The hair-trigger effect for a class of nonlocal nonlinear equations. (English) Zbl 1390.35151
Summary: We prove the hair-trigger effect for a class of nonlocal nonlinear evolution equations on \(\mathbb{R}^d\) which have only two constant stationary solutions, 0 and \(\theta > 0\). The effect consists in that the solution with an initial condition non identical to zero converges (when time goes to \(\infty\)) to {\(\theta\)} locally uniformly in \(\mathbb{R}^d\). We also find sufficient conditions for existence, uniqueness and comparison principle in the considered equations.
MSC:
| 35K57 | Reaction-diffusion equations |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 47G20 | Integro-differential operators |
| 45G10 | Other nonlinear integral equations |
Keywords:
hair-trigger effect; nonlocal diffusion; reaction-diffusion equation; monostable equation; long-time behaviour; integral equation; nonlocal nonlinearityReferences:
| [1] | Aguerrea M, Gomez C and Trofimchuk S 2012 On uniqueness of semi-wavefronts Math. Ann.354 73-109 · Zbl 1311.45008 |
| [2] | Alfaro M 2017 Fujita blow up phenomena and hair trigger effect: the role of dispersal tails Ann. Inst. Henri Poincaré Anal. Non Linéaire 34 1309-27 · Zbl 1379.35037 |
| [3] | Alfaro M and Coville J 2017 Propagation phenomena in monostable integro-differential equations: acceleration or not? J. Differ. Equ.263 5727-58 · Zbl 1409.35204 |
| [4] | Andreu-Vaillo F, Mazón J M, Rossi J D and Toledo-Melero J J 2010 Nonlocal Diffusion Problems(Mathematical Surveys and Monographs vol 165) (Providence, RI: American Mathematical Society) · Zbl 1214.45002 |
| [5] | Aronson D G 1977 The asymptotic speed of propagation of a simple epidemic Nonlinear Diffusion NSF-CBMS Regional Conf. Nonlinear Diffusion Equations(Univ. Houston, Houston, TX, 1976) (London: Pitman) pp 1-23 · Zbl 0361.35011 |
| [6] | Aronson D G and Weinberger H F 1975 Nonlinear diffusion in population genetics, combustion and nerve pulse propagation Partial Differential Equations and Related Topics (Program, Tulane Univ., New Orleans, LA, 1974)(Lecture Notes in Mathematics vol 446) (Berlin: Springer) pp 5-49 · Zbl 0325.35050 |
| [7] | Aronson D G and Weinberger H F 1978 Multidimensional nonlinear diffusion arising in population genetics Adv. Math.30 33-76 · Zbl 0407.92014 |
| [8] | Aydogmus O 2018 Phase transitions in a logistic metapopulation model with nonlocal interactions Bull. Math. Biol.80 228-53 · Zbl 1385.92037 |
| [9] | Brändle C, Chasseigne E and Ferreira R 2011 Unbounded solutions of the nonlocal heat equation Commun. Pure Appl. Anal.10 1663-86 · Zbl 1233.35044 |
| [10] | Berestycki H, Coville J and Vo H-H 2016 Persistence criteria for populations with non-local dispersion J. Math. Biol.72 1693-745 · Zbl 1346.35202 |
| [11] | Bolker B and Pacala S W 1997 Using moment equations to understand stochastically driven spatial pattern formation in ecological systems Theor. Popul. Biol.52 179-97 · Zbl 0890.92020 |
| [12] | Bolker B and Pacala S W 1999 Spatial moment equations for plant competitions: understanding spatial strategies and the advantages of short dispersal Am. Nat.153 575-602 |
| [13] | Bonnefon O, Coville J, Garnier J and Roques L 2014 Inside dynamics of solutions of integro-differential equations Discrete Contin. Dyn. Syst. B 19 3057-85 · Zbl 1310.35235 |
| [14] | Clark J, Lewis M, McLachlan J and HilleRisLambers J 2003 Estimating population spread: what can we forecast and how well? Ecology 84 1979-88 |
| [15] | Cousens R, Dytham C and Law R 2008 Dispersal in Plants: a Population Perspective (Oxford: Oxford University Press) p 240 |
| [16] | Coville J 2007 Maximum principles, sliding techniques and applications to nonlocal equations Electron. J. Differ. Equ.68 1-23 (https://ejde.math.txstate.edu/Volumes/2007/68/coville.pdf) · Zbl 1137.35321 |
| [17] | Coville J, Dávila J and Martínez S 2008 Nonlocal anisotropic dispersal with monostable nonlinearity J. Differ. Equ.244 3080-118 · Zbl 1148.45011 |
| [18] | Coville J, Dávila J and Martínez S 2013 Pulsating fronts for nonlocal dispersion and KPP nonlinearity Ann. Inst. Henri Poincaré Anal. Non Linéaire 30 179-223 · Zbl 1288.45007 |
| [19] | Coville J and Dupaigne L 2007 On a non-local equation arising in population dynamics Proc. R. Soc. Edinburgh A 137 727-55 · Zbl 1133.35056 |
| [20] | Crandall M G, Ishii H and Lions P-L 1992 User’s guide to viscosity solutions of second order partial differential equations Bull. Am. Math. Soc.27 1-67 · Zbl 0755.35015 |
| [21] | Dancer E N and Du Y 2003 Some remarks on Liouville type results for quasilinear elliptic equations Proc. Am. Math. Soc.131 1891-9 · Zbl 1076.35038 |
| [22] | Dieckmann U and Law R 2000 Relaxation projections and the method of moments The Geometry of Ecological Interactions (Cambridge: Cambridge University Press) pp 412-55 |
| [23] | Diekmann O 1978 On a nonlinear integral equation arising in mathematical epidemiology Differential Equations and Applications (Proc. 3rd Scheveningen Conf. (Scheveningen, 1977))(North-Holland Mathematics Studies vol 31) (Amsterdam: North-Holland) pp 133-40 · Zbl 0386.45006 |
| [24] | Diekmann O 1978 Thresholds and travelling waves for the geographical spread of infection J. Math. Biol.6 109-30 · Zbl 0415.92020 |
| [25] | Durrett R 1988 Crabgrass, measles and gypsy moths: an introduction to modern probability Bull. Am. Math. Soc.18 117-43 · Zbl 0653.60095 |
| [26] | Fife P C 1979 Mathematical Aspects of Reacting and Diffusing Systems(Lecture Notes in Biomathematics vol 28) (Berlin: Springer) · Zbl 0403.92004 |
| [27] | Finkelshtein D, Kondratiev Y and Kutoviy O 2012 Semigroup approach to birth-and-death stochastic dynamics in continuum J. Funct. Anal.262 1274-308 · Zbl 1250.47090 |
| [28] | Finkelshtein D, Kondratiev Y and Tkachov P 2015 Traveling waves and long-time behavior in a doubly nonlocal Fisher-KPP equation (arXiv:1508.02215) |
| [29] | Finkelshtein D, Kondratiev Y and Tkachov P 2016 Accelerated front propagation for monostable equations with nonlocal diffusion (arXiv:1611.09329) |
| [30] | Finkelshtein D and Tkachov P 2017 Accelerated nonlocal nonsymmetric dispersion for monostable equations on the real line Appl. Anal. (https://doi.org/10.1080/00036811.2017.1400537) · Zbl 1407.35027 |
| [31] | Finkelshtein D and Tkachov P 2018 Kesten’s bound for sub-exponential densities on the real line and its multi-dimensional analogues Adv. Appl. Probab. to appear · Zbl 1390.35151 |
| [32] | Fisher R 1937 The wave of advance of advantageous genes Ann. Eugenics 7 355-69 · JFM 63.1111.04 |
| [33] | Fournier N and Méléard S 2004 A microscopic probabilistic description of a locally regulated population and macroscopic approximations Ann. Appl. Probab.1 1880-919 · Zbl 1060.92055 |
| [34] | Garnier J 2011 Accelerating solutions in integro-differential equations SIAM J. Math. Anal.43 1955-74 · Zbl 1232.47058 |
| [35] | Giga Y, Goto S, Ishii H and Sato M-H 1991 Comparison principle and convexity preserving properties for singular degenerate parabolic equations on unbounded domains Indiana Univ. Math. J.40 443-70 · Zbl 0836.35009 |
| [36] | Grigoryan A, Kondratiev Yu, Piatnitski A and Zhizhina E 2017 Point-wise estimates for nonlocal heat kernel of convolution type operators (arXiv:1707.06709) |
| [37] | Hagan P S 1981 The instability of nonmonotonic wave solutions of parabolic equations Stud. Appl. Math.64 57-88 · Zbl 0452.35068 |
| [38] | Hamel F and Ryzhik L 2014 On the nonlocal Fisher-KPP equation: steady states, spreading speed and global bounds Nonlinearity 27 2735 · Zbl 1317.35122 |
| [39] | Kanel’ J I 1964 Stabilization of the solutions of the equations of combustion theory with finite initial functions Mat. Sb.65 398-413 |
| [40] | Kendall D G 1965 Mathematical models of the spread of infection Mathematics and Computer Science in Biology and Medicine (London: H.M. Stationery Off.) pp 213-24 |
| [41] | Kolmogorov A N, Petrovsky I G and Piskunov N S 1937 Étude de l’équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique Bull. Univ. État Moscou Sér. Inter. A 1 1-26 |
| [42] | Levin S, Muller-Landau H, Nathan R and Chave J 2003 The ecology and evolution of seed dispersal: a theoretical perspective Ann. Rev. Ecol. Evol. Systematics 34 575-604 |
| [43] | Lutscher F, Pachepsky E and Lewis M A 2005 The effect of dispersal patterns on stream populations SIAM J. Appl. Math.65 1305-27 · Zbl 1068.92002 |
| [44] | Mollison D 1972 Possible velocities for a simple epidemic Adv. Appl. Probab.4 233-57 · Zbl 0251.92012 |
| [45] | Mollison D 1972 The rate of spatial propagation of simple epidemics Proc. of the 6th Berkeley Symp. on Mathematical Statistics and Probability vol III: Probability Theory(Univ. California Berkeley, CA, 1970/1971) (Berkeley, CA: University California Press) pp 579-614 · Zbl 0266.92008 |
| [46] | Murray J D 2003 Mathematical Biology. II(Interdisciplinary Applied Mathematics vol 18) 3rd edn (New York: Springer) (Spatial models and biomedical applications) · Zbl 1006.92002 |
| [47] | Ninomiya H, Tanaka Y and Yamamoto H 2017 Reaction, diffusion and non-local interaction J. Math. Biol.75 1203-33 · Zbl 1386.35209 |
| [48] | Pazy A 1983 Semigroups of Linear Operators and Applications to Partial Differential Equations(Applied Mathematical Sciences vol 44) (New York: Springer) · Zbl 0516.47023 |
| [49] | Perthame B and Souganidis P E 2005 Front propagation for a jump process model arising in spatial ecology Discrete Contin. Dyn. Syst.13 1235-46 · Zbl 1085.35017 |
| [50] | Raghib M, Hill N A and Dieckmann U 2011 A multiscale maximum entropy moment closure for locally regulated space-time point process models of population dynamics J. Math. Biol.62 605-53 · Zbl 1232.92074 |
| [51] | Schumacher K 1980 Travelling-front solutions for integro-differential equations. I J. Reine Angew. Math.316 54-70 · Zbl 0419.45004 |
| [52] | Shigesada N and Kawasaki K 1997 Biological Invasions: Theory and Practice (Oxford: Oxford University Press) |
| [53] | Sun Y-J, Li W-T and Wang Z-C 2011 Traveling waves for a nonlocal anisotropic dispersal equation with monostable nonlinearity Nonlinear Anal.74 814-26 · Zbl 1211.35068 |
| [54] | Weinberger H 1982 Long-time behavior of a class of biological models SIAM J. Math. Anal.13 353-96 · Zbl 0529.92010 |
| [55] | Yagisita H 2009 Existence and nonexistence of traveling waves for a nonlocal monostable equation Publ. Res. Inst. Math. Sci.45 925-53 · Zbl 1191.35093 |
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