• Previous Article
    Positive solutions for perturbations of the Robin eigenvalue problem plus an indefinite potential
  • DCDS Home
  • This Issue
  • Next Article
    Existence and uniqueness of positive solutions for a class of logistic type elliptic equations in $\mathbb{R}^N$ involving fractional Laplacian
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.

[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.

[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.

[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.

[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.

[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.

[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.

[8]

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

[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.

[10]

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

[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.

[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.

[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.

[14]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[25]

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

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.

[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.

[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.

[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.

[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.

[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.

[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.

[8]

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

[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.

[10]

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

[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.

[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.

[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.

[14]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[25]

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

[1]

Heinz Schättler, Urszula Ledzewicz. Lyapunov-Schmidt reduction for optimal control problems. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 2201-2223. doi: 10.3934/dcdsb.2012.17.2201

[2]

Christian Pötzsche. Nonautonomous bifurcation of bounded solutions I: A Lyapunov-Schmidt approach. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 739-776. doi: 10.3934/dcdsb.2010.14.739

[3]

Jingzhi Li, Jun Zou. A direct sampling method for inverse scattering using far-field data. Inverse Problems & Imaging, 2013, 7 (3) : 757-775. doi: 10.3934/ipi.2013.7.757

[4]

Qi Wang, Yanren Hou. Determining an obstacle by far-field data measured at a few spots. Inverse Problems & Imaging, 2015, 9 (2) : 591-600. doi: 10.3934/ipi.2015.9.591

[5]

Huai-An Diao, Peijun Li, Xiaokai Yuan. Inverse elastic surface scattering with far-field data. Inverse Problems & Imaging, 2019, 13 (4) : 721-744. doi: 10.3934/ipi.2019033

[6]

Yuqian Zhou, Qian Liu. Reduction and bifurcation of traveling waves of the KdV-Burgers-Kuramoto equation. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 2057-2071. doi: 10.3934/dcdsb.2016036

[7]

Maria Schonbek, Tomas Schonbek. Moments and lower bounds in the far-field of solutions to quasi-geostrophic flows. Discrete & Continuous Dynamical Systems - A, 2005, 13 (5) : 1277-1304. doi: 10.3934/dcds.2005.13.1277

[8]

Heping Dong, Deyue Zhang, Yukun Guo. A reference ball based iterative algorithm for imaging acoustic obstacle from phaseless far-field data. Inverse Problems & Imaging, 2019, 13 (1) : 177-195. doi: 10.3934/ipi.2019010

[9]

Afaf Bouharguane. On the instability of a nonlocal conservation law. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 419-426. doi: 10.3934/dcdss.2012.5.419

[10]

Roland Griesmaier, Nuutti Hyvönen, Otto Seiskari. A note on analyticity properties of far field patterns. Inverse Problems & Imaging, 2013, 7 (2) : 491-498. doi: 10.3934/ipi.2013.7.491

[11]

Alberto Bressan, Graziano Guerra. Shift-differentiabilitiy of the flow generated by a conservation law. Discrete & Continuous Dynamical Systems - A, 1997, 3 (1) : 35-58. doi: 10.3934/dcds.1997.3.35

[12]

Alberto Bressan, Khai T. Nguyen. Conservation law models for traffic flow on a network of roads. Networks & Heterogeneous Media, 2015, 10 (2) : 255-293. doi: 10.3934/nhm.2015.10.255

[13]

Robert I. McLachlan, G. R. W. Quispel. Discrete gradient methods have an energy conservation law. Discrete & Continuous Dynamical Systems - A, 2014, 34 (3) : 1099-1104. doi: 10.3934/dcds.2014.34.1099

[14]

Julien Jimenez. Scalar conservation law with discontinuous flux in a bounded domain. Conference Publications, 2007, 2007 (Special) : 520-530. doi: 10.3934/proc.2007.2007.520

[15]

Xiao-Biao Lin, Stephen Schecter. Traveling waves and shock waves. Discrete & Continuous Dynamical Systems - A, 2004, 10 (4) : i-ii. doi: 10.3934/dcds.2004.10.4i

[16]

Zhigang Wang. Vanishing viscosity limit of the rotating shallow water equations with far field vacuum. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 311-328. doi: 10.3934/dcds.2018015

[17]

Olha Ivanyshyn. Shape reconstruction of acoustic obstacles from the modulus of the far field pattern. Inverse Problems & Imaging, 2007, 1 (4) : 609-622. doi: 10.3934/ipi.2007.1.609

[18]

Huey-Er Lin, Jian-Guo Liu, Wen-Qing Xu. Effects of small viscosity and far field boundary conditions for hyperbolic systems. Communications on Pure & Applied Analysis, 2004, 3 (2) : 267-290. doi: 10.3934/cpaa.2004.3.267

[19]

Giovanni Alessandrini, Eva Sincich, Sergio Vessella. Stable determination of surface impedance on a rough obstacle by far field data. Inverse Problems & Imaging, 2013, 7 (2) : 341-351. doi: 10.3934/ipi.2013.7.341

[20]

Georges Bastin, B. Haut, Jean-Michel Coron, Brigitte d'Andréa-Novel. Lyapunov stability analysis of networks of scalar conservation laws. Networks & Heterogeneous Media, 2007, 2 (4) : 751-759. doi: 10.3934/nhm.2007.2.751

2017 Impact Factor: 1.179

Metrics

  • PDF downloads (12)
  • HTML views (2)
  • Cited by (0)

Other articles
by authors

[Back to Top]