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Nonconvergent bounded solutions of semilinear heat equations on arbitrary domains. (English) Zbl 1024.35046
The authors study the following heat equation: \[ u_t=\Delta u+g(x,u),\quad x\in\Omega,\quad u\big|_{\partial\Omega}=0 \tag{1} \] where \(\Omega\) a bounded \(C^2\)-smooth domain of \(\mathbb R^N\).
We recall that equation (1) possesses a global Lyapunov function and, consequently, the \(\omega\)-limit set of every bounded solution of (1) consists only of equilibria. Nevertheless, for every domain \(\Omega\), the authors find a \(C^\infty\)-function \(g(x,u)\) such that equation (1) possesses nonconvergent (as \(t\to\infty\)) bounded solutions. Moreover, for the nonlinearity constructed, equation (1) possesses an infinite-dimensional manifold of nonconvergent solutions.

MSC:
35K57 Reaction-diffusion equations
37L10 Normal forms, center manifold theory, bifurcation theory for infinite-dimensional dissipative dynamical systems
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[1] Abraham, R.; Marsden, J.E.; Ratiu, T., Manifolds, tensor analysis and applications, (1988), Springer New York · Zbl 0875.58002
[2] Amann, H., Linear and quasilinear parabolic problems, (1995), Birkäuser Berlin
[3] Bates, P.; Jones, C., Invariant manifolds for semilinear partial differential equations, Dyn. reported, 2, 1-38, (1989)
[4] Brunovský, P.; Poláčik, P., The morse – smale structure of a generic reaction – diffusion equation in higher space dimension, J. differential equations, 135, 129-181, (1997) · Zbl 0868.35062
[5] Chen, X.-Y.; Hale, J.K.; Tan, B., Invariant foliations for C1 semigroups in Banach spaces, J. differential equations, 139, 283-318, (1997) · Zbl 0994.34047
[6] Chow, S.-N.; Lin, X.-B.; Lu, K., Smooth invariant foliations in infinite-dimensional spaces, J. differential equations, 94, 266-291, (1991) · Zbl 0749.58043
[7] Chow, S.-N.; Lu, K., Invariant manifolds for flows in Banach spaces, J. differential equations, 74, 285-317, (1988) · Zbl 0691.58034
[8] Daners, D.; Koch Medina, P., Abstract evolution equations, periodic problems and applications, (1992), Longman Scientific & Technical Harlow · Zbl 0789.35001
[9] Golubitsky, M.; Guillemin, V., Stable mappings and their singularities, (1974), Springer New York · Zbl 0294.58004
[10] Hale, J.K.; Raugel, G., Convergence in gradient-like systems with applications to PDE, J. appl. math. phys. (ZAMP), 43, 63-124, (1992) · Zbl 0751.58033
[11] Henry, D., Geometric theory of semilinear parabolic equations, (1981), Springer New York · Zbl 0456.35001
[12] D. Henry, Perturbation of the Boundary for Boundary Value Problems of Partial Differential Operators, Cambridge University Press, Cambridge, to appear.
[13] Jendoubi, M.A., A simple unified approach to some convergence theorems of L. Simon, J. funct. anal., 153, 187-202, (1998) · Zbl 0895.35012
[14] M.-A. Jendoubi, P. Poláčik, Nonstabilizing bounded solutions of semilinear hyperbolic and elliptic equations with damping, preprint.
[15] Lions, P.L., Structure of the set of steady-state solutions and asymptotic behavior of semilinear heat equations, J. differential equations, 53, 362-386, (1984) · Zbl 0491.35057
[16] Lunardi, A., Analytic semigroups and optimal regularity in parabolic problems, (1995), Birkhäuser Berlin · Zbl 0816.35001
[17] Matano, H., Convergence of solutions of one-dimensional semilinear parabolic equations, J. math. Kyoto univ., 18, 221-227, (1978) · Zbl 0387.35008
[18] Mielke, A., A reduction principle for nonautonomous systems in infinite-dimensional space, J. differential equations, 65, 66-88, (1986) · Zbl 0601.35018
[19] Palis, J.; de Melo, W., Geometric theory of dynamical systems, (1982), Springer New York
[20] Poláčik, P., Parabolic equations: asymptotic behavior and dynamics on invariant manifolds, () · Zbl 1002.35001
[21] Poláčik, P.; Rybakowski, K.P., Nonconvergent bounded trajectories in semilinear heat equations, J. differential equations, 124, 472-494, (1996) · Zbl 0845.35054
[22] Prizzi, M., Perturbation of elliptic operators and complex dynamics of parabolic partial differential equations, Proc. R. soc. edinb., sect. A, math., 130, 397-418, (2000) · Zbl 0968.35064
[23] Prizzi, M.; Rybakowski, K.P., Inverse problems and chaotic dynamics of parabolic equations on arbitrary spatial domains, J. differential equations, 142, 17-53, (1998) · Zbl 0915.35059
[24] Rybakowski, K.P., An abstract approach to smoothness of invariant manifolds, Appl. anal., 49, 119-150, (1993) · Zbl 0736.35016
[25] Saut, J.; Temam, R., Generic properties of nonlinear boundary value problems, Comm. partial differential equations, 4, 293-319, (1979) · Zbl 0462.35016
[26] Simon, L., Asymptotics for a class of nonlinear evolution equations, with applications to geometric problems, Ann. math., 118, 525-571, (1983) · Zbl 0549.35071
[27] Smith, H.L., Monotone dynamical systems, (1995), American Mathematical Society Providence, RI
[28] A. Vanderbauwhede, G. Iooss, Center manifold theory in infinite dimensions, in: C. Jones, H.-O. Walther, and U. Kirchgraber (Eds.), Dynamics Reported: Expositions in Dynamical Systems, Springer, Berlin, 1992, pp. 125-163. · Zbl 0751.58025
[29] Zelenyak, T.I., Stabilization of solutions of boundary value problems for a second order parabolic equation with one space variable, Differential equations (transl. from differencialnye uravnenia), 4, 17-22, (1968) · Zbl 0232.35053
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