. Biol., 65 (2012), 349. doi: 10.1007/s00285-011-0461-1. [4] M. Kyed, Existence of travelling wave solutions for the heat equation in infinite cylinders with a nonlinear boundary condition,, Math. Nachr., 281 (2008), 253. doi: 10.1002/mana.200710599. [5] A. Volpert, Vit. Volpert and Vl. Volpert, "Traveling Wave Solutions of Parabolic Systems,", Translation of Mathematical Monographs, 140 (1994). [6] V. Volpert and A. Volpert, Spectrum of elliptic operators and stability of travelling waves,, Asymptotic Analysis, 23 (2000), 111. [7] V. Volpert, "Elliptic Partial Differential Equations. Volume 1. Fredholm Theory of Elliptic Problems in Unbounded Domains,", Birkhäuser, (2011). doi: 10.1007/978-3-0346-0537-3.

show all references

##### References:
 [1] A. Fabiato, Calcium-induced release of calcium from the cardiac sarcoplasmic reticulum,, Am. J. Physiol. Cell. Physiol., 245 (1983), 1. doi: 10.1016/0022-2828(92)90114-F. [2] A. Friedman, "Partial Differential Equations of Parabolic Type,", Prentice-Hall, (1964). [3] N. El Khatib, S. Genieys, B. Kazmierczak and V. Volpert, Reaction-diffusion model of atherosclerosis development,, J. Math. Biol., 65 (2012), 349. doi: 10.1007/s00285-011-0461-1. [4] M. Kyed, Existence of travelling wave solutions for the heat equation in infinite cylinders with a nonlinear boundary condition,, Math. Nachr., 281 (2008), 253. doi: 10.1002/mana.200710599. [5] A. Volpert, Vit. Volpert and Vl. Volpert, "Traveling Wave Solutions of Parabolic Systems,", Translation of Mathematical Monographs, 140 (1994). [6] V. Volpert and A. Volpert, Spectrum of elliptic operators and stability of travelling waves,, Asymptotic Analysis, 23 (2000), 111. [7] V. Volpert, "Elliptic Partial Differential Equations. Volume 1. Fredholm Theory of Elliptic Problems in Unbounded Domains,", Birkhäuser, (2011). doi: 10.1007/978-3-0346-0537-3.
 [1] Daniel Franco, Donal O'Regan. Existence of solutions to second order problems with nonlinear boundary conditions. Conference Publications, 2003, 2003 (Special) : 273-280. doi: 10.3934/proc.2003.2003.273 [2] Shu Luan. On the existence of optimal control for semilinear elliptic equations with nonlinear neumann boundary conditions. Mathematical Control & Related Fields, 2017, 7 (3) : 493-506. doi: 10.3934/mcrf.2017018 [3] Le Thi Phuong Ngoc, Nguyen Thanh Long. Existence and exponential decay for a nonlinear wave equation with nonlocal boundary conditions. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2001-2029. doi: 10.3934/cpaa.2013.12.2001 [4] R.A. Satnoianu, Philip K. Maini, F.S. Garduno, J.P. Armitage. Travelling waves in a nonlinear degenerate diffusion model for bacterial pattern formation. Discrete & Continuous Dynamical Systems - B, 2001, 1 (3) : 339-362. doi: 10.3934/dcdsb.2001.1.339 [5] Michal Fečkan, Vassilis M. Rothos. Travelling waves of forced discrete nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - S, 2011, 4 (5) : 1129-1145. doi: 10.3934/dcdss.2011.4.1129 [6] Boumediene Abdellaoui, Daniela Giachetti, Ireneo Peral, Magdalena Walias. Elliptic problems with nonlinear terms depending on the gradient and singular on the boundary: Interaction with a Hardy-Leray potential. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 1747-1774. doi: 10.3934/dcds.2014.34.1747 [7] Dmitry Treschev. Travelling waves in FPU lattices. Discrete & Continuous Dynamical Systems - A, 2004, 11 (4) : 867-880. doi: 10.3934/dcds.2004.11.867 [8] Yosra Boukari, Houssem Haddar. The factorization method applied to cracks with impedance boundary conditions. Inverse Problems & Imaging, 2013, 7 (4) : 1123-1138. doi: 10.3934/ipi.2013.7.1123 [9] José Raúl Quintero, Juan Carlos Muñoz Grajales. On the existence and computation of periodic travelling waves for a 2D water wave model. Communications on Pure & Applied Analysis, 2018, 17 (2) : 557-578. doi: 10.3934/cpaa.2018030 [10] Yong Jung Kim, Wei-Ming Ni, Masaharu Taniguchi. Non-existence of localized travelling waves with non-zero speed in single reaction-diffusion equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3707-3718. doi: 10.3934/dcds.2013.33.3707 [11] Matthew H. Chan, Peter S. Kim, Robert Marangell. Stability of travelling waves in a Wolbachia invasion. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 609-628. doi: 10.3934/dcdsb.2018036 [12] Alexandre Nolasco de Carvalho, Marcos Roberto Teixeira Primo. Spatial homogeneity in parabolic problems with nonlinear boundary conditions. Communications on Pure & Applied Analysis, 2004, 3 (4) : 637-651. doi: 10.3934/cpaa.2004.3.637 [13] Gennaro Infante. Positive solutions of differential equations with nonlinear boundary conditions. Conference Publications, 2003, 2003 (Special) : 432-438. doi: 10.3934/proc.2003.2003.432 [14] Soohyun Bae, Jaeyoung Byeon. Standing waves of nonlinear Schrödinger equations with optimal conditions for potential and nonlinearity. Communications on Pure & Applied Analysis, 2013, 12 (2) : 831-850. doi: 10.3934/cpaa.2013.12.831 [15] Thomas Y. Hou, Pingwen Zhang. Convergence of a boundary integral method for 3-D water waves. Discrete & Continuous Dynamical Systems - B, 2002, 2 (1) : 1-34. doi: 10.3934/dcdsb.2002.2.1 [16] Frederic Rousset. The residual boundary conditions coming from the real vanishing viscosity method. Discrete & Continuous Dynamical Systems - A, 2002, 8 (3) : 605-625. doi: 10.3934/dcds.2002.8.606 [17] Harunori Monobe, Hirokazu Ninomiya. Multiple existence of traveling waves of a free boundary problem describing cell motility. Discrete & Continuous Dynamical Systems - B, 2014, 19 (3) : 789-799. doi: 10.3934/dcdsb.2014.19.789 [18] B. Abdellaoui, E. Colorado, I. Peral. Existence and nonexistence results for a class of parabolic equations with mixed boundary conditions. Communications on Pure & Applied Analysis, 2006, 5 (1) : 29-54. doi: 10.3934/cpaa.2006.5.29 [19] Tae Gab Ha. Global existence and general decay estimates for the viscoelastic equation with acoustic boundary conditions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 6899-6919. doi: 10.3934/dcds.2016100 [20] Mariane Bourgoing. Viscosity solutions of fully nonlinear second order parabolic equations with $L^1$ dependence in time and Neumann boundary conditions. Existence and applications to the level-set approach. Discrete & Continuous Dynamical Systems - A, 2008, 21 (4) : 1047-1069. doi: 10.3934/dcds.2008.21.1047

2017 Impact Factor: 1.187