Regularity and asymptotic behavior of solutions of nonautonomous differential equations (Q1898806)

Regularity and asymptotic behavior of solutions and the existence of a nonautonomous approximate inertial manifold is obtained for abstract evolution equations of the form \[ u'+ Au+ N(t, u)=0, \] in which \(A\) is a selfadjoint operator with compact resolvent in a Hilbert space \(H\). The operator \(N(t, u)= G(u)+ F(t, u)\) is nonlinear with \(G\) a monotone gradient that is locally Lipschitz from \(D(A^{1/2})\) into \(H\), and \(F:\mathbb{R}^+\times H\to H\) a Lipschitz perturbation which is Hölder continuous in \(t\). Weak solutions are shown to be uniformly locally Hölder into \(D(A)\) with equicontinuity in families of solutions with \(|u(0) |\leq r\). A priori estimates of \(|Au(t)|\) are also verified, and there is a global attractor whose component elements form an equicontinuous family of solutions.