American Institute of Mathematical Sciences

Advanced
March  2013, 8(1): 191-209. doi: 10.3934/nhm.2013.8.191

Effect of boundary conditions on the dynamics of a pulse solution for reaction-diffusion systems

Shin-Ichiro Ei 1, and  Toshio Ishimoto 2,

1. 

Institute of Mathematics for Industry, Kyusyu University, 744 Motooka, Nishi-ku, Fukuoka, 819-0395
2. 

Opinion Poll Research Center, The Asahi Shimbun Company, Tokyo 104-8011, Japan

Received  February 2012 Revised  January 2013 Published  April 2013

We consider pulse-like localized solutions for reaction-diffusion systems on a half line and impose various boundary conditions at one end of it. It is shown that the movement of a pulse solution with the homogeneous Neumann boundary condition is completely opposite from that with the Dirichlet boundary condition. As general cases, Robin type boundary conditions are also considered. Introducing one parameter connecting the Neumann and the Dirichlet boundary conditions, we clarify the transition of motions of solutions with respect to boundary conditions.
Keywords: Reaction-diffusion systems, front solutions., effect of boundary conditions, pulse solutions.
Mathematics Subject Classification: Primary: 35K57; Secondary: 35B25, 35K5.
Citation: Shin-Ichiro Ei, Toshio Ishimoto. Effect of boundary conditions on the dynamics of a pulse solution for reaction-diffusion systems. Networks & Heterogeneous Media, 2013, 8 (1) : 191-209. doi: 10.3934/nhm.2013.8.191
References:
[1]

J. Carr and R. L. Pego, Metastable patterns in solutions of $u_t = epsilon^2 u_{x x} + f(u)$,, Comm. Pure Appl. Math., 42 (1989), 523.  doi: 10.1002/cpa.3160420502.  Google Scholar

[2]

A. Doelman, R. A. Gardner and T. J. Kaper, Stability analysis of singular patterns in the 1-D Gray-Scott model,, Physica D, 122 (1998), 1.  doi: 10.1016/S0167-2789(98)00180-8.  Google Scholar

[3]

S.-I. Ei, The motion of weakly interacting pulses in reaction-diffusion systems,, J. Dynam. Differential Equations, 14 (2002), 85.  doi: 10.1023/A:1012980128575.  Google Scholar

[4]

P. C. Fife and J. B. Mcleod, The approach of solutions of nonlinear diffusion equations to travelling front solutions,, Arch. Ration. Mech. Anal., 65 (1977), 335.   Google Scholar

[5]

G. Fusco and J. Hale, Slow motion manifold, dormant instability and singular perturbations,, J. Dynamics and Differential Equations, 1 (1989), 75.  doi: 10.1007/BF01048791.  Google Scholar

[6]

K. Kawasaki and T. Ohta, Kink dynamics in one-dimensional nonlinear systems,, Physica A, 116 (1982), 573.  doi: 10.1016/0378-4371(82)90178-9.  Google Scholar

[7]

J. M. Murray, "Mathematical Biology,", Springer-Verlag, (1989).   Google Scholar

[8]

Y. Nishiura, "Far-From-Equilibrium Dynamics,", (Translations of Mathematical Monographs), (2002).   Google Scholar

show all references

References:
[1]

J. Carr and R. L. Pego, Metastable patterns in solutions of $u_t = epsilon^2 u_{x x} + f(u)$,, Comm. Pure Appl. Math., 42 (1989), 523.  doi: 10.1002/cpa.3160420502.  Google Scholar

[2]

A. Doelman, R. A. Gardner and T. J. Kaper, Stability analysis of singular patterns in the 1-D Gray-Scott model,, Physica D, 122 (1998), 1.  doi: 10.1016/S0167-2789(98)00180-8.  Google Scholar

[3]

S.-I. Ei, The motion of weakly interacting pulses in reaction-diffusion systems,, J. Dynam. Differential Equations, 14 (2002), 85.  doi: 10.1023/A:1012980128575.  Google Scholar

[4]

P. C. Fife and J. B. Mcleod, The approach of solutions of nonlinear diffusion equations to travelling front solutions,, Arch. Ration. Mech. Anal., 65 (1977), 335.   Google Scholar

[5]

G. Fusco and J. Hale, Slow motion manifold, dormant instability and singular perturbations,, J. Dynamics and Differential Equations, 1 (1989), 75.  doi: 10.1007/BF01048791.  Google Scholar

[6]

K. Kawasaki and T. Ohta, Kink dynamics in one-dimensional nonlinear systems,, Physica A, 116 (1982), 573.  doi: 10.1016/0378-4371(82)90178-9.  Google Scholar

[7]

J. M. Murray, "Mathematical Biology,", Springer-Verlag, (1989).   Google Scholar

[8]

Y. Nishiura, "Far-From-Equilibrium Dynamics,", (Translations of Mathematical Monographs), (2002).   Google Scholar

[1]

A. Dall'Acqua. Positive solutions for a class of reaction-diffusion systems. Communications on Pure & Applied Analysis, 2003, 2 (1) : 65-76. doi: 10.3934/cpaa.2003.2.65
[2]

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
[3]

Wei-Jian Bo, Guo Lin, Shigui Ruan. Traveling wave solutions for time periodic reaction-diffusion systems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4329-4351. doi: 10.3934/dcds.2018189
[4]

Wei Feng, Weihua Ruan, Xin Lu. On existence of wavefront solutions in mixed monotone reaction-diffusion systems. Discrete & Continuous Dynamical Systems - B, 2016, 21 (3) : 815-836. doi: 10.3934/dcdsb.2016.21.815
[5]

Oleksiy V. Kapustyan, Pavlo O. Kasyanov, José Valero. Regular solutions and global attractors for reaction-diffusion systems without uniqueness. Communications on Pure & Applied Analysis, 2014, 13 (5) : 1891-1906. doi: 10.3934/cpaa.2014.13.1891
[6]

Marek Fila, Hirokazu Ninomiya, Juan-Luis Vázquez. Dirichlet boundary conditions can prevent blow-up in reaction-diffusion equations and systems. Discrete & Continuous Dynamical Systems - A, 2006, 14 (1) : 63-74. doi: 10.3934/dcds.2006.14.63
[7]

Jihoon Lee, Vu Manh Toi. Attractors for a class of delayed reaction-diffusion equations with dynamic boundary conditions. Discrete & Continuous Dynamical Systems - B, 2020, 25 (8) : 3135-3152. doi: 10.3934/dcdsb.2020054
[8]

Luisa Malaguti, Cristina Marcelli, Serena Matucci. Continuous dependence in front propagation of convective reaction-diffusion equations. Communications on Pure & Applied Analysis, 2010, 9 (4) : 1083-1098. doi: 10.3934/cpaa.2010.9.1083
[9]

Wei Wang, Wanbiao Ma. Global dynamics and travelling wave solutions for a class of non-cooperative reaction-diffusion systems with nonlocal infections. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3213-3235. doi: 10.3934/dcdsb.2018242
[10]

Klemens Fellner, Evangelos Latos, Takashi Suzuki. Global classical solutions for mass-conserving, (super)-quadratic reaction-diffusion systems in three and higher space dimensions. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3441-3462. doi: 10.3934/dcdsb.2016106
[11]

Shi-Liang Wu, Yu-Juan Sun, San-Yang Liu. Traveling fronts and entire solutions in partially degenerate reaction-diffusion systems with monostable nonlinearity. Discrete & Continuous Dynamical Systems - A, 2013, 33 (2) : 921-946. doi: 10.3934/dcds.2013.33.921
[12]

Ciprian G. Gal, Mahamadi Warma. Reaction-diffusion equations with fractional diffusion on non-smooth domains with various boundary conditions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1279-1319. doi: 10.3934/dcds.2016.36.1279
[13]

Yuri Latushkin, Roland Schnaubelt, Xinyao Yang. Stable foliations near a traveling front for reaction diffusion systems. Discrete & Continuous Dynamical Systems - B, 2017, 22 (8) : 3145-3165. doi: 10.3934/dcdsb.2017168
[14]

Rui Li, Yuan Lou. Some monotone properties for solutions to a reaction-diffusion model. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 4445-4455. doi: 10.3934/dcdsb.2019126
[15]

Jiang Liu, Xiaohui Shang, Zengji Du. Traveling wave solutions of a reaction-diffusion predator-prey model. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1063-1078. doi: 10.3934/dcdss.2017057
[16]

Bang-Sheng Han, Zhi-Cheng Wang. Traveling wave solutions in a nonlocal reaction-diffusion population model. Communications on Pure & Applied Analysis, 2016, 15 (3) : 1057-1076. doi: 10.3934/cpaa.2016.15.1057
[17]

Tarik Mohammed Touaoula, Mohammed Nor Frioui, Nikolay Bessonov, Vitaly Volpert. Dynamics of solutions of a reaction-diffusion equation with delayed inhibition. Discrete & Continuous Dynamical Systems - S, 2020, 13 (9) : 2425-2442. doi: 10.3934/dcdss.2020193
[18]

Jong-Shenq Guo, Yoshihisa Morita. Entire solutions of reaction-diffusion equations and an application to discrete diffusive equations. Discrete & Continuous Dynamical Systems - A, 2005, 12 (2) : 193-212. doi: 10.3934/dcds.2005.12.193
[19]

Shin-Ichiro Ei, Shyuh-Yaur Tzeng. Spike solutions for a mass conservation reaction-diffusion system. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3357-3374. doi: 10.3934/dcds.2020049
[20]

Qiang Liu, Zhichang Guo, Chunpeng Wang. Renormalized solutions to a reaction-diffusion system applied to image denoising. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1839-1858. doi: 10.3934/dcdsb.2016025

2019 Impact Factor: 1.053

Tools

Metrics

  • PDF downloads (28)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences