Transactions of the American Mathematical Society

A priori bounds, nodal equilibria and connecting orbits in indefinite superlinear parabolic problems

Author(s): Nils Ackermann; Thomas Bartsch; Petr Kaplicky; Pavol Quittner
Journal: Trans. Amer. Math. Soc. 360 (2008), 3493-3539.
MSC (2000): Primary 37L05; Secondary 35K20, 35K55, 37L10, 47H20
Posted: February 13, 2008
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Abstract: We consider the dynamics of the semiflow associated with a class of semilinear parabolic problems on a smooth bounded domain, posed with homogeneous Dirichlet boundary conditions. The distinguishing feature of this class is the indefinite superlinear (but subcritical) growth of the nonlinearity at infinity. We present new a priori bounds for global semiorbits that enable us to give dynamical proofs of known and new existence results for equilibria. In addition, we can prove the existence of connecting orbits in many cases.

One advantage of our approach is that the parabolic semiflow is naturally order preserving, in contrast to pseudo-gradient flows considered when using variational methods. Therefore we can obtain much information on nodal properties of equilibria that was not known before.

1.: N. Ackermann and T. Bartsch, Superstable manifolds of semilinear parabolic problems, J. Dynam. Differential Equations 17 (2005), no. 1, 115-173. MR 2157843
2.: S. Alama and M. Del Pino, Solutions of elliptic equations with indefinite nonlinearities via Morse theory and linking, Ann. Inst. H. Poincaré Anal. Non Linéaire 13 (1996), no. 1, 95-115. MR 96m:35091
3.: S. Alama and G. Tarantello, On semilinear elliptic equations with indefinite nonlinearities, Calc. Var. Partial Differential Equations 1 (1993), no. 4, 439-475. MR 97a:35057
4.: H. Amann, Linear and quasilinear parabolic problems, Vol. I abstract linear theory, Monographs in Mathematics, vol. 89, Birkhäuser Boston, Inc., Boston, MA, 1995. MR 96g:34088
5.: H. Amann and J. López-Gómez, A priori bounds and multiple solutions for superlinear indefinite elliptic problems, J. Differential Equations 146 (1998), no. 2, 336-374. MR 99e:35057
6.: H. Amann and P. Quittner, Optimal control problems with final observation governed by explosive parabolic equations, SIAM J. Control Optim. 44 (2005), no. 4, 1215-1238 (electronic). MR 2177310
7.: J. M. Arrieta, A. N. Carvalho, and A. Rodrıguez-Bernal, Parabolic problems with nonlinear boundary conditions and critical nonlinearities, J. Differential Equations 156 (1999), no. 2, 376-406. MR 2000f:35075
8.: T. Bartsch, Critical point theory on partially ordered Hilbert spaces, J. Funct. Anal. 186 (2001), no. 1, 117-152. MR 2002i:58011
9.: T. Bartsch, K.-C. Chang, and Z.-Q. Wang, On the Morse indices of sign changing solutions of nonlinear elliptic problems, Math. Z. 233 (2000), no. 4, 655-677. MR 2001c:35079
10.: T. Bartsch and Z.-Q. Wang, On the existence of sign changing solutions for semilinear Dirichlet problems, Topol. Methods Nonlinear Anal. 7 (1996), no. 1, 115-131. MR 97m:35076
11.: T. Bartsch and T. Weth, A note on additional properties of sign changing solutions to superlinear elliptic equations, Topol. Methods Nonlinear Anal. 22 (2003), no. 1, 1-14. MR 2037264
12.: P. W. Bates and C. K. R. T. Jones, Invariant manifolds for semilinear partial differential equations, Dynamics reported, Vol. 2, Dynam. Report. Ser. Dynam. Systems Appl., vol. 2, Wiley, Chichester, 1989, pp. 1-38. MR 90g:58017
13.: H. Berestycki, I. Capuzzo-Dolcetta, and L. Nirenberg, Superlinear indefinite elliptic problems and nonlinear Liouville theorems, Topol. Methods Nonlinear Anal. 4 (1994), no. 1, 59-78. MR 96d:35041
14.: -, Variational methods for indefinite superlinear homogeneous elliptic problems, NoDEA Nonlinear Differential Equations Appl. 2 (1995), no. 4, 553-572. MR 96i:35033
15.: H. Brezis and T. Cazenave, A nonlinear heat equation with singular initial data, J. Anal. Math. 68 (1996), 277-304. MR 97f:35092
16.: T. Cazenave and P.-L. Lions, Solutions globales d'équations de la chaleur semi linéaires, Comm. Partial Differential Equations 9 (1984), no. 10, 955-978. MR 87k:35125
17.: K.-C. Chang, Infinite-dimensional Morse theory and multiple solution problems, Progress in Nonlinear Differential Equations and their Applications, 6, Birkhäuser Boston, Inc., Boston, MA, 1993. MR 94e:58023
18.: K.-C. Chang and M.-Y. Jiang, Dirichlet problem with indefinite nonlinearities, Calc. Var. Partial Differential Equations 20 (2004), 257-282. MR 2062944 (2005h:35058)
19.: W. Chen and C. Li, Indefinite elliptic problems in a domain, Discrete Contin. Dynam. Systems 3 (1997), no. 3, 333-340. MR 98a:35035
20.: E. N. Dancer and Y. Du, The generalized Conley index and multiple solutions of semilinear elliptic problems, Abstr. Appl. Anal. 1 (1996), no. 1, 103-135. MR 97i:35048
21.: Yihong Du and Shujie Li, Nonlinear Liouville theorems and a priori estimates for indefinite superlinear elliptic equations, Adv. Differential Equations 10 (2005), no. 8, 841-860. MR 2150868 (2006d:35073)
22.: B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations 6 (1981), no. 8, 883-901. MR 82h:35033
23.: Y. Giga, A bound for global solutions of semilinear heat equations, Comm. Math. Phys. 103 (1986), no. 3, 415-421. MR 87j:35187
24.: Y. Giga, S. Matsui, and S. Sasayama, Blow up rate for semilinear heat equation with subcritical nonlinearity, Indiana Univ. Math. J. 53 (2004), no. 2, 483-514. MR 2060042 (2005g:35153)
25.: M. Grossi, P. Magrone, and M. Matzeu, Linking type solutions for elliptic equations with indefinite nonlinearities up to the critical growth, Discrete Contin. Dynam. Systems 7 (2001), no. 4, 703-718. MR 2002h:35086
26.: H. Hofer, A geometric description of the neighbourhood of a critical point given by the mountain-pass theorem, J. London Math. Soc. (2) 31 (1985), no. 3, 566-570. MR 87e:58041
27.: M. A. Krasnosel'skii, Topological methods in the theory of nonlinear integral equations, Translated by A. H. Armstrong; translation edited by J. Burlak. A Pergamon Press Book, The Macmillan Co., New York, 1964. MR 28:2414
28.: H. A. Levine, Some nonexistence and instability theorems for solutions of formally parabolic equations of the form $Pu\sb{t}=-Au+\mathcal{F}(u)$ , Arch. Rational Mech. Anal. 51 (1973), 371-386. MR 50:714
29.: S. Li and Z.-Q. Wang, Mountain pass theorem in order intervals and multiple solutions for semilinear elliptic Dirichlet problems, J. Anal. Math. 81 (2000), 373-396. MR 2001h:35063
30.: S. Li and M. Willem, Applications of local linking to critical point theory, J. Math. Anal. Appl. 189 (1995), no. 1, 6-32. MR 96a:58045
31.: J. López-Gómez and P. Quittner, Complete and energy blow-up in indefinite superlinear parabolic problems, Discrete Contin. Dyn. Syst. 14 (2006), no. 1, 169-186. MR 2170308
32.: P. Magrone, On a class of semilinear elliptic equations with potential changing sign, Dynam. Systems Appl. 9 (2000), no. 4, 459-467. MR 2002d:35070
33.: M. Marcus and L. Véron, Uniqueness and asymptotic behavior of solutions with boundary blow-up for a class of nonlinear elliptic equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 14 (1997), no. 2, 237-274. MR 97m:35068
34.: H. Matano, Existence of nontrivial unstable sets for equilibriums of strongly order-preserving systems, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 30 (1984), no. 3, 645-673. MR 85d:35014
35.: J. Mawhin and M. Willem, Critical point theory and Hamiltonian systems, Applied Mathematical Sciences, vol. 74, Springer-Verlag, New York, 1989. MR 90e:58016
36.: W.-M. Ni, P. E. Sacks, and J. Tavantzis, On the asymptotic behavior of solutions of certain quasilinear parabolic equations, J. Differential Equations 54 (1984), no. 1, 97-120. MR 87j:35057
37.: R. S. Palais, Homotopy theory of infinite dimensional manifolds, Topology 5 (1966), 1-16. MR 32:6455
38.: P. Quittner, Global existence of solutions of parabolic problems with nonlinear boundary conditions, Singularities and differential equations (Warsaw, 1993), Banach Center Publ., vol. 33, Polish Acad. Sci., Warsaw, 1996, pp. 309-314. MR 98h:35127
39.: -, A priori bounds for global solutions of a semilinear parabolic problem, Acta Math. Univ. Comenian. (N.S.) 68 (1999), no. 2, 195-203. MR 2001d:35026
40.: -, Continuity of the blow-up time and a priori bounds for solutions in superlinear parabolic problems, Houston J. Math. 29 (2003), no. 3, 757-799 (electronic). MR 1998164
41.: -, Multiple equilibria, periodic solutions and a priori bounds for solutions in superlinear parabolic problems, NoDEA Nonlinear Differential Equations Appl. 11 (2004), 237-258. MR 2210288 (2006k:35140)
42.: M. Ramos, S. Terracini, and C. Troestler, Superlinear indefinite elliptic problems and Pohožaev type identities, J. Funct. Anal. 159 (1998), no. 2, 596-628. MR 2000h:35053
43.: K. P. Rybakowski, The homotopy index and partial differential equations, Universitext, Springer-Verlag, Berlin, 1987. MR 89d:58025
44.: F. B. Weissler, Semilinear evolution equations in Banach spaces, J. Funct. Anal. 32 (1979), no. 3, 277-296. MR 80i:47091
45.: M. Willem, Minimax theorems, Progress in Nonlinear Differential Equations and their Applications, 24, Birkhäuser Boston, Inc., Boston, MA, 1996. MR 97h:58037

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 37L05, 35K20, 35K55, 37L10, 47H20

Nils Ackermann
Affiliation: Instituto de Matemáticas, Universidad Nacional Autónoma de México, México, D.F. C.P. 04510, México
Email: nils@ackermath.info

Thomas Bartsch
Affiliation: Mathematisches Institut, University of Giessen, Arndtstr. 2, 35392 Giessen, Germany
Email: thomas.bartsch@math.uni-giessen.de

Petr Kaplicky
Affiliation: Faculty of Mathematics and Physics, Charles University Prague, Sokolovská 83, 186 75 Praha 8, Czech Republic
Email: kaplicky@karlin.mff.cuni.cz

Pavol Quittner
Affiliation: Department of Applied Mathematics and Statistics, Comenius University, Mlynská dolina, 84248 Bratislava, Slovakia
Email: quittner@pc2.iam.fmph.uniba.sk

DOI: 10.1090/S0002-9947-08-04404-8
PII: S 0002-9947(08)04404-8
Received by editor(s): July 29, 2004
Received by editor(s) in revised form: April 7, 2006
Posted: February 13, 2008
Additional Notes: The first and second authors were supported by DFG Grants BA 1009/15-1, BA 1009/15-2
The third author was supported by the GACR Grant 201/03/0934
The fourth author was supported by the DFG Grant Gi 30/76-1
Copyright of article: Copyright 2008, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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