Approximate inertial manifolds for retarded semilinear parabolic equations. (Q1399861)

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.