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May  2017, 37(5): 2619-2651. doi: 10.3934/dcds.2017112

Traveling fronts bifurcating from stable layers in the presence of conservation laws

1. 

Miami University, Department of Mathematics, 301 S. Patterson Ave. Oxford, OH 45056, USA

2. 

University of Minnesota, School of Mathematics 206 Church St. S.E. Minneapolis, MN 55455, USA

* Corresponding author: Alin Pogan

Received  December 2015 Revised  December 2016 Published  February 2017

Fund Project: AS acknowledges support under grants NSF DMS-1311740 and DMS-1612441. AP acknowledges support by through a Summer Research Grant by College of Arts and Science, Miami University.

We study traveling waves bifurcating from stable standing layers in systems where a reaction-diffusion equation couples to a scalar conservation law. We prove the existence of weekly decaying traveling fronts that emerge in the presence of a weakly stable direction on a center manifold. Moreover, we show the existence of bifurcating traveling waves of constant mass. The main difficulty is to prove the smoothness of the ansatz in exponentially weighted spaces required to apply the Lyapunov-Schmidt methods.

Citation: Alin Pogan, Arnd Scheel. Traveling fronts bifurcating from stable layers in the presence of conservation laws. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2619-2651. doi: 10.3934/dcds.2017112
References:
[1]

D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, Partial differential equations and related topics, 446 (2006), 5-49. doi: 10.1007/BFb0070595. Google Scholar

[2]

G. Caginalp, An analysis of a phase field model of a free boundary, Arch. Ration. Mech. Anal., 92 (1986), 205-245. doi: 10.1007/BF00254827. Google Scholar

[3]

P. CoulletJ. LegaB. Houchmanzadeh and J. Lajzerowicz, Breaking chirality in nonequilibrium systems, Phys. Rev. Lett., 65 (1990), 1352-1355. doi: 10.1103/PhysRevLett.65.1352. Google Scholar

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S.-I. EiH. Ikeda and T. Kawana, Dynamics of front solutions in a specific reaction-diffusion system in one dimension, Japan J. Indust. Appl. Math., 25 (2008), 117-147. doi: 10.1007/BF03167516. Google Scholar

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R. GohS. Mesuro and A. Scheel, Coherent structures in reaction-diffusion models for precipitation, Special volume on "Precipitation patterns in reaction-diffusion systems", Research Signpost, (2010), 73-93. Google Scholar

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R. GohS. Mesuro and A. Scheel, Spatial wavenumber selection in recurrent precipitation, SIAM J. Appl. Dyn. Sys., 10 (2011), 360-402. doi: 10.1137/100793086. Google Scholar

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A. Hagberg and E. Meron, Pattern formation in non-gradient reaction-diffusion systems: The effects of front bifurcation, Nonlinearity, 7 (1994), 805-835. doi: 10.1088/0951-7715/7/3/006. Google Scholar

[8]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lect. Notes Math. , 840, Springer-Verlag, New York, 1981. doi: 10.1007/BFb0089647. Google Scholar

[9]

T. Hillen and K. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217. doi: 10.1007/s00285-008-0201-3. Google Scholar

[10]

P. Howard, Stability of Transition Front Solutions in Multidimensional Cahn-Hilliard Systems, J. Nonlinear Science, 26 (2016), 619-661. Google Scholar

[11]

H. IkedaM. Mimura and Y. Nishiura, Global bifurcation phenomena of traveling wave solutions for some bistable reaction-diffusion systems, Nonlinear Anal., 13 (1989), 507-526. Google Scholar

[12]

G. Jaramillo and A. Scheel, Deformation of Striped Patterns by Inhomogeneities, Math. Meth. Appl. Sci., 38 (2015), 51-65. doi: 10.1002/mma.3049. Google Scholar

[13]

G. Jaramillo and A. Scheel, Pacemakers in large arrays of oscillators with nonlocal coupling, J. Diff. Eqns., 260 (2016), 2060-2090. doi: 10.1016/j.jde.2015.09.054. Google Scholar

[14]

A. JilkineL. Keshet and Y. Mori, Wave-pinning and cell polarity from a bistable reaction-diffusion system, Bioph. J., 94 (2008), 3684-3697. Google Scholar

[15]

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

[16]

S. NasunoN. Yoshimo and S. Kai, Structural transition andmotion of domain walls in liquidcrystals under a rotating magnetic field, Phys. Rev. E, 51 (1995), 1598. Google Scholar

[17]

K. J. Palmer, Exponential dichotomies and transversal homoclinic points, J. Diff. Eq., 55 (1984), 225-256. doi: 10.1016/0022-0396(84)90082-2. Google Scholar

[18]

K. J. Palmer, Exponential dichotomies and Fredholm operators, Proc. Amer. Math. Soc., 104 (1988), 149-156. doi: 10.1090/S0002-9939-1988-0958058-1. Google Scholar

[19]

A. Pogan and A. Scheel, Instability of spikes in the presence of conservation laws, Z. Angew. Math. Phys., 61 (2010), 979-998. doi: 10.1007/s00033-010-0058-3. Google Scholar

[20]

A. Pogan and A. Scheel, Instability of radially-symmetric spikes in systems with a conserved quantity, Fields Inst. Comm., 64 (2013), 119-140. doi: 10.1007/978-1-4614-4523-4_4. Google Scholar

[21]

A. Pogan and A. Scheel, Fredholm properties of radially symmetric, second order differential operators, Int. J. Dyn. Sys. Diff. Eqns., 3 (2011), 289-327. doi: 10.1504/IJDSDE.2011.041878. Google Scholar

[22]

A. Pogan and A. Scheel, Layers in the presence of conservation laws, J. Dyn. Diff. Eq., 24 (2012), 249-287. doi: 10.1007/s10884-012-9248-3. Google Scholar

[23]

A. PoganA. Scheel and K. Zumbrun, Quasi-gradient systems, modulational dichotomies, and stability of spatially periodic patterns, Diff. Int. Eqns., 26 (2013), 389-438. Google Scholar

[24]

J. Rinzel and D. Terman, Propagation phenomena in a bistable reaction-diffusion system, SIAM J. Appl. Math., 42 (1982), 1111-1137. doi: 10.1137/0142077. Google Scholar

[25]

B. Sandstede and A. Scheel, Relative Morse indices, Fredholm indices, and group velocities, Discr. Cont. Dynam. Syst., 20 (2008), 139-158. Google Scholar

show all references

References:
[1]

D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, Partial differential equations and related topics, 446 (2006), 5-49. doi: 10.1007/BFb0070595. Google Scholar

[2]

G. Caginalp, An analysis of a phase field model of a free boundary, Arch. Ration. Mech. Anal., 92 (1986), 205-245. doi: 10.1007/BF00254827. Google Scholar

[3]

P. CoulletJ. LegaB. Houchmanzadeh and J. Lajzerowicz, Breaking chirality in nonequilibrium systems, Phys. Rev. Lett., 65 (1990), 1352-1355. doi: 10.1103/PhysRevLett.65.1352. Google Scholar

[4]

S.-I. EiH. Ikeda and T. Kawana, Dynamics of front solutions in a specific reaction-diffusion system in one dimension, Japan J. Indust. Appl. Math., 25 (2008), 117-147. doi: 10.1007/BF03167516. Google Scholar

[5]

R. GohS. Mesuro and A. Scheel, Coherent structures in reaction-diffusion models for precipitation, Special volume on "Precipitation patterns in reaction-diffusion systems", Research Signpost, (2010), 73-93. Google Scholar

[6]

R. GohS. Mesuro and A. Scheel, Spatial wavenumber selection in recurrent precipitation, SIAM J. Appl. Dyn. Sys., 10 (2011), 360-402. doi: 10.1137/100793086. Google Scholar

[7]

A. Hagberg and E. Meron, Pattern formation in non-gradient reaction-diffusion systems: The effects of front bifurcation, Nonlinearity, 7 (1994), 805-835. doi: 10.1088/0951-7715/7/3/006. Google Scholar

[8]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lect. Notes Math. , 840, Springer-Verlag, New York, 1981. doi: 10.1007/BFb0089647. Google Scholar

[9]

T. Hillen and K. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217. doi: 10.1007/s00285-008-0201-3. Google Scholar

[10]

P. Howard, Stability of Transition Front Solutions in Multidimensional Cahn-Hilliard Systems, J. Nonlinear Science, 26 (2016), 619-661. Google Scholar

[11]

H. IkedaM. Mimura and Y. Nishiura, Global bifurcation phenomena of traveling wave solutions for some bistable reaction-diffusion systems, Nonlinear Anal., 13 (1989), 507-526. Google Scholar

[12]

G. Jaramillo and A. Scheel, Deformation of Striped Patterns by Inhomogeneities, Math. Meth. Appl. Sci., 38 (2015), 51-65. doi: 10.1002/mma.3049. Google Scholar

[13]

G. Jaramillo and A. Scheel, Pacemakers in large arrays of oscillators with nonlocal coupling, J. Diff. Eqns., 260 (2016), 2060-2090. doi: 10.1016/j.jde.2015.09.054. Google Scholar

[14]

A. JilkineL. Keshet and Y. Mori, Wave-pinning and cell polarity from a bistable reaction-diffusion system, Bioph. J., 94 (2008), 3684-3697. Google Scholar

[15]

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

[16]

S. NasunoN. Yoshimo and S. Kai, Structural transition andmotion of domain walls in liquidcrystals under a rotating magnetic field, Phys. Rev. E, 51 (1995), 1598. Google Scholar

[17]

K. J. Palmer, Exponential dichotomies and transversal homoclinic points, J. Diff. Eq., 55 (1984), 225-256. doi: 10.1016/0022-0396(84)90082-2. Google Scholar

[18]

K. J. Palmer, Exponential dichotomies and Fredholm operators, Proc. Amer. Math. Soc., 104 (1988), 149-156. doi: 10.1090/S0002-9939-1988-0958058-1. Google Scholar

[19]

A. Pogan and A. Scheel, Instability of spikes in the presence of conservation laws, Z. Angew. Math. Phys., 61 (2010), 979-998. doi: 10.1007/s00033-010-0058-3. Google Scholar

[20]

A. Pogan and A. Scheel, Instability of radially-symmetric spikes in systems with a conserved quantity, Fields Inst. Comm., 64 (2013), 119-140. doi: 10.1007/978-1-4614-4523-4_4. Google Scholar

[21]

A. Pogan and A. Scheel, Fredholm properties of radially symmetric, second order differential operators, Int. J. Dyn. Sys. Diff. Eqns., 3 (2011), 289-327. doi: 10.1504/IJDSDE.2011.041878. Google Scholar

[22]

A. Pogan and A. Scheel, Layers in the presence of conservation laws, J. Dyn. Diff. Eq., 24 (2012), 249-287. doi: 10.1007/s10884-012-9248-3. Google Scholar

[23]

A. PoganA. Scheel and K. Zumbrun, Quasi-gradient systems, modulational dichotomies, and stability of spatially periodic patterns, Diff. Int. Eqns., 26 (2013), 389-438. Google Scholar

[24]

J. Rinzel and D. Terman, Propagation phenomena in a bistable reaction-diffusion system, SIAM J. Appl. Math., 42 (1982), 1111-1137. doi: 10.1137/0142077. Google Scholar

[25]

B. Sandstede and A. Scheel, Relative Morse indices, Fredholm indices, and group velocities, Discr. Cont. Dynam. Syst., 20 (2008), 139-158. Google Scholar

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