American Institute of Mathematical Sciences

Advanced
October  2004, 10(4): 941-964. doi: 10.3934/dcds.2004.10.941

Evans function and blow-up methods in critical eigenvalue problems

Björn Sandstede 1, and  Arnd Scheel 2,

1. 

Department of Mathematics, The Ohio State University, Columbus, OH 43210
2. 

Department of Mathematics, University of Minnesota, Minneapolis, MN 55455, United States

Received  November 2002 Revised  September 2003 Published  March 2004

Contact defects are one of several types of defects that arise generically in oscillatory media modelled by reaction-diffusion systems. An interesting property of these defects is that the asymptotic spatial wavenumber is approached only with algebraic order O$(1/x)$ (the associated phase diverges logarithmically). The essential spectrum of the PDE linearization about a contact defect always has a branch point at the origin. We show that the Evans function can be extended across this branch point and discuss the smoothness properties of the extension. The construction utilizes blow-up techniques and is quite general in nature. We also comment on known relations between roots of the Evans function and the temporal asymptotics of Green's functions, and discuss applications to algebraically decaying solitons.
Keywords: radial Laplacian., Evans function, algebraic decay.
Mathematics Subject Classification: 35P05, 35B35, 35K5.
Citation: Björn Sandstede, Arnd Scheel. Evans function and blow-up methods in critical eigenvalue problems. Discrete & Continuous Dynamical Systems - A, 2004, 10 (4) : 941-964. doi: 10.3934/dcds.2004.10.941
[1]

Todd Kapitula, Björn Sandstede. Eigenvalues and resonances using the Evans function. Discrete & Continuous Dynamical Systems - A, 2004, 10 (4) : 857-869. doi: 10.3934/dcds.2004.10.857
[2]

Yuri Latushkin, Alim Sukhtayev. The Evans function and the Weyl-Titchmarsh function. Discrete & Continuous Dynamical Systems - S, 2012, 5 (5) : 939-970. doi: 10.3934/dcdss.2012.5.939
[3]

Peter Howard, K. Zumbrun. The Evans function and stability criteria for degenerate viscous shock waves. Discrete & Continuous Dynamical Systems - A, 2004, 10 (4) : 837-855. doi: 10.3934/dcds.2004.10.837
[4]

Ramon Plaza, K. Zumbrun. An Evans function approach to spectral stability of small-amplitude shock profiles. Discrete & Continuous Dynamical Systems - A, 2004, 10 (4) : 885-924. doi: 10.3934/dcds.2004.10.885
[5]

Jan Burczak, P. Kaplický. Evolutionary, symmetric $p$-Laplacian. Interior regularity of time derivatives and its consequences. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2401-2445. doi: 10.3934/cpaa.2016042
[6]

Martin D. Buhmann, Slawomir Dinew. Limits of radial basis function interpolants. Communications on Pure & Applied Analysis, 2007, 6 (3) : 569-585. doi: 10.3934/cpaa.2007.6.569
[7]

Peter Giesl. Construction of a global Lyapunov function using radial basis functions with a single operator. Discrete & Continuous Dynamical Systems - B, 2007, 7 (1) : 101-124. doi: 10.3934/dcdsb.2007.7.101
[8]

Tomás Sanz-Perela. Regularity of radial stable solutions to semilinear elliptic equations for the fractional Laplacian. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2547-2575. doi: 10.3934/cpaa.2018121
[9]

Elisa Calzolari, Roberta Filippucci, Patrizia Pucci. Existence of radial solutions for the $p$-Laplacian elliptic equations with weights. Discrete & Continuous Dynamical Systems - A, 2006, 15 (2) : 447-479. doi: 10.3934/dcds.2006.15.447
[10]

Rossella Bartolo, Anna Maria Candela, Addolorata Salvatore. Infinitely many radial solutions of a non--homogeneous $p$--Laplacian problem. Conference Publications, 2013, 2013 (special) : 51-59. doi: 10.3934/proc.2013.2013.51
[11]

Xi Wang, Zuhan Liu, Ling Zhou. Asymptotic decay for the classical solution of the chemotaxis system with fractional Laplacian in high dimensions. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 4003-4020. doi: 10.3934/dcdsb.2018121
[12]

Mohammad A. Rammaha, Daniel Toundykov, Zahava Wilstein. Global existence and decay of energy for a nonlinear wave equation with $p$-Laplacian damping. Discrete & Continuous Dynamical Systems - A, 2012, 32 (12) : 4361-4390. doi: 10.3934/dcds.2012.32.4361
[13]

Yaping Wu, Xiuxia Xing, Qixiao Ye. Stability of travelling waves with algebraic decay for $n$-degree Fisher-type equations. Discrete & Continuous Dynamical Systems - A, 2006, 16 (1) : 47-66. doi: 10.3934/dcds.2006.16.47
[14]

Lihua Min, Xiaoping Yang. Finite speed of propagation and algebraic time decay of solutions to a generalized thin film equation. Communications on Pure & Applied Analysis, 2014, 13 (2) : 543-566. doi: 10.3934/cpaa.2014.13.543
[15]

Trad Alotaibi, D. D. Hai, R. Shivaji. Existence and nonexistence of positive radial solutions for a class of $p$-Laplacian superlinear problems with nonlinear boundary conditions. Communications on Pure & Applied Analysis, 2020, 19 (9) : 4655-4666. doi: 10.3934/cpaa.2020131
[16]

Raúl Ferreira, Julio D. Rossi. Decay estimates for a nonlocal $p-$Laplacian evolution problem with mixed boundary conditions. Discrete & Continuous Dynamical Systems - A, 2015, 35 (4) : 1469-1478. doi: 10.3934/dcds.2015.35.1469
[17]

Olga Bernardi, Matteo Dalla Riva. Analytic dependence on parameters for Evans' approximated Weak KAM solutions. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 4625-4636. doi: 10.3934/dcds.2017199
[18]

Laura Luzzi, Ghaya Rekaya-Ben Othman, Jean-Claude Belfiore. Algebraic reduction for the Golden Code. Advances in Mathematics of Communications, 2012, 6 (1) : 1-26. doi: 10.3934/amc.2012.6.1
[19]

Javier de la Cruz, Michael Kiermaier, Alfred Wassermann, Wolfgang Willems. Algebraic structures of MRD codes. Advances in Mathematics of Communications, 2016, 10 (3) : 499-510. doi: 10.3934/amc.2016021
[20]

Ville Salo, Ilkka Törmä. Recoding Lie algebraic subshifts. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020307

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (35)
  • HTML views (0)
  • Cited by (28)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences