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Center manifolds for semilinear equations with non-dense domain and applications to Hopf bifurcation in age structured models

Author(s): Pierre Magal; Shigui Ruan
Journal: Memoirs of the AMS 202 (2009), no. 951.
MSC (2000): Primary 35K90, 37L10; Secondary 92D25
Posted: July 22, 2009
MathSciNet review: 2559965
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Abstract: Several types of differential equations, such as delay differential equations, age-structure models in population dynamics, evolution equations with boundary conditions, can be written as semilinear Cauchy problems with an operator which is not densely defined in its domain. The goal of this paper is to develop a center manifold theory for semilinear Cauchy problems with non-dense domain. Using Liapunov-Perron method and following the techniques of Vanderbauwhede et al. in treating infinite dimensional systems, we study the existence and smoothness of center manifolds for semilinear Cauchy problems with non-dense domain. As an application, we use the center manifold theorem to establish a Hopf bifurcation theorem for age structured models.

References:

1.: R. M. Anderson, Discussion: The Kermack-McKendrick epidemic threshold theorem, Bull. Math. Biol. 53 (1991), 3-32.
2.: V. Andreasen, Instability in an SIR-model with age-dependent susceptibility, in ``Mathematical Population Dynamics'', Vol. 1, eds. by O. Arino, D. Axelrod, M. Kimmel and M. Langlais, Wuerz Publishing, Winnipeg, 1995, pp. 3-14.
3.: W. Arendt, Resolvent positive operators, Proc. London Math. Soc. 54 (1987), 321-349. MR 872810 (88c:47074)
4.: W. Arendt, Vector valued Laplace transforms and Cauchy problems, Israel J. Math. 59 (1987), 327-352. MR 920499 (89a:47064)
5.: W. Arendt, C. J. K. Batty, M. Hieber, and F. Neubrander, Vector-Valued Laplace Transforms and Cauchy Problems, Birkhä user, Basel, 2001. MR 1886588 (2003g:47072)
6.: A. Avez, Calcul différentiel, Masson, Paris, 1983. MR 700398 (85c:58001)
7.: J. M. Ball, Saddle point analysis for an ordinary differential equation in a Banach space and an application to dynamic buckling of a beam, in ``Nonlinear Elasticity'', ed. by R. W. Dickey, Academic Press, New York, 1973, pp. 93-160.
8.: P. W. Bates and C. K. R. T. Jones, Invariant manifolds for semilinear partial differential equations, Dynamics Reported, ed. by U. Kirchgraber and H. O. Walther, Vol. 2, John Wiley & Sons, 1989, pp. 1-38. MR 1000974 (90g:58017)
9.: P. W. Bates, K. Lu and C. Zeng, Existence and persistence of invariant manifolds for semi flows in Banach space, Mem. Amer. Math. Soc. 135 (1998), No. 645. MR 1445489 (99b:58210)
10.: S. Bertoni, Periodic solutions for non-linear equations of structure populations, J. Math. Anal. Appl. 220 (1998), 250-267. MR 1613952 (98m:35209)
11.: F. E. Browder, On the spectral theory of elliptic differential operators, Math. Ann. 142 (1961), 22-130. MR 0209909 (35:804)
12.: J. Carr, Applications of Centre Manifold Theory, Springer-Verlag, New York, 1981. MR 635782 (83g:34039)
13.: C. Castillo-Chavez, H. W. Hethcote, V. Andreasen, S. A. Levin, and W. M. Liu, Epidemiological models with age structure, proportionate mixing, and cross-immunity, J. Math. Biol. 27 (1989), 233-258. MR 1000090 (90g:92056)
14.: Y. Cha, M. Iannelli and F. A. Milner, Stability change of an epidemic model, Dynam. Syst. Appl. 9 (2000), 361-376. MR 1844637 (2002f:92028)
15.: C. Chicone and Y. Latushkin, Center manifolds for infinite dimensional nonautonomous differential equations, J. Differential Equations 141 (1997), 356-399. MR 1488358 (2000i:34117)
16.: S.-N. Chow and J. K. Hale, Methods of Bifurcation Theory, Springer-Verlag, New York, 1982. MR 660633 (84e:58019)
17.: C.-N. Chow, C. Li and D. Wang, Normal Forms and Bifurcation of Planar Vector Fields, Cambridge Univ. Press, Cambridge, 1994. MR 1290117 (95i:58161)
18.: C.-N. Chow, X.-B. Lin and K. Lu, Smooth invariant foliations in infinite dimensional spaces, J. Differential Equations 94 (1991), 266-291. MR 1137616 (92k:58210)
19.: S.-N. Chow, W. Liu, and Y. Yi, Center manifolds for smooth invariant manifolds, Trans. Amer. Math. Soc. 352 (2000), 5179-5211. MR 1650077 (2001b:37032)
20.: S.-N. Chow, W. Liu and Y. Yi, Center manifolds for invariant sets, J. Differential Equations 168 (2000), 355-385. MR 1808454 (2002a:37028)
21.: S.-N. Chow and K. Lu, Invariant manifolds for flows in Banach spaces, J. Differential Equations 74 (1988), 285-317. MR 952900 (89h:58163)
22.: S.-N. Chow and K. Lu, centre unstable maniflds, Proc. Royal Soc. Edinburgh 108A (1988), 303-320. MR 943805 (90a:58148)
23.: S.-N. Chow and K. Lu, Invariant manifolds and foliations for quasiperiodic systems, J. Differential Equations 117 (1995), 1-27. MR 1320181 (96b:34064)
24.: S. N. Chow and Y. Yi, Center manifold and stability for skew-product flows, J. Dynam. Differential Equations 6 (1994), 543-582. MR 1303274 (95k:58142)
25.: J. M. Cushing, Model stability and instability in age structured populations, J. Theoret. Biol. 86 (1980), 709-730. MR 596373 (81m:92039)
26.: J. M. Cushing, Bifurcation of time periodic solutions of the McKendrick equations with applications to population dynamics, Comput. Math. Appl. 9 (1983), 459-478. MR 702665 (84g:92028)
27.: J. M. Cushing, An Introduction to Structured Population Dynamics, SIAM, Philadelphia, 1998. MR 1636703 (99k:92024)
28.: E. M. C. D'Agata, P. Magal, D. Olivier, S. Ruan and G. F. Webb, Modeling antibiotic resistance in hospitals: The impact of minimizing treatment duration, J. Theoretical Biology 249 (2007), 487-499.
29.: E. M. C. D'Agata, P. Magal, S. Ruan and G. F. Webb, Asymptotic behavior in nosocomial epidemi models with antibiotic resistance, Differential Integral Equations 19 (2006), 573-600. MR 2235142 (2008a:35153)
30.: G. Da Prato and A. Lunardi, Stability, instability and center manifold theorem for fully nonlinear autonomous parabolic equations in Banach spaces, Arch. Rational Mech. Anal. 101 (1988), 115-141. MR 921935 (89e:35019)
31.: G. Da Prato and E. Sinestrari, Differential operators with non-dense domain, Ann. Scuola. Norm. Sup. Pisa Cl. Sci. 14 (1987), 285-344. MR 939631 (89f:47062)
32.: O. Diekmann, H. Heesterbeek and H. Metz, The legacy of Kermack and McKendrick, in ``Epidemic Models: Their Strcture and Relation to Data'', ed. by. D. Mollison, Cambridge University Press, Cambridge, 1995, pp. 95-115.
33.: O. Diekmann, P. Getto and M. Gyllenberg, Stability and bifurcation analysis of Volterra functional equations in the light of suns and stars, SIAM J. Math. Anal. 34 (2007), 1023-1069. MR 2368893 (2008j:45002)
34.: O. Diekmann and S. A. van Gils, Invariant manifold for Volterra integral equations of convolution type, J. Differential Equations 54 (1984), 139-180. MR 757290 (85h:45026)
35.: O. Diekmann and S. A. van Gils, The center manifold for delay equations in the light of suns and stars, in ``Singularity Theory and its Applications,'', Lect. Notes Math. Vol. 1463, Springer, Berlin, 1991, pp. 122-141. MR 1129051 (92m:47151)
36.: O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel, and H.-O. Walther, Delay Equations. Functional-, Complex-, and Nonlinear Analysis, Springer-Verlag, New York, 1995. MR 1345150 (97a:34001)
37.: P. Dolbeault, Analyse Complexe, Masson, Paris, 1990. MR 1059456 (91h:30001)
38.: A. Ducrot, Z. Liu and P. Magal, Essential growth rate for bounded linear perturbation of non densely defined Cauchy problems, J. Math. Anal. Appl. 341 (2008), 501-518. MR 2394101 (2008m:34130)
39.: A. Ducrot, Z. Liu and P. Magal, Projectors on the generalized eigenspaces for neutral functional differential equations in spaces, Can. J. Math. (to appear).
40.: N. Dunford and J. T. Schwartz, Linear Operator, Part I: General Theory, Interscience, New York, 1958. MR 1009162 (90g:47001a)
41.: K.-J. Engel and R. Nagel, One Parameter Semigroups for Linear Evolution Equations, Springer-Verlag, New York, 2000. MR 1721989 (2000i:47075)
42.: K. Ezzinbi and M. Adimy, The basic theory of abstract semilinear functional differential equations with non-dense domain, in ``Delay Differential Equations and Applications'', eds. by O. Arino, M. L. Hbid and E. Ait Dads, Springer, Berlin, 2006, pp. 347-397. MR 2337821
43.: T. Faria, W. Huang and J. Wu, Smoothness of center manifolds for maps and formal adjoints for semilinear FDES in general Banach spaces, SIAM J. Math. Anal. 34 (2002), 173-203. MR 1950831 (2003k:34146)
44.: N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, J. Differential Equations 31 (1979), 53-98. MR 524817 (80m:58032)
45.: Th. Gallay, A center-stable manifold theorem for differential equations in Banach spaces, Comm. Math. Phys. 152 (1993), 249-268. MR 1210168 (94i:34126)
46.: M. E. Gurtin and R. C. MacCamy, Nonlinear age-dependent population dynamics, Arch. Rational Mech. Anal. 54 (1974), 28l-300. MR 0354068 (50:6550)
47.: J. Hadamard, Sur l'iteration et les solutions asymptotiques des equations differentielles, Bull. Soc. Math. France 29 (1901), 224-228.
48.: J. K. Hale, Integral manifolds of perturbated differential equations, Ann. Math. 73 (1961), 496-531. MR 0123786 (23:A1108)
49.: J. K. Hale, Ordinary Differential Equations, 2nd Ed., Krieger Pub., Huntington, NY, 1980. MR 587488 (82e:34001)
50.: J. K. Hale, Flows on center manifolds for scalar functional differential equations, Proc. Roy. Soc. Edinburgh 101A (1985), 193-201. MR 824220 (87d:34117)
51.: J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993. MR 1243878 (94m:34169)
52.: B. D. Hassard, N. D. Kazarinoff and Y.-H. Wan, Theory and Applications of Hopf Bifurcaton, Cambridge Univ. Press, Cambridge, 1981. MR 603442 (82j:58089)
53.: K. P. Hadeler and K. Dietz, Nonlinear hyperbolic partial differential equations for the dynamics of parasite populations, Comput. Math. Appl. 9 (1983), 415-430. MR 702661 (84h:92031)
54.: D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lect. Notes Math. Vol. 840, Springer-Verlag, Berlin, 1981. MR 610244 (83j:35084)
55.: M. Hirsch, C. Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Math. Vol. 583, Springer-Verlag, New York, 1976. MR 0501173 (58:18595)
56.: A. Homburg, Global aspects of homoclinic bifurcations of vector fields, Mem. Amer. Math. Soc. 121 (1996), No. 578. MR 1327210 (96i:58125)
57.: F. Hoppensteadt, Mathematical Theories of Populations: Demographics, Genetics, and Epidemics, SIAM, Philadelphia, 1975. MR 0526771 (58:26164)
58.: H. J. Hupkes and S. M. Verduyn Lunel, Center manifold theory for functional differential equations of mixed type, J. Dynam. Differential Equations 19 (2007), 497-560. MR 2333418 (2008c:34156)
59.: M. Iannelli, Mathematical Theory of Age-Structured Population Dynamics, Appl. Math. Monographs C. N. R., Vol. 7, Giadini Editori e Stampatori, Pisa, 1994.
60.: H. Inaba, Mathematical analysis for an evolutionary epidemic model, in ``Mathematical Models in Medical and Health Sciences'', eds. by M. A. Horn, G. Simonett and G. F. Webb, Vanderbilt Univ. Press, Nashville, TN, 1998, pp. 213-236. MR 1741583 (2001g:92037)
61.: H. Inaba, Kermack and McKendrick revisted: The variable susceptibility model for infectious diseases, Japan J. Indust. Appl. Math. 18 (2001), 273-292. MR 1842912 (2002e:92025)
62.: H. Inaba, Endemic threshold and stability in an evolutionary epidemic model, in ``Mathematical Approaches for Emerging and Reemerging Infectious Diseases: Models, Methods, and Theory'', eds. by C. Castillo-Chavez et al., Springer-Verlag, New York, 2002, pp. 337-359. MR 1938912
63.: T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, 1995. MR 1335452 (96a:47025)
64.: H. Kellermann and M. Hieber, Integrated semigroups, J. Funct. Anal. 84 (1989), 160-180. MR 999494 (90h:47072)
65.: A. Kelley, The stable, center-stable, center, center-unstable, unstable manifolds. J. Differential Equations 3 (1967), 546-570. MR 0221044 (36:4096)
66.: B. L. Keyfitz and N. Keyfitz, The McKendrick partial differential equation and its uses in epidemiology and population study, Math. Comput. Modelling 26 (1997), No. 6, l-9. MR 1601714
67.: T. Kostava and J. Li, Oscillations and stability due to juvenile competitive effects on adult fertility, Comput. Math. Appl. 32 (1996), No. 11, 57-70.
68.: T. Krisztin, Invariance and noninvarince of center manifolds of time- maps with respect to the semiflow, SIAM J. Math. Anal. 36 (2004), 717-739. MR 2111913 (2005h:34211)
69.: N. Krylov and N. N. Bogoliubov, The Application of Methods of Nonlinear Mechanics to the Theory of Stationary Oscillations, Pub. 8 Ukrainian Acad. Sci., Kiev, 1934.
70.: S. Lang, Real Analysis, 2nd Ed., Addison-Wesley, Reading, MA, 1983. MR 783635 (87b:00001)
71.: A. M. Liapunov, Probléme génerale de la stabilité du mouvement, Ann. Fac. Sci. Toulouse 2 (1907), 203-474. MR 0021186 (9:34j)
72.: X.-B. Lin, Homoclinic bifurcations with weakly expanding center manifolds, Dynamics Reported (new series), ed. by C. K. R. T. Jones, U. Kirchgraber and H. O. Walther, Vol. 5, Springer-Verlag, 1996, pp. 99-189. MR 1393487 (97k:58116)
73.: X. Lin, J. So and J. Wu, Center manifolds for partial differential equations with delays, Proc. Roy. Soc. Edinburgh 122A (1992), 237-254. MR 1200199 (93j:34116)
74.: Z. Liu, P. Magal and S. Ruan, Projectors on the generalized eigenspaces for functional differential equations using integrated semigroup, J. Differential Equations 244 (2008), 1784-1809. MR 2404439 (2009b:34191)
75.: P. Magal, Compact attractors for time periodic age-structured population models, Electr. J. Differential Equations 2001 (2001), No. 65, 1-35. MR 1863784 (2002k:37160)
76.: P. Magal and S. Ruan, On integrated semigroups and age-structured models in $L^{p}$ space, Differential Integral Equations 20 (2007), 197-239. MR 2294465 (2008c:47066)
77.: P. Magal and S. Ruan (eds.), Structured Population Models in Biology and Epidemiology, Lect. Notes Math. Vol. 1936, Springer-Verlag, Berlin, 2008. MR 2445337
78.: P. Magal and S. Ruan, On semilinear Cauchy problems with non-dense domain, Adv. Diff. Equations (in press).
79.: P. Magal and S. Ruan, Sustained oscillations in an evolutionary epidemiological model of influenza A drift (submitted).
80.: P. Magal and H. R. Thieme, Eventual compactness for semiflows generated by nonlinear age-structured models, Comm. Pure Appl. Anal. 3 (2004), 695-727. MR 2106296 (2005h:34161)
81.: A. Mielke, A reduction principle for nonautonomous systems in infinite-dimensional spaces, J. Differential Equations 65 (1986), 68-88. MR 859473 (87k:47140)
82.: A. Mielke, Normal hyperbolicity of center manifolds and Saint-Vernant's principle, Arch. Rational Mech. Anal. 110 (1990), 353-372. MR 1049211 (91e:58171)
83.: Nguyen Van Minh and J. Wu, Invariant manifolds of partial functional differential equations, J. Differential Equations 198 (2004), 381-421. MR 2039148 (2005a:34102)
84.: F. Neubrander, Integrated semigroups and their application to the abstract Cauchy problem, Pac. J. Math. 135 (1988), 111-155. MR 965688 (90b:47073)
85.: A. Pazy, Semigroups of Linear Operator and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. MR 710486 (85g:47061)
86.: C. M. Pease, An evolutionary epidemiological mechanism with application to type A influenza, Theoret. Pop. Biol. 31 (1987), 422-452.
87.: O. Perron, Über stabilität und asymptotische verhalten der integrale von differentialgleichungssystemen, Math. Z. 29 (1928), 129-160. MR 1544998
88.: V. A. Pliss, Principal reduction in the theory of stability of motion, Izv. Akad. Nauk. SSSR Mat. Ser. 28 (1964), 1297-1324. MR 0190449 (32:7861)
89.: J. Prüss, On the qualitative behavior of populations with age-speciific interactions, Comput. Math. Appl. 9 (1983), 327-339. MR 702651 (84h:92035)
90.: B. Sandstede, Center manifolds for homoclinic solutions, J. Dynam. Differential Equations 12 (2000), 449-510. MR 1800130 (2001m:37167)
91.: B. Scarpellini, Center manifolds of infinite dimensions I: Main results and applications, ZAMP 42 (1991), 1-32. MR 1102229 (92i:58170)
92.: H. H. Schaefer, Banach Lattice and Positive Operator, Springer-Verlag, Berlin, 1974. MR 0423039 (54:11023)
93.: A. Scheel, Radially symmetric patterns of reaction-diffusion systems, Mem. Amer. Math. Soc. 165 (2003), No. 786. MR 1997690 (2005c:35160)
94.: G. R. Sell and Y. You, Dynamics of Evolutionary Equations, Springer-Verlag, New York, 2002. MR 1873467 (2003f:37001b)
95.: J. Sijbrand, Properties of center manifolds, Trans. Amer. Math. Soc. 289 (1985), 431-469. MR 783998 (86i:58099)
96.: J. H. Swart, Hopf bifurcation and the stability of non-linear age-depedent population models, Comput. Math. Appl. 15 (1988), 555-564. MR 953565 (89g:92052)
97.: A. E. Taylor and D. C. Lay, Introduction to Functional Analysis, John Wiley & Sons, New York, 1980. MR 564653 (81b:46001)
98.: H. R. Thieme, Semiflows generated by Lipschitz perturbations of non-densely defined operators, Differential Integral Equations 3 (1990), 1035-1066. MR 1073056 (92e:47121)
99.: H. R. Thieme, ``Integrated semigroups'' and integrated solutions to abstract Cauchy problems, J. Math. Anal. Appl. 152 (1990), 416-447. MR 1077937 (91k:47093)
100.: H. R. Thieme, Stability change for the endemic equilibrium in age-structured models for the spread of S-I-R type infectious diseases, in ``Differential Equation Models in Biology, Epidemiology and Ecology'', eds. by S. N. Busenberg and M. Martelli, Lect. Notes in Biomath. 92, Springer, Berlin, 1991, pp. 139-158. MR 1193478 (93h:92034)
101.: H. R. Thieme, Quasi-compact semigroups via bounded perturbation, in ``Advances in Mathematical Population Dynamics-Molecules, Cells and Man'', eds. by O. Arino, D. Axelrod and M. Kimmel, World Sci. Publ., River Edge, NJ, 1997, pp. 691-713. MR 1634223 (99i:47070)
102.: H. R. Thieme, Positive perturbation of operator semigroups: Growth bounds, essential compactness, and asynchronous exponential growth, Discrete Contin. Dynam. Systems 4 (1998), 735-764. MR 1641201 (2000e:47069)
103.: A. Vanderbauwhede, Invariant manifolds in infinite dimensions, in ``Dynamics of Infinite Dimensional Systems'' , ed. by S. N. Chow and J. K. Hale, Springer-Verlag, Berlin, 1987, pp. 409-420. MR 921925 (89e:47098)
104.: A. Vanderbauwhede, Center manifold, normal forms and elementary bifurcations, Dynamics Reported, ed. by U. Kirchgraber and H. O. Walther, Vol. 2, John Wiley & Sons, 1989, pp. 89-169. MR 1000977 (90g:58092)
105.: A. Vanderbauwhede and S. A. van Gils, Center manifolds and contractions on a scale of Banach spaces, J. Funct. Anal. 72 (1987), 209-224. MR 886811 (88d:58085)
106.: A. Vanderbauwhede and G. Iooss, Center manifold theory in infinite dimensions, Dynamics Reported (new series), ed. by C. K. R. T. Jones, U. Kirchgraber and H. O. Walther, Vol. 1, Springer-Verlag, Berlin, 1992, pp. 125-163. MR 1153030 (93f:58174)
107.: G. F. Webb, An age-dependent epidenuc model with spatial diffusion, Arch. Rational Mech. Anal. 75 (1980), 91-102. MR 592106 (81k:92052)
108.: G. F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics, Marcel Dekker, New York, 1985. MR 772205 (86e:92032)
109.: G. F. Webb, E. M. C. D'Agata, P. Magal and S. Ruan, A model of antibiotic resistant bacterial epidemics in hospitals, Proc. Natl. Acad. Sci. USA 102 (2005), 13343-13348.
110.: S. Wiggins, Normally Hyperbolic Invariant Manifolds in Dynamical Systems, Springer-Verlag, New York, 1994. MR 1278264 (95g:58163)
111.: J. Wu, Theory and Applications of Partial Differential Equations, Springer-Verlag, New York, 1996. MR 1415838 (98a:35135)
112.: Y. Yi, A generalized integral manifold theorem, J. Differential Equations 102 (1993), 153-187. MR 1209981 (94c:58148)
113.: K. Yosida, Functional Analysis, Springer-Verlag, Berlin, 1980. MR 617913 (82i:46002)
114.: P. Zhang, Z. Feng and F. Milner, A schistosomiasis model with an age-structure in human hosts and its applicationo to treatment strategies, Math. Biosci. 205 (2007), 83-107. MR 2290375 (2007i:92062)

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