Exponential separation and principal Floquet bundles for linear parabolic equations on general bounded domains: Nondivergence case

Húska, Juraj

doi:10.1090/S0002-9947-08-04413-9

Exponential separation and principal Floquet bundles for linear parabolic equations on general bounded domains: Nondivergence case
HTML articles powered by AMS MathViewer

by Juraj Húska PDF

Trans. Amer. Math. Soc. 360 (2008), 4639-4679 Request permission

Abstract:

We consider the Dirichlet problem for linear nonautonomous second order parabolic equations of nondivergence type on general bounded domains with bounded measurable coefficients. Under such minimal regularity assumptions, we establish the existence of a principal Floquet bundle exponentially separated from a complementary invariant bundle. As a special case of our main theorem, assuming the coefficients are time-periodic, we obtain a new result on the existence of a principal eigenvalue of an associated (time-periodic) parabolic eigenvalue problem. We also show the existence of a uniform spectral gap between the principal eigenvalue and the rest of the spectrum for a class of time-periodic uniformly parabolic operators. Finally, we prove the uniqueness of positive entire solutions in the class of solutions whose supremum norms do not grow superexponentially as time goes to negative infinity.

References

Jose M. Arrieta, Elliptic equations, principal eigenvalue and dependence on the domain, Comm. Partial Differential Equations 21 (1996), no. 5-6, 971–991. MR 1391529, DOI 10.1080/03605309608821213
H. Berestycki and L. Nirenberg, On the method of moving planes and the sliding method, Bol. Soc. Brasil. Mat. (N.S.) 22 (1991), no. 1, 1–37. MR 1159383, DOI 10.1007/BF01244896
H. Berestycki, L. Nirenberg, and S. R. S. Varadhan, The principal eigenvalue and maximum principle for second-order elliptic operators in general domains, Comm. Pure Appl. Math. 47 (1994), no. 1, 47–92. MR 1258192, DOI 10.1002/cpa.3160470105
Isabeau Birindelli, Hopf’s lemma and anti-maximum principle in general domains, J. Differential Equations 119 (1995), no. 2, 450–472. MR 1340547, DOI 10.1006/jdeq.1995.1098
Shui-Nee Chow, Kening Lu, and John Mallet-Paret, Floquet theory for parabolic differential equations, J. Differential Equations 109 (1994), no. 1, 147–200. MR 1272403, DOI 10.1006/jdeq.1994.1047
Shui-Nee Chow, Kening Lu, and John Mallet-Paret, Floquet bundles for scalar parabolic equations, Arch. Rational Mech. Anal. 129 (1995), no. 3, 245–304. MR 1328478, DOI 10.1007/BF00383675
Daniel Daners, Existence and perturbation of principal eigenvalues for a periodic-parabolic problem, Proceedings of the Conference on Nonlinear Differential Equations (Coral Gables, FL, 1999) Electron. J. Differ. Equ. Conf., vol. 5, Southwest Texas State Univ., San Marcos, TX, 2000, pp. 51–67. MR 1799044
D. E. Edmunds and L. A. Peletier, Removable singularities of solutions of quasi-linear parabolic equations, J. London Math. Soc. (2) 2 (1970), 273–283. MR 259365, DOI 10.1112/jlms/s2-2.2.273
T. Godoy and U. Kaufmann, On principal eigenvalues for periodic parabolic problems with optimal condition on the weight function, J. Math. Anal. Appl. 262 (2001), no. 1, 208–220. MR 1857223, DOI 10.1006/jmaa.2001.7559
Manfred Gruber, Harnack inequalities for solutions of general second order parabolic equations and estimates of their Hölder constants, Math. Z. 185 (1984), no. 1, 23–43. MR 724044, DOI 10.1007/BF01214972
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
Peter Hess and Peter Poláčik, Boundedness of prime periods of stable cycles and convergence to fixed points in discrete monotone dynamical systems, SIAM J. Math. Anal. 24 (1993), no. 5, 1312–1330. MR 1234018, DOI 10.1137/0524075
Juraj Húska, Exponential separation and principal Floquet bundles for linear parabolic equations on general bounded domains: the divergence case, Indiana Univ. Math. J. 55 (2006), no. 3, 1015–1043. MR 2244596, DOI 10.1512/iumj.2006.55.2868
Juraj Húska, Harnack inequality and exponential separation for oblique derivative problems on Lipschitz domains, J. Differential Equations 226 (2006), no. 2, 541–557. MR 2237690, DOI 10.1016/j.jde.2006.02.008
Juraj Húska and Peter Poláčik, The principal Floquet bundle and exponential separation for linear parabolic equations, J. Dynam. Differential Equations 16 (2004), no. 2, 347–375. MR 2105779, DOI 10.1007/s10884-004-2784-8
J. Húska, P. Poláčik, and M. V. Safonov, Principal eigenvalues, spectral gaps and exponential separation between positive and sign-changing solutions of parabolic equations, Discrete Contin. Dyn. Syst. suppl. (2005), 427–435. MR 2192700
Juraj Húska, Peter Poláčik, and Mikhail V. Safonov, Harnack inequalities, exponential separation, and perturbations of principal Floquet bundles for linear parabolic equations, Ann. Inst. H. Poincaré C Anal. Non Linéaire 24 (2007), no. 5, 711–739 (English, with English and French summaries). MR 2348049, DOI 10.1016/j.anihpc.2006.04.006
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
N. V. Krylov and M. V. Safonov, A property of the solutions of parabolic equations with measurable coefficients, Izv. Akad. Nauk SSSR Ser. Mat. 44 (1980), no. 1, 161–175, 239 (Russian). MR 563790
N. V. Krylov, Nonlinear elliptic and parabolic equations of the second order, Mathematics and its Applications (Soviet Series), vol. 7, D. Reidel Publishing Co., Dordrecht, 1987. Translated from the Russian by P. L. Buzytsky [P. L. Buzytskiĭ]. MR 901759, DOI 10.1007/978-94-010-9557-0
Gary M. Lieberman, Second order parabolic differential equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. MR 1465184, DOI 10.1142/3302
J. Mierczyński. Flows on order bundles, unpublished.
Janusz Mierczyński, $p$-arcs in strongly monotone discrete-time dynamical systems, Differential Integral Equations 7 (1994), no. 5-6, 1473–1494. MR 1269666
Janusz Mierczyński, Globally positive solutions of linear parabolic PDEs of second order with Robin boundary conditions, J. Math. Anal. Appl. 209 (1997), no. 1, 47–59. MR 1444510, DOI 10.1006/jmaa.1997.5323
Janusz Mierczyński, Globally positive solutions of linear parabolic partial differential equations of second order with Dirichlet boundary conditions, J. Math. Anal. Appl. 226 (1998), no. 2, 326–347. MR 1650236, DOI 10.1006/jmaa.1998.6065
Janusz Mierczyński, The principal spectrum for linear nonautonomous parabolic PDEs of second order: basic properties, J. Differential Equations 168 (2000), no. 2, 453–476. Special issue in celebration of Jack K. Hale’s 70th birthday, Part 2 (Atlanta, GA/Lisbon, 1998). MR 1808456, DOI 10.1006/jdeq.2000.3893
Janusz Mierczyński and Wenxian Shen, Exponential separation and principal Lyapunov exponent/spectrum for random/nonautonomous parabolic equations, J. Differential Equations 191 (2003), no. 1, 175–205. MR 1973287, DOI 10.1016/S0022-0396(03)00016-0
Keith Miller, Barriers on cones for uniformly elliptic operators, Ann. Mat. Pura Appl. (4) 76 (1967), 93–105 (English, with Italian summary). MR 221087, DOI 10.1007/BF02412230
Masaharu Nishio, The uniqueness of positive solutions of parabolic equations of divergence form on an unbounded domain, Nagoya Math. J. 130 (1993), 111–121. MR 1223732, DOI 10.1017/S0027763000004451
Peter Poláčik, Estimates of solutions and asymptotic symmetry for parabolic equations on bounded domains, Arch. Ration. Mech. Anal. 183 (2007), no. 1, 59–91. MR 2259340, DOI 10.1007/s00205-006-0004-x
P. Poláčik, On uniqueness of positive entire solutions and other properties of linear parabolic equations, Discrete Contin. Dyn. Syst. 12 (2005), no. 1, 13–26. MR 2121246, DOI 10.3934/dcds.2005.12.13
P. Poláčik and I. Tereščák, Convergence to cycles as a typical asymptotic behavior in smooth strongly monotone discrete-time dynamical systems, Arch. Rational Mech. Anal. 116 (1992), no. 4, 339–360. MR 1132766, DOI 10.1007/BF00375672
P. Poláčik and Ignác Tereščák, Exponential separation and invariant bundles for maps in ordered Banach spaces with applications to parabolic equations, J. Dynam. Differential Equations 5 (1993), no. 2, 279–303. MR 1223450, DOI 10.1007/BF01053163
Wenxian Shen and Yingfei Yi, Almost automorphic and almost periodic dynamics in skew-product semiflows, Mem. Amer. Math. Soc. 136 (1998), no. 647, x+93. MR 1445493, DOI 10.1090/memo/0647
I. Tereščák. Dynamics of ${C}^1$ smooth strongly monotone discrete-time dynamical systems. preprint.
I. Tereščák. Dynamical systems with discrete Lyapunov functionals. Ph.D. thesis, Comenius University, 1994.