Superstable manifolds of semilinear parabolic problems

被引：0
作者：
Ackermann N. [1 ]
Bartsch T. [1 ]
机构：
[1] Mathematisches Institut, Justus-Liebig-Universität, D-35392 Gießen
来源：
Journal of Dynamics and Differential Equations | 2005年 / 17卷 / 1期
关键词：
Connecting orbits; Invariant manifolds; Nodal properties;
D O I：
10.1007/s10884-005-3144-z
中图分类号：
学科分类号：
摘要：
We investigate the dynamics of the semiflow φ induced on H 0 1(Ω) by the Cauchy problem of the semilinear parabolic equation ∂Tu-Δ= f on Ω. Here Ω⊂ℝ n is a bounded smooth domain, and f: Ω × ℝ→ℝ has subcritical growth in u and satisfies f (x, 0) ≡.0. In particular we are interested in the case when f is definite superlinear in u. The set A: = {u ε H01(Ω)|φt (u) → 0 as t → ∞} of attraction of 0 contains a decreasing family of invariant sets W1 ⊇W2⊇ 3⊇... distinguished by the rate of convergence φt (u) → 0. We prove that the W k's are global submanifolds of H01 (Ω), and we find equilibria in the boundaries W̄kWk. We also obtain results on nodal and comparison properties of these equilibria. In addition the paper contains a detailed exposition of the semigroup approach for semilinear equations, improving earlier results on stable manifolds and asymptotic behavior near an equilibrium. © Springer Science+Business Media, Inc. 2005.
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页码：115 / 173
页数：58
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