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**Spectral barriers and inertial manifolds for dissipative partial differential equations.**
*(English)*
Zbl 0701.35024

The authors consider the dissipative partial differential equation (1) \(u_ t+A(u)+R(u)=0\), where A is the dissipative part which is usually a self-adjoint unbounded positive operator, in a Hilbert space H and the nonlinear inhomogeneous term R(u) is defined, the domain of A, and has some local Lipschitz property there.

An inertial manifold is an invariant manifold that is finite dimensional, Lipschitz, and attracts exponentially all trajectories. The authors introduce the notion of a spectral barrier for a nonlinear dissipative partial differential equation. Using this notion, the authors present a proof of existence of inertial manifolds that requires easily verifiable conditions, namely the existence of large enough spectral barriers.

Most equations of the type (1) possess absorbing sets, i.e., sets Y in which all the trajectories enter in finite time, never to leave again. Some of the proofs of existence of inertial manifolds assume that the nonlinear term R(u) is equal to zero outside Y. This procedure is called “Preparation”.

In section 3 the authors consider Kuramoto-Sivashinskij equation \[ (2)\quad u_ t+u_{xxxx}+u_{xx}+uu_ x=0, \] while in section 6 they consider reaction-diffusion systems \[ (3)\quad \partial u/\partial t- \Delta u+f(u)=0. \]

An inertial manifold is an invariant manifold that is finite dimensional, Lipschitz, and attracts exponentially all trajectories. The authors introduce the notion of a spectral barrier for a nonlinear dissipative partial differential equation. Using this notion, the authors present a proof of existence of inertial manifolds that requires easily verifiable conditions, namely the existence of large enough spectral barriers.

Most equations of the type (1) possess absorbing sets, i.e., sets Y in which all the trajectories enter in finite time, never to leave again. Some of the proofs of existence of inertial manifolds assume that the nonlinear term R(u) is equal to zero outside Y. This procedure is called “Preparation”.

In section 3 the authors consider Kuramoto-Sivashinskij equation \[ (2)\quad u_ t+u_{xxxx}+u_{xx}+uu_ x=0, \] while in section 6 they consider reaction-diffusion systems \[ (3)\quad \partial u/\partial t- \Delta u+f(u)=0. \]

Reviewer: F.M.Ragab

### MSC:

35B40 | Asymptotic behavior of solutions to PDEs |

35G10 | Initial value problems for linear higher-order PDEs |

35K25 | Higher-order parabolic equations |

### Keywords:

dissipative partial differential equation; inertial manifold; spectral barriers; Kuramoto-Sivashinskij equation; reaction-diffusion systems
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\textit{P. Constantin} et al., J. Dyn. Differ. Equations 1, No. 1, 45--73 (1989; Zbl 0701.35024)

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### References:

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