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Approximate inertial manifolds for retarded semilinear parabolic equations. (English) Zbl 1039.35133
The author studies the following abstract version of a semilinear retarded parabolic equation $\frac {du}{dt}+Au=B(u_t), \tag{1}$ where $$A$$ is a linear positively defined selfadjoint operator in a Hilbert space $$H$$ whose inverse is compact, $$u_t(\theta):=u(t+\theta)$$ and the nonlinear operator $$B=B_r$$ maps the space $$C((-r,0),D(A^\alpha))$$, $$0\leq\alpha\leq1/2$$ to $$H$$ (thus, the number $$r>0$$ plays the role of a retardation time).
The main result of the paper is that, under the natural assumptions on the nonlinear term $$B$$ equation (1) possesses a family of approximate inertial manifolds. We however note that, in contrast to the well-known results on the nonretarded case, in the present paper the error of approximation $$\theta>0$$ cannot be taken arbitrarily small and should satisfy the inequality $$\theta\geq \theta_0(r)$$ where $$\theta_0(r)>0$$ depends on the retardation time $$r$$. Nevertheless, the author shows that $$\theta_0(r) \to0$$ as the retardation time $$r$$ tends to zero and the associated approximate manifolds tend as $$r\to0$$ to the approximate inertial manifolds of the corresponding non-retarded problem.

##### MSC:
 35R10 Functional partial differential equations 37L25 Inertial manifolds and other invariant attracting sets of infinite-dimensional dissipative dynamical systems
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