Families of stable manifolds for one-dimensional parabolic equations

Abstract

ABSTRACT. We prove that to any invariant subset of the dynamical system generated by a one-dimensional quasilinear parabolic equation there corresponds an invariant family of stable manifolds of finite codimension.K~.v WORDS: infinite-dimensional dynamical systems, asymptotic behavior of trajectories, invariant families of stable manifolds. w IntroductionThe asymptotic behavior of trajectories is one of the main topics in the theory of evolution differential equations. In particular, it can be studied by constructing invariant families of manifolds. In the present paper, we prove the existence of a family of stable manifolds associated with an invariant subset of the dynamical system generated by the one-dimensional parabolic equation The existence of invariant manifolds for infinite-dimensional dynamical systems generated by equations of the form (1) was studied in [2][3][4] under the assumption that f does not depend on uz. This restriction is removed in the present paper. To make the exposition shorter, we consider only autonomous equations; this simplification is not essential, and all results can readily be generalized to the nonautonomous case. To be definite, let us consider the homogeneous Dirichlet boundary conditions. w The dynamical system Consider the Hilbert space W of square integrable functions on [0,1]. The space W is equipped with the standard inner product (., 9 ), which generates the corresponding norm [ 9 ] [1]. The operator L is densely defined and self-adjoint in W. Furthermore, there exists an orthonormal basis {vk(z)}, k E N, in W such that Lvk = A~v~, A~ < A~+l, and A~ --* oo.For each integer k we define a Hilbert space W k as the completion of the domain of L k/2 with respect to the norm I lk = ILk/ l, generated by the inner product (u, v)k = (Lk/2u, L~/2v). We set W ~ and W~176 N Wk k>_l by definition. By the Rellich theorem, the natural embedding W k --* W k-1 is compact.

Finite-dimensional dynamics on attractors of non-linear parabolic equations

“…In this definition the B(u, v) are operators of scalar type (see [19]) for all u, v ∈ A. In the case when T 0 = 0 and u = v the representation B = S −1 HS in Definition 2.2 was actually used by Kamaev [15], [16] in his study of phase dynamics near the attractors of (scalar or vector) equations of a somewhat wider class than (2). In these papers either the conventional Liouville transformation or [16] a modification of it was applied to the right-hand side of the linearized equation.…”

“…Consequently, |q( • ; u, v)| α const. Hence, the multipliers T 0 (u, v) belong to L(X α ) and T 0 (u, v) α const for u, v ∈ A. Formulae (14b) and (15)…”