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June  2020, 40(6): 3357-3374. doi: 10.3934/dcds.2020049

Spike solutions for a mass conservation reaction-diffusion system

Shin-Ichiro Ei 1,, and  Shyuh-Yaur Tzeng 2,

1. 

Department of Mathematics, Faculty of Science, Hokkaido University, Sapporo 060-0810, Japan
2. 

Department of Mathematics, National Changhua University of Education, Changhua, Taiwan

* Corresponding author: Shin-Ichiro Ei

-dedicated to 70th birthday of Prof. Wei-Ming Ni-

Received  March 2019 Revised  July 2019 Published  October 2019

This article deals with a mass conservation reaction-diffusion system. As a model for studying cell polarity, we are interested in the existence of spike solutions and some properties related to its dynamics. Variational arguments will be employed to investigate the existence questions. The profile of a spike solution looks like a standing pulse. In addition, the motion of such spikes in heterogeneous media will be derived.

Keywords: Reaction-diffusion system, mass conservation, spike solution, cell polarity, localized pattern, standing pulse, invariant manifold.
Mathematics Subject Classification: Primary: 35B25, 35J505, 35K57; Secondary: 37L6.
Citation: 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
References:
[1]

A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 394-381.  doi: 10.1016/0022-1236(73)90051-7.  Google Scholar

[2]

C.-N. Chen and Y. S. Choi, Standing pulse solutions to FitzHugh-Nagumo equations, Arch. Rational Mech. Anal., 206 (2012), 741-777.  doi: 10.1007/s00205-012-0542-3.  Google Scholar

[3]

C.-N. Chen and Y. S. Choi, Traveling pulse solutions to FitzHugh-Nagumo equations, Calculus of Variations and Partial Differential Equations, 54 (2015), 1-45.  doi: 10.1007/s00526-014-0776-z.  Google Scholar

[4]

C.-N. Chen, Y. S. Choi and N. Fusco, The $\Gamma$-limit of traveling waves in the FitzHugh-Nagumo system, J. Differential Equations, 267 (2019), 1805-1835.  doi: 10.1016/j.jde.2019.02.023.  Google Scholar

[5]

C.-N. Chen, Y.-S. Choi and X. F. Ren, Bubbles and droplets in a singular limit of the FitzHugh-Nagumo system, Interfaces and Free Boundaries, 20 (2018), 165-210.  doi: 10.4171/IFB/400.  Google Scholar

[6]

C.-N. Chen, Y.-S. Choi, Y. Y. Hu and X. F. Ren, Higher dimensional bubble profiles in a singular limit of the FitzHugh-Nagumo system, SIAM J. Math. Anal., 50 (2018), 5072-5095.  doi: 10.1137/17M1144933.  Google Scholar

[7]

C.-N. Chen, S.-I. Ei and S.-Y. Tzeng, Heterogeneity induced effects for pulse dynamics in FitzHugh-Nagumo type systems, Physica D: Nonlinear Phenomena, 382/383 (2018), 22-32.  doi: 10.1016/j.physd.2018.07.001.  Google Scholar

[8]

C.-N. Chen and X. J. Hu, Maslov index for homoclinic orbits of Hamiltonian systems, Ann. Inst. H. Poincaré Anal. Non Linéarie, 24 (2007), 589-603.  doi: 10.1016/j.anihpc.2006.06.002.  Google Scholar

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C.-N. Chen and X. J. Hu, Stability criteria for reaction-diffusion systems with skew-gradient structure, Communications in Partial Differential Equations, 33 (2008), 189-208.  doi: 10.1080/03605300601188755.  Google Scholar

[10]

C.-N. Chen and X. J. Hu, Stability analysis for standing pulse solutions to FitzHugh-Nagumo equations, Calculus of Variations and Partial Differential Equations, 49 (2014), 827-845.  doi: 10.1007/s00526-013-0601-0.  Google Scholar

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C.-N. Chen and K. Tanaka, A variational approach for standing waves of FitzHugh-Nagumo type systems, J. Differential Equations, 257 (2014), 109-144.  doi: 10.1016/j.jde.2014.03.013.  Google Scholar

[12]

C.-N. Chen and S.-Y. Tzeng, Periodic solutions and their connecting orbits of Hamiltonian systems, J. Differential Equations, 177 (2001), 121-145.  doi: 10.1006/jdeq.2000.3996.  Google Scholar

[13]

A. Doelman, P. van Heijster and T. J. Kaper, Pulse dynamics in a three-component system: Existence analysis, J. Dynam. Differential Equations, 21 (2009), 73-115.  doi: 10.1007/s10884-008-9125-2.  Google Scholar

[14]

S.-I. Ei, The motion of weakly interacting pulses in reaction-diffusion systems, J. Dynam. Differential Equations, 14 (2002), 85-137.  doi: 10.1023/A:1012980128575.  Google Scholar

[15]

S.-I. Ei and J. C. Wei, Dynamics of metastable localized patterns and its application to the interaction of spike solutions for the Gierer-Meinhardt systems in two spatial dimension, Japan J. Ind. Appl. Math., 19 (2002), 181-226.  doi: 10.1007/BF03167453.  Google Scholar

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S. Ishihara, M. Otsuji and A. Mochizuki, Transient and steady state of mass-conserved reaction-diffusion systems, Phys. Rev. E, 75 (2007), 015203(R). doi: 10.1103/PhysRevE.75.015203.  Google Scholar

[17]

S. Jimbo and Y. Morita, Lyapunov function and spectrum comparison for a reaction-diffusion system with mass conservation, J. Differential Equations, 255 (2013), 1657-1683.  doi: 10.1016/j.jde.2013.05.021.  Google Scholar

[18]

J. Keener and J. Sneyd, Mathematical Physiology. Vol. I: Cellular Physiology, Second edition, Interdisciplinary Applied Mathematics, 8/I. Springer, New York, 2009. doi: 10.1007/978-0-387-79388-7.  Google Scholar

[19]

M. Kuwamura, S. Seirin-Lee and S.-I. Ei, Dynamics of localized unimodal patterns in reaction-diffusion systems for cell polarization by extracellular signaling, SIAM J. APPL. MATH., 78 (2018), 3238-3257.  doi: 10.1137/18M1163749.  Google Scholar

[20]

E. Latos and T. Suzuki, Global dynamics of a reaction-diffusion system with mass conservation, J. Math. Anal. Appl., 411 (2014), 107-118.  doi: 10.1016/j.jmaa.2013.09.039.  Google Scholar

[21]

Y. Morita and T. Ogawa, Stability and bifurcation of nonconstant solutions to a reaction-diffusion system with conservation of mass, Nonlinearity, 23 (2010), 1387-1411.  doi: 10.1088/0951-7715/23/6/007.  Google Scholar

[22]

W.-M. Ni and J. C. Wei, On the location and profile of spike-layer solutions to singularly perturbed semilinear Dirichlet problems, Comm. Pure Appl. Math., 48 (1995), 731-768.  doi: 10.1002/cpa.3160480704.  Google Scholar

[23]

M. Otsuji, S. Ishihara, C. Co, K. Kaibuchi, A. Mochizuki and S. Kuroda, A mass conserved reaction-diffusion system captures properties of cell polarity, PLoS Comput. Biol., 3 (2007), 1040-1054.  doi: 10.1371/journal.pcbi.0030108.  Google Scholar

[24]

P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conference Series in Mathematics, 65. Published for the Conference Board of the Mathematical Sciences, Washington, DC, by the American Mathematical Society, Providence, RI, 1986. doi: 10.1090/cbms/065.  Google Scholar

[25]

C. Reinecke and G. Sweers, A positive solution on ${\mathbb R}^n$ to a equations of FitzHugh-Nagumo type, J. Differential Equations, 153 (1999), 292-312.  doi: 10.1006/jdeq.1998.3560.  Google Scholar

[26]

T. Suzuki and S. Tasaki, Stationary Fix-Caginalp equation with non-local term, Nonlinear Anal., 71 (2009), 1329-1349.  doi: 10.1016/j.na.2008.12.007.  Google Scholar

[27]

P. van Heijster, C.-N. Chen, Y. Nishiura and T. Teramoto, Localized patterns in a three-component FitzHugh-Nagumo model revisited via an action functional, J. Dynam. Differential Equations, 30 (2018), 521-555.  doi: 10.1007/s10884-016-9557-z.  Google Scholar

[28]

P. van Heijster, C.-N. Chen, Y. Nishiura and T. Teramoto, Pinned solutions in a heterogeneous three-component FitzHugh-Nagumo model, J. Dyn. Differ. Equ., 31 (2019), 153-203.  doi: 10.1007/s10884-018-9694-7.  Google Scholar

[29]

E. Yanagida, Stability of fast travelling pulse solutions of the FitzHugh-Nagumo equations, J. Math. Biol., 22 (1985), 81-104.  doi: 10.1007/BF00276548.  Google Scholar

show all references

References:
[1]

A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 394-381.  doi: 10.1016/0022-1236(73)90051-7.  Google Scholar

[2]

C.-N. Chen and Y. S. Choi, Standing pulse solutions to FitzHugh-Nagumo equations, Arch. Rational Mech. Anal., 206 (2012), 741-777.  doi: 10.1007/s00205-012-0542-3.  Google Scholar

[3]

C.-N. Chen and Y. S. Choi, Traveling pulse solutions to FitzHugh-Nagumo equations, Calculus of Variations and Partial Differential Equations, 54 (2015), 1-45.  doi: 10.1007/s00526-014-0776-z.  Google Scholar

[4]

C.-N. Chen, Y. S. Choi and N. Fusco, The $\Gamma$-limit of traveling waves in the FitzHugh-Nagumo system, J. Differential Equations, 267 (2019), 1805-1835.  doi: 10.1016/j.jde.2019.02.023.  Google Scholar

[5]

C.-N. Chen, Y.-S. Choi and X. F. Ren, Bubbles and droplets in a singular limit of the FitzHugh-Nagumo system, Interfaces and Free Boundaries, 20 (2018), 165-210.  doi: 10.4171/IFB/400.  Google Scholar

[6]

C.-N. Chen, Y.-S. Choi, Y. Y. Hu and X. F. Ren, Higher dimensional bubble profiles in a singular limit of the FitzHugh-Nagumo system, SIAM J. Math. Anal., 50 (2018), 5072-5095.  doi: 10.1137/17M1144933.  Google Scholar

[7]

C.-N. Chen, S.-I. Ei and S.-Y. Tzeng, Heterogeneity induced effects for pulse dynamics in FitzHugh-Nagumo type systems, Physica D: Nonlinear Phenomena, 382/383 (2018), 22-32.  doi: 10.1016/j.physd.2018.07.001.  Google Scholar

[8]

C.-N. Chen and X. J. Hu, Maslov index for homoclinic orbits of Hamiltonian systems, Ann. Inst. H. Poincaré Anal. Non Linéarie, 24 (2007), 589-603.  doi: 10.1016/j.anihpc.2006.06.002.  Google Scholar

[9]

C.-N. Chen and X. J. Hu, Stability criteria for reaction-diffusion systems with skew-gradient structure, Communications in Partial Differential Equations, 33 (2008), 189-208.  doi: 10.1080/03605300601188755.  Google Scholar

[10]

C.-N. Chen and X. J. Hu, Stability analysis for standing pulse solutions to FitzHugh-Nagumo equations, Calculus of Variations and Partial Differential Equations, 49 (2014), 827-845.  doi: 10.1007/s00526-013-0601-0.  Google Scholar

[11]

C.-N. Chen and K. Tanaka, A variational approach for standing waves of FitzHugh-Nagumo type systems, J. Differential Equations, 257 (2014), 109-144.  doi: 10.1016/j.jde.2014.03.013.  Google Scholar

[12]

C.-N. Chen and S.-Y. Tzeng, Periodic solutions and their connecting orbits of Hamiltonian systems, J. Differential Equations, 177 (2001), 121-145.  doi: 10.1006/jdeq.2000.3996.  Google Scholar

[13]

A. Doelman, P. van Heijster and T. J. Kaper, Pulse dynamics in a three-component system: Existence analysis, J. Dynam. Differential Equations, 21 (2009), 73-115.  doi: 10.1007/s10884-008-9125-2.  Google Scholar

[14]

S.-I. Ei, The motion of weakly interacting pulses in reaction-diffusion systems, J. Dynam. Differential Equations, 14 (2002), 85-137.  doi: 10.1023/A:1012980128575.  Google Scholar

[15]

S.-I. Ei and J. C. Wei, Dynamics of metastable localized patterns and its application to the interaction of spike solutions for the Gierer-Meinhardt systems in two spatial dimension, Japan J. Ind. Appl. Math., 19 (2002), 181-226.  doi: 10.1007/BF03167453.  Google Scholar

[16]

S. Ishihara, M. Otsuji and A. Mochizuki, Transient and steady state of mass-conserved reaction-diffusion systems, Phys. Rev. E, 75 (2007), 015203(R). doi: 10.1103/PhysRevE.75.015203.  Google Scholar

[17]

S. Jimbo and Y. Morita, Lyapunov function and spectrum comparison for a reaction-diffusion system with mass conservation, J. Differential Equations, 255 (2013), 1657-1683.  doi: 10.1016/j.jde.2013.05.021.  Google Scholar

[18]

J. Keener and J. Sneyd, Mathematical Physiology. Vol. I: Cellular Physiology, Second edition, Interdisciplinary Applied Mathematics, 8/I. Springer, New York, 2009. doi: 10.1007/978-0-387-79388-7.  Google Scholar

[19]

M. Kuwamura, S. Seirin-Lee and S.-I. Ei, Dynamics of localized unimodal patterns in reaction-diffusion systems for cell polarization by extracellular signaling, SIAM J. APPL. MATH., 78 (2018), 3238-3257.  doi: 10.1137/18M1163749.  Google Scholar

[20]

E. Latos and T. Suzuki, Global dynamics of a reaction-diffusion system with mass conservation, J. Math. Anal. Appl., 411 (2014), 107-118.  doi: 10.1016/j.jmaa.2013.09.039.  Google Scholar

[21]

Y. Morita and T. Ogawa, Stability and bifurcation of nonconstant solutions to a reaction-diffusion system with conservation of mass, Nonlinearity, 23 (2010), 1387-1411.  doi: 10.1088/0951-7715/23/6/007.  Google Scholar

[22]

W.-M. Ni and J. C. Wei, On the location and profile of spike-layer solutions to singularly perturbed semilinear Dirichlet problems, Comm. Pure Appl. Math., 48 (1995), 731-768.  doi: 10.1002/cpa.3160480704.  Google Scholar

[23]

M. Otsuji, S. Ishihara, C. Co, K. Kaibuchi, A. Mochizuki and S. Kuroda, A mass conserved reaction-diffusion system captures properties of cell polarity, PLoS Comput. Biol., 3 (2007), 1040-1054.  doi: 10.1371/journal.pcbi.0030108.  Google Scholar

[24]

P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conference Series in Mathematics, 65. Published for the Conference Board of the Mathematical Sciences, Washington, DC, by the American Mathematical Society, Providence, RI, 1986. doi: 10.1090/cbms/065.  Google Scholar

[25]

C. Reinecke and G. Sweers, A positive solution on ${\mathbb R}^n$ to a equations of FitzHugh-Nagumo type, J. Differential Equations, 153 (1999), 292-312.  doi: 10.1006/jdeq.1998.3560.  Google Scholar

[26]

T. Suzuki and S. Tasaki, Stationary Fix-Caginalp equation with non-local term, Nonlinear Anal., 71 (2009), 1329-1349.  doi: 10.1016/j.na.2008.12.007.  Google Scholar

[27]

P. van Heijster, C.-N. Chen, Y. Nishiura and T. Teramoto, Localized patterns in a three-component FitzHugh-Nagumo model revisited via an action functional, J. Dynam. Differential Equations, 30 (2018), 521-555.  doi: 10.1007/s10884-016-9557-z.  Google Scholar

[28]

P. van Heijster, C.-N. Chen, Y. Nishiura and T. Teramoto, Pinned solutions in a heterogeneous three-component FitzHugh-Nagumo model, J. Dyn. Differ. Equ., 31 (2019), 153-203.  doi: 10.1007/s10884-018-9694-7.  Google Scholar

[29]

E. Yanagida, Stability of fast travelling pulse solutions of the FitzHugh-Nagumo equations, J. Math. Biol., 22 (1985), 81-104.  doi: 10.1007/BF00276548.  Google Scholar

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