Previous |  Up |  Next

Article

Keywords:
parabolic equations; elliptic equations; hyperbolic equations; asymptotic behavior; center manifold
Summary:
We consider three types of semilinear second order PDEs on a cylindrical domain $\Omega \times (0,\infty )$, where $\Omega $ is a bounded domain in ${{\mathbb{R}}}^N$, $N\ge 2$. Among these, two are evolution problems of parabolic and hyperbolic types, in which the unbounded direction of $\Omega \times (0,\infty )$ is reserved for time $t$, the third type is an elliptic equation with a singled out unbounded variable $t$. We discuss the asymptotic behavior, as $t\rightarrow \infty $, of solutions which are defined and bounded on $\Omega \times (0,\infty )$.
References:
[1] A. V. Babin, M. I. Vishik: Attractors of Evolution Equations. North-Holland, Amsterdam, 1992. MR 1156492
[2] P. Brunovský, X. Mora, P. Poláčik, J. Solà-Morales: Asymptotic behavior of solutions of semilinear elliptic equations on an unbounded strip. Acta Math. Univ. Comenian. (N.S.) 60 (1991), 163–183. MR 1155242
[3] P. Brunovský, P. Poláčik, B. Sandstede: Convergence in general periodic parabolic equations in one space dimension. Nonlinear Anal. 18 (1992), 209–215. MR 1148285
[4] A. Calsina, X. Mora, J. Solà-Morales: The dynamical approach to elliptic problems in cylindrical domains, and a study of their parabolic singular limit. J. Differ. Equations 102 (1993), 244–304. MR 1216730
[5] X.-Y. Chen: Uniqueness of the $\omega $-limit point of solutions of a semilinear heat equation on the circle. Proc. Japan Acad. Ser. A Math. Sci. 62 (1986), 335–337. MR 0888140 | Zbl 0641.35028
[6] X.-Y. Chen, H. Matano: Convergence, asymptotic periodicity, and finite-point blow-up in one-dimensional semilinear heat equations. J. Differ. Equations 78 (1989), 160–190. MR 0986159
[7] X.-Y. Chen, P. Poláčik: Asymptotic periodicity of positive solutions of reaction diffusion equations on a ball. J. Reine Angew. Math. 472 (1996), 17–51. MR 1384905
[8] E. Feireisl, P. Poláčik: Structure of periodic solutions and asymptotic behavior for time-periodic reaction-diffusion equations on R. Adv. Differ. Equations 5 (2000), 583–622. MR 1750112
[9] E. Feireisl, F. Simondon: Convergence for degenerate parabolic equations. J. Differ. Equations 152 (1999), 439–466. MR 1674569
[10] J. K. Hale: Asymptotic Behavior of Dissipative Systems. American Mathematical Society, Providence, RI, 1988. MR 0941371 | Zbl 0642.58013
[11] J. K. Hale, G. Raugel: Convergence in gradient-like systems with applications to PDE. J. Applied Math. Physics (ZAMP) 43 (1992), 63–124. MR 1149371
[12] A. Haraux, M. A. Jendoubi: Convergence of solutions of second-order gradient-like systems with analytic nonlinearities. J. Differ. Equations 144 (1998), 313–320. MR 1616968
[13] A. Haraux, P. Poláčik: Convergence to a positive equilibrium for some nonlinear evolution equations in a ball. Acta Math. Univ. Comenian. (N.S.) 61 (1992), 129–141. MR 1205867
[14] M. A. Jendoubi: Convergence of global and bounded solutions of the wave equation with linear dissipation and analytic nonlinearity. J. Differ. Equations 144 (1998), 302–312. MR 1616964 | Zbl 0912.35028
[15] M. A. Jendoubi: A simple unified approach to some convergence theorems of L. Simon. J. Funct. Anal. 153 (1998), 187–202. MR 1609269 | Zbl 0895.35012
[16] M. A. Jendoubi, P. Poláčik: Nonstabilizing solutions of semilinear hyperbolic and elliptic equations with damping. Preprint.
[17] K. Kirchgässner: Wave-solutions of reversible systems and applications. J. Differ. Equations 45 (1982), 113–127.
[18] H. Matano: Convergence of solutions of one-dimensional semilinear parabolic equations. J. Math. Kyoto Univ. 18 (1978), 221–227. MR 0501842 | Zbl 0387.35008
[19] A. Mielke: Hamiltonian and Lagrangian Flows on Center Manifolds with Applications to Elliptic Variational Problems. Springer, Berlin, 1991. MR 1165943 | Zbl 0747.58001
[20] A. Mielke: Essential manifolds for an elliptic problem in infinite strip. J. Differ. Equations 110 (1994), 322–355. MR 1278374
[21] J. Palis, W. de Melo: Geometric Theory of Dynamical Systems. Springer, New York, 1982. MR 0669541
[22] P. Poláčik: Parabolic equations: asymptotic behavior and dynamics on invariant manifolds. Handbook on Dynamical Systems III: Towards Applications. Elsevier, B. Fiedler (ed.), to appear.
[23] M. A. Jendoubi, P. Poláčik: Nonstabilizing solutions of semilinear hyperbolic and elliptic equations with damping.
[24] P. Poláčik, K. P. Rybakowski: Nonconvergent bounded trajectories in semilinear heat equations. J. Differ. Equations 124 (1996), 472–494. MR 1370152
[25] P. Poláčik, F. Simondon: Nonconvergent bounded solutions of semilinear heat equations on arbitrary domains.
[26] L. Simon: Asymptotics for a class of nonlinear evolution equations, with applications to geometric problems. Annals Math. 118 (1983), 525–571. MR 0727703
[27] T. Valent: Boundary value problems of finite elasticity. Springer, New York, 1988. MR 0917733 | Zbl 0648.73019
[28] P. Takáč: Stabilization of positive solutions of analytic gradient-like systems.
[29] R. Temam: Infinite-dimensional dynamical systems in mechanics and physics. Springer, New York, 1988. MR 0953967 | Zbl 0662.35001
[30] A. Vanderbauwhede, G. Iooss: Center manifold theory in infinite dimensions. Dynamics Reported: Expositions in Dynamical Systems, Springer, Berlin, 1992, pp. 125–163. MR 1153030
[31] T. I. Zelenyak: Stabilization of solutions of boundary value problems for a second order parabolic equation with one space variable. Differ. Equations 4 (1968), 17–22. MR 0223758
Partner of
EuDML logo