Asymptotics of the principal eigenvalue for a linear time-periodic parabolic operator II: Small diffusion

Liu, Shuang; Lou, Yuan; Peng, Rui; Zhou, Maolin

doi:10.1090/tran/8364

Asymptotics of the principal eigenvalue for a linear time-periodic parabolic operator II: Small diffusion
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by Shuang Liu, Yuan Lou, Rui Peng and Maolin Zhou PDF

Trans. Amer. Math. Soc. 374 (2021), 4895-4930 Request permission

Abstract:

We investigate the effect of small diffusion on the principal eigenvalues of linear time-periodic parabolic operators with zero Neumann boundary conditions in one dimensional space. The asymptotic behaviors of the principal eigenvalues, as the diffusion coefficients tend to zero, are established for non-degenerate and degenerate spatial-temporally varying environments. A new finding is the dependence of these asymptotic behaviors on the periodic solutions of a specific ordinary differential equation induced by the drift. The proofs are based upon delicate constructions of super/sub-solutions and the applications of comparison principles.

References

Alex Beltramo and Peter Hess, On the principal eigenvalue of a periodic-parabolic operator, Comm. Partial Differential Equations 9 (1984), no. 9, 919–941. MR 749652, DOI 10.1080/03605308408820351
H. Berestycki and P.-L. Lions, Some applications of the method of super and subsolutions, Bifurcation and nonlinear eigenvalue problems (Proc., Session, Univ. Paris XIII, Villetaneuse, 1978) Lecture Notes in Math., vol. 782, Springer, Berlin, 1980, pp. 16–41. MR 572249
Robert Stephen Cantrell and Chris Cosner, Spatial ecology via reaction-diffusion equations, Wiley Series in Mathematical and Computational Biology, John Wiley & Sons, Ltd., Chichester, 2003. MR 2191264, DOI 10.1002/0470871296
Cécile Carrère and Grégoire Nadin, Influence of mutations in phenotypically-structured populations in time periodic environment, Discrete Contin. Dyn. Syst. Ser. B 25 (2020), no. 9, 3609–3630. MR 4128352, DOI 10.3934/dcdsb.2020075
Xinfu Chen and Yuan Lou, Principal eigenvalue and eigenfunctions of an elliptic operator with large advection and its application to a competition model, Indiana Univ. Math. J. 57 (2008), no. 2, 627–658. MR 2414330, DOI 10.1512/iumj.2008.57.3204
Xinfu Chen and Yuan Lou, Effects of diffusion and advection on the smallest eigenvalue of an elliptic operator and their applications, Indiana Univ. Math. J. 61 (2012), no. 1, 45–80. MR 3029392, DOI 10.1512/iumj.2012.61.4518
Allen Devinatz, Richard Ellis, and Avner Friedman, The asymptotic behavior of the first real eigenvalue of second order elliptic operators with a small parameter in the highest derivatives. II, Indiana Univ. Math. J. 23 (1973/74), 991–1011. MR 344709, DOI 10.1512/iumj.1974.23.23081
Susely Figueroa Iglesias and Sepideh Mirrahimi, Long time evolutionary dynamics of phenotypically structured populations in time-periodic environments, SIAM J. Math. Anal. 50 (2018), no. 5, 5537–5568. MR 3864685, DOI 10.1137/18M1175185
S. Figueroa Iglesias and S. Mirrahimi, Selection and mutation in a shifting and fluctuating environment, preprint, 2019, https://hal.archives-ouvertes.fr/hal-02320525.
Avner Friedman, The asymptotic behavior of the first real eigenvalue of a second order elliptic operator with a small parameter in the highest derivatives, Indiana Univ. Math. J. 22 (1972/73), 1005–1015. MR 320551, DOI 10.1512/iumj.1973.22.22084
M. I. Freidlin and A. D. Wentzell, Random perturbations of dynamical systems, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 260, Springer-Verlag, New York, 1984. Translated from the Russian by Joseph Szücs. MR 722136, DOI 10.1007/978-1-4684-0176-9
Peter Hess, Periodic-parabolic boundary value problems and positivity, Pitman Research Notes in Mathematics Series, vol. 247, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1991. MR 1100011
V. Hutson, W. Shen, and G. T. Vickers, Estimates for the principal spectrum point for certain time-dependent parabolic operators, Proc. Amer. Math. Soc. 129 (2001), no. 6, 1669–1679. MR 1814096, DOI 10.1090/S0002-9939-00-05808-1
V. Hutson, K. Mischaikow, and P. Poláčik, The evolution of dispersal rates in a heterogeneous time-periodic environment, J. Math. Biol. 43 (2001), no. 6, 501–533. MR 1874400, DOI 10.1007/s002850100106
King-Yeung Lam and Yuan Lou, Asymptotic behavior of the principal eigenvalue for cooperative elliptic systems and applications, J. Dynam. Differential Equations 28 (2016), no. 1, 29–48. MR 3455183, DOI 10.1007/s10884-015-9504-4
S. Liu and Y. Lou, A functional approach towards eigenvalue problems associated with incompressible flow, Discrete Contin. Dyn. Syst.-A, 40 (2020) 3715–3736.
Shuang Liu, Yuan Lou, Rui Peng, and Maolin Zhou, Monotonicity of the principal eigenvalue for a linear time-periodic parabolic operator, Proc. Amer. Math. Soc. 147 (2019), no. 12, 5291–5302. MR 4021088, DOI 10.1090/proc/14653
S. Liu, Y. Lou, R. Peng, M. Zhou, Asymptotics of the principal eigenvalue for a linear time-periodic parabolic operator I: Large advection, submitted.
Gregoire Nadin, The principal eigenvalue of a space-time periodic parabolic operator, Ann. Mat. Pura Appl. (4) 188 (2009), no. 2, 269–295. MR 2491804, DOI 10.1007/s10231-008-0075-4
Grégoire Nadin, Some dependence results between the spreading speed and the coefficients of the space-time periodic Fisher-KPP equation, European J. Appl. Math. 22 (2011), no. 2, 169–185. MR 2774781, DOI 10.1017/S0956792511000027
Rui Peng, Guanghui Zhang, and Maolin Zhou, Asymptotic behavior of the principal eigenvalue of a linear second order elliptic operator with small/large diffusion coefficient, SIAM J. Math. Anal. 51 (2019), no. 6, 4724–4753. MR 4033689, DOI 10.1137/18M1217577
Rui Peng and Xiao-Qiang Zhao, Effects of diffusion and advection on the principal eigenvalue of a periodic-parabolic problem with applications, Calc. Var. Partial Differential Equations 54 (2015), no. 2, 1611–1642. MR 3396426, DOI 10.1007/s00526-015-0838-x
Rui Peng and Maolin Zhou, Effects of large degenerate advection and boundary conditions on the principal eigenvalue and its eigenfunction of a linear second-order elliptic operator, Indiana Univ. Math. J. 67 (2018), no. 6, 2523–2568. MR 3900377, DOI 10.1512/iumj.2018.67.7547
A. D. Wentzell, On the asymptotic behavior of the first eigenvalue of a second order differential operator with small parameter in higher derivatives, Theory Probab. Appl., 20 (1975) 599–602.