Asymptotic Stability of Critical Pulled Fronts via Resolvent Expansions Near the Essential Spectrum

We study nonlinear stability of pulled fronts in scalar parabolic equations on the real line of arbitrary order, under conceptual assumptions on existence and spectral stability of fronts. In this general setting, we establish sharp algebraic decay rates and temporal asymptotics of perturbations to the front. Some of these results are known for the specific example of the Fisher-KPP equation, and our results can thus be viewed as establishing universality of some aspects of this simple model. We also give a precise description of how the spatial localization of perturbations to the front affects the temporal decay rate, across the full range of localizations for which asymptotic stability holds. Technically, our approach is based on a detailed study of the resolvent operator for the linearized problem, through which we obtain sharp linear time decay estimates that allow for a direct nonlinear analysis.

[1]  J. Eckmann,et al.  The non-linear stability of front solutions for parabolic partial differential equations , 1994 .

[2]  A. Scheel,et al.  Instability of spikes in the presence of conservation laws , 2010 .

[3]  Todd Kapitula,et al.  Spectral and Dynamical Stability of Nonlinear Waves , 2013 .

[4]  Todd Kapitula,et al.  STABILITY OF TRAVELING WAVES FOR NONCONVEX SCALAR VISCOUS CONSERVATION-LAWS , 1993 .

[5]  M. Avery,et al.  Spectral stability of the critical front in the extended Fisher-KPP equation , 2020, Zeitschrift für angewandte Mathematik und Physik.

[6]  T. Gallay,et al.  Local stability of critical fronts in nonlinear parabolic partial differential equations , 1994 .

[7]  David H. Sattinger.  Weighted norms for the stability of traveling waves , 1977 .

[8]  Matt Holzer,et al.  Anomalous spreading in a system of coupled Fisher-KPP equations , 2012, 1211.6129.

[9]  Arnd Scheel,et al.  Criteria for Pointwise Growth and Their Role in Invasion Processes , 2013, J. Nonlinear Sci..

[10]  K. P. Hadeler,et al.  Travelling fronts in nonlinear diffusion equations , 1975 .

[11]  K. Elder,et al.  Marginal stability analysis of the phase field crystal model in one spatial dimension , 2011 .

[12]  W. Saarloos.  Front propagation into unstable states , 2003, cond-mat/0308540.

[13]  Martin Grant,et al.  Modeling elasticity in crystal growth. , 2001, Physical review letters.

[14]  V. Rottschäfer,et al.  Existence and Stability of Traveling Fronts in the Extended Fisher-Kolmogorov Equation , 2001 .

[15]  Björn Sandstede,et al.  Evans function and blow-u[ methods in critical eienvalue problems , 2004 .

[16]  J. Bricmont,et al.  Stability of moving fronts in the Ginzburg-Landau equation , 1993, chao-dyn/9306009.

[17]  Daniel B. Henry.  Geometric Theory of Semilinear Parabolic Equations , 1989 .

[18]  Arnd Scheel,et al.  Pattern‐forming fronts in a Swift–Hohenberg equation with directional quenching — parallel and oblique stripes , 2017, J. Lond. Math. Soc..

[19]  A. Lunardi.  Analytic Semigroups and Optimal Regularity in Parabolic Problems , 2003 .

[20]  Kenneth J. Palmer,et al.  Exponential dichotomies and Fredholm operators , 1988 .

[21]  Matt Holzer,et al.  Asymptotic stability of the critical Fisher–KPP front using pointwise estimates , 2017, Zeitschrift für angewandte Mathematik und Physik.

[22]  Antti Kupiainen,et al.  Renormalization Group and the Ginzburg-Landau equation , 1992 .

[23]  Björn Sandstede,et al.  Relative Morse indices, Fredholm indices, and group velocities , 2007 .

[24]  K. Lau.  On the nonlinear diffusion equation of Kolmogorov, Petrovsky, and Piscounov , 1985 .

[25]  D. Aronson,et al.  Multidimensional nonlinear di u-sion arising in population genetics , 1978 .

[26]  Björn Sandstede,et al.  Absolute and Convective Instabilities of Waves on Unbounded and Large Bounded Domains , 2022 .

[27]  A. Scheel,et al.  Diffusive stability of oscillations in reaction-diffusion systems , 2008, 0806.4915.

[28]  Kenneth J. Palmer,et al.  Exponential dichotomies and transversal homoclinic points , 1984 .

[29]  Matt Holzer,et al.  Linear spreading speeds from nonlinear resonant interaction , 2016, 1609.05757.

[30]  David H. Sattinger,et al.  On the stability of waves of nonlinear parabolic systems , 1976 .

[31]  Lenya Ryzhik,et al.  A short proof of the logarithmic Bramson correction in Fisher-KPP equations , 2013, Networks Heterog. Media.

[32]  On the Stability of Traveling Waves in Weighted L∞ Spaces , 1994 .

[33]  Maury Bramson,et al.  Maximal displacement of branching brownian motion , 1978 .

[34]  M. Bramson.  Convergence of solutions of the Kolmogorov equation to travelling waves , 1983 .

[35]  Bernold Fiedler,et al.  Spatio-Temporal Dynamics of Reaction-Diffusion Patterns , 2003 .

[36]  Arjen Doelman,et al.  On the transition from the Ginzburg-Landau equation to the extended Fisher-Kolmogorov equation , 1998 .

[37]  Matt Holzer,et al.  Asymptotic stability of the critical pulled front in a Lotka-Volterra competition model , 2019, Journal of Differential Equations.

[38]  Tosio Kato.  Perturbation theory for linear operators , 1966 .

[39]  Klaus Kirchgässner,et al.  On the nonlinear dynamics of travelling fronts , 1992 .

[40]  A. Jensen,et al.  Schrödinger operators on the half line: Resolvent expansions and the Fermi golden rule at thresholds , 2006, 0707.2146.