## Some remarks on inertial manifolds.(English)Zbl 0815.35037

The aim of the author is to construct inertial manifolds for a class of evolution equations of the form (1) $$u_ t + Au + R(t,u) = 0$$ which takes place on a Hilbert space $$H$$ with norm $$\|\;\|$$. In order to assure the existence of such manifolds the author imposes some assumptions on (1) which are to some extent weaker than those usually encountered. $$A$$ is assumed to be a positive selfadjoint operator with compact resolvents and spectrum $$0 < \lambda_ 1 \leq \lambda_ 2 \dots$$; we denote by $$w_ j$$ an eigenvector of $$\lambda_ j$$. Crucial to the theory is a spectral gap condition of the form (2) $$\lambda_{N+1} - \lambda_ N > K_ 1(\lambda^{\gamma/2}_{N+1} + \lambda_ N^{\gamma/2})^ 2$$ for some $$N > 0$$. The constant $$K_ 1$$ and $$\gamma \in (0,{1 \over 2})$$ are determined by the nonlinearity $$R(t,u)$$ which is assumed to satisfy the following conditions:
(3) $$\| R(t,u) \| \leq K_ 0$$, (4) $$\| R(t,u) - R(t,v) \| \leq K_ 1 \| A^ \gamma (u - v) \|$$, (5) $$\| R(t + h,u) - R(t,u) \| \leq K_ 2 | h |$$,
where $$u,v \in \text{dom} (A^ \gamma)$$. Under these assumptions the evolution equation (1) generates a semiflow $$S(t,t_ 0)$$ which maps $$\text{dom} (A^ \gamma)$$ into itself and which has some smoothness properties which are proved in the course of the paper. Let $$P_ N$$ be the orthogonal projection onto $$\text{span} (w_ 1, \dots, w_ N)$$. The first main result (theorem 1) states that there is a Lipschitz mapping $$\varphi (\cdot, \cdot)$$ from $$\mathbb{R} \times P_ N \text{dom} (A^ \gamma)$$ into $$(1-P_ N) \text{dom} (A^ \gamma)$$ such that for each $$t \in\mathbb{R}$$ the manifold $$M_ t = \text{graph} (\varphi (t, \cdot))$$ has the following properties: (a) $$M_ t = S(t,t_ 0) M_{t_ 0}$$, $$t,t_ 0 \in\mathbb{R}$$, (b) if $$u(t) = S(t,t_ 0) u_ 0$$ then there exists $$v_ 0 \in M_{t_ 0}$$ such that $\Bigl \| A^ \gamma \bigl( S(t + t_ 0, t_ 0) u_ 0 - S(t + t_ 0, t_ 0) v_ 0 \bigr) \Bigr \| \leq C_ 1e^{-t\nu},\;t \geq 0$ for suitable $$C_ 1$$, $$\nu > 0$$ independent of $$t_ 0,t$$. There are further results of perturbational nature which compare the manifolds $$M_ t$$ with $$\widetilde M_ t$$, $$t \in\mathbb{R}$$ which emerge from nonlinearities $$R(t,u)$$ and $$\widetilde R(t,u)$$ respectively, and which loosely speaking express that $$M_ t$$ and $$\widetilde M_ t$$ are asymptotically close if $$R(t,u)$$ and $$\widetilde R(t,u)$$ are asymptotically close. The proofs are based on a series of technical lemmas. Among these, lemma 3.3 in particular expresses that the flow $$S(t,t_ 0)$$ has a certain squeezing property which differs from that usually encountered.

### MSC:

 35G10 Initial value problems for linear higher-order PDEs 35K25 Higher-order parabolic equations 35B40 Asymptotic behavior of solutions to PDEs 58D25 Equations in function spaces; evolution equations 47H20 Semigroups of nonlinear operators
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