Positive unstable periodic solutions for superlinear parabolic equations
HTML articles powered by AMS MathViewer
- by Norimichi Hirano and Noriko Mizoguchi PDF
- Proc. Amer. Math. Soc. 123 (1995), 1487-1495 Request permission
Abstract:
In this paper, we are concerned with a superlinear parabolic equation \[ \left \{ {\begin {array}{*{20}{c}} {\frac {{\partial u}}{{\partial t}} - \Delta u = {u^p} + h(t,x),} \hfill & {(t,x) \in {{\mathbf {R}}_ + } \times \Omega ,} \hfill \\ {{u = 0,}} \hfill & {(t,x) \in {{\mathbf {R}}_ + } \times \partial \Omega ,} \hfill \\ {{u > 0,}} \hfill & {(t,x) \in {{\mathbf {R}}_ + } \times \partial \Omega ,} \hfill \\ \end {array} } \right .\] where $\Omega \subset {{\mathbf {R}}^N}$ is a bounded domain with smooth boundary $\partial \Omega$, h is T-periodic with respect to the first variable, and $1 < p < \frac {{N + 2}}{{N - 2}}$ if $N \geq 3$ and $1 < p < + \infty$ if $N \leq 2$. It is shown that there exist a stable and an unstable positive T-periodic solution for this problem if h is sufficiently small in ${L^\infty }$.References
- Nicholas D. Alikakos, Peter Hess, and Hiroshi Matano, Discrete order preserving semigroups and stability for periodic parabolic differential equations, J. Differential Equations 82 (1989), no. 2, 322–341. MR 1027972, DOI 10.1016/0022-0396(89)90136-8
- Herbert Amann, Periodic solutions of semilinear parabolic equations, Nonlinear analysis (collection of papers in honor of Erich H. Rothe), Academic Press, New York, 1978, pp. 1–29. MR 499089
- 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
- Alfonso Castro and Alan C. Lazer, Results on periodic solutions of parabolic equations suggested by elliptic theory, Boll. Un. Mat. Ital. B (6) 1 (1982), no. 3, 1089–1104 (English, with Italian summary). MR 683495
- Klaus Deimling, Nonlinear functional analysis, Springer-Verlag, Berlin, 1985. MR 787404, DOI 10.1007/978-3-662-00547-7
- E. N. Dancer and P. Hess, On stable solutions of quasilinear periodic-parabolic problems, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 14 (1987), no. 1, 123–141. MR 937539
- Maria J. Esteban, On periodic solutions of superlinear parabolic problems, Trans. Amer. Math. Soc. 293 (1986), no. 1, 171–189. MR 814919, DOI 10.1090/S0002-9947-1986-0814919-8
- Maria J. Esteban, A remark on the existence of positive periodic solutions of superlinear parabolic problems, Proc. Amer. Math. Soc. 102 (1988), no. 1, 131–136. MR 915730, DOI 10.1090/S0002-9939-1988-0915730-7
- B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math. 34 (1981), no. 4, 525–598. MR 615628, DOI 10.1002/cpa.3160340406
- Yoshikazu Giga, A bound for global solutions of semilinear heat equations, Comm. Math. Phys. 103 (1986), no. 3, 415–421. MR 832917, DOI 10.1007/BF01211756
- Daniel Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics, vol. 840, Springer-Verlag, Berlin-New York, 1981. MR 610244, DOI 10.1007/BFb0089647
- P. Hess, On positive solutions of semilinear periodic-parabolic problems, Infinite-dimensional systems (Retzhof, 1983) Lecture Notes in Math., vol. 1076, Springer, Berlin, 1984, pp. 101–114. MR 763357, DOI 10.1007/BFb0072770
- Norimichi Hirano, Existence of unstable periodic solutions for semilinear parabolic equations, Nonlinear Anal. 23 (1994), no. 6, 731–744. MR 1298565, DOI 10.1016/0362-546X(94)90215-1 N. Hirano and N. Mizoguchi, Existence of unstable periodic solutions for semilinear parabolic equations, preprint.
- Morris W. Hirsch, Differential equations and convergence almost everywhere in strongly monotone semiflows, Nonlinear partial differential equations (Durham, N.H., 1982) Contemp. Math., vol. 17, Amer. Math. Soc., Providence, R.I., 1983, pp. 267–285. MR 706104
- Mitsuharu Ôtani, Nonmonotone perturbations for nonlinear parabolic equations associated with subdifferential operators, periodic problems, J. Differential Equations 54 (1984), no. 2, 248–273. MR 757295, DOI 10.1016/0022-0396(84)90161-X J. Prüss, Periodic solutions of semilinear evolution equations, Nonlinear Anal. 3 (1979), 601-612.
- Ioan I. Vrabie, Periodic solutions for nonlinear evolution equations in a Banach space, Proc. Amer. Math. Soc. 109 (1990), no. 3, 653–661. MR 1015686, DOI 10.1090/S0002-9939-1990-1015686-4
Additional Information
- © Copyright 1995 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 123 (1995), 1487-1495
- MSC: Primary 35K55; Secondary 35B10, 35B35
- DOI: https://doi.org/10.1090/S0002-9939-1995-1234627-2
- MathSciNet review: 1234627