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Pointwise stability of reaction diffusion fronts

Part of: Parabolic equations and systems

Published online by Cambridge University Press:  25 March 2019

Yingwei Li
Yingwei Li*
Affiliation:
Indiana University, 831 East Third Street, Rawles Hall, Bloomington, Indiana47405, USA ([email protected])
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Abstract

Using pointwise semigroup techniques, we establish sharp rates of decay in space and time of a perturbed reaction diffusion front to its time-asymptotic limit. This recovers results of Sattinger, Henry and others of time-exponential convergence in weighted Lp and Sobolev norms, while capturing the new feature of spatial diffusion at Gaussian rate. Novel features of the argument are a pointwise Green function decomposition reconciling spectral decomposition and short-time Nash-Aronson estimates and an instantaneous tracking scheme similar to that used in the study of stability of viscous shock waves.

Keywords

pointwise semigroup methodreaction diffusion equationnonlinear stability

MSC classification

Primary: 35K57: Reaction-diffusion equations
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 5 , October 2020 , pp. 2216 - 2254
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Copyright
Copyright © Royal Society of Edinburgh 2019

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References

1Aronson, D. G.. Bounds for the fundamental solution of a parabolic equation. Bull. Amer. Math. Soc. 73 (1967), 890–896.CrossRefGoogle Scholar
2Cronin, J.. Ordinary differential equations: introduction and qualitative theory, Third Edition (Boca Raton: Chapman & Hall/CRC Pure and Applied Mathematics) (2008).Google Scholar
3Gardner, R. and Zumbrun, K.. The gap lemma and geometric criteria for instability of viscous shock profiles. Comm. Pure Appl. Math. 51 (1998), 797–855.3.0.CO;2-1>CrossRefGoogle Scholar
4Henry, D.. Geometric theory of semilinear parabolic equations. Lecture Notes in Mathematics 840 (New York: Springer-Verlag, 1981).Google Scholar
5Johnson, M. and Zumbrun, K.. Nonlinear stability of spatially-periodic traveling-wave solutions of systems of reaction diffusion equations. Annales de l'Institut Henri Poincaré - Analyse non linéaire 28 (2011), 471–483.CrossRefGoogle Scholar
6Johnson, M., Noble, P., Rodrigues, L. M. and Zumbrun, K.. Nonlocalized modulation of periodic reaction diffusion waves: nonlinear stability. Arch. Ration. Mech. Anal. 207 (2013), 693–715.CrossRefGoogle Scholar
7Li, Y.. Scalar Green function bounds for instantaneous shock location and one-dimensional stability of viscous shock waves. Quart. Appl. Math. 74 (2016), 499–538.CrossRefGoogle Scholar
8Mascia, C. and Zumbrun, K.. Stability of large-amplitude viscous shock profiles of hyperbolic-parabolic systems. Arch. Ration. Mech. Anal. 172 (2004), 93–131.CrossRefGoogle Scholar
9Nash, J.. Continuity of solutions of parabolic and elliptic equations. Amer. J. Math. 80 (1958), 931–954.CrossRefGoogle Scholar
10Sattinger, D. H.. On the stability of waves of nonlinear parabolic systems. Adv. Math. 22 (1976), 312–355.CrossRefGoogle Scholar
11Texier, B. and Zumbrun, K.. Relative Poincare-Hopf bifurcation and galloping instability of traveling waves. Meth. Appl. Anal. 12 (2005), 349–380.Google Scholar
12Zumbrun, K.. Instantaneous shock location and one-dimensional nonlinear stability of viscous shock waves. Quart. Appl. Math. 69 (2011), 177–202.CrossRefGoogle Scholar
13Zumbrun, K. and Howard, P.. Pointwise semigroup methods and stability of viscous shock waves. Indiana. Univ. Math. J. 47 (1998), 741–871.Google Scholar