SUFFICIENT CONDITIONS FOR THE CONTINUITY OF INERTIAL MANIFOLDS FOR SINGULARLY PERTURBED PROBLEMS

We consider a nonlinear evolution equation in the form U-t + A(epsilon)U + N(epsilon)G(epsilon)(U) = 0, (E-epsilon) together with its singular limit problem as epsilon -> 0 U-t + AU + NG(U) = 0, (E) where epsilon is an element of (0, 1] (possibly epsilon = 0), A(epsilon) and A are linear closed (possibly) un-bounded operators, N-epsilon and N are linear (possibly) unbounded operators, G(epsilon) and G are nonlinear functions. We give sufficient conditions on A(epsilon), N-epsilon and G(epsilon) (and also A, N and G) that guarantee not only the existence of an inertial man-ifold of dimension independent of epsilon for (E-epsilon) on a Banach space H, but also the Holder continuity, lower and upper semicontinuity at epsilon = 0 of the intersection of the inertial manifold with a bounded absorbing set. Applications to higher order viscous Cahn-Hilliard-Oono equations, the hyperbolic type equations and the phase-field systems, subject to either Neumann or Dirichlet boundary conditions (BC) (in which case Omega subset of R-d is a bounded domain with smooth boundary) or periodic BC (in which case Omega = Pi(d)(i=1) (0, L-i), L-i > 0), d = 1,2 or 3, are considered. These three classes of dissipative equations read phi(t) + N(epsilon phi(t) + N alpha+1 phi + N phi + g(phi)) + sigma phi = 0, alpha is an element of N, (P-epsilon) epsilon phi(tt) + phi(t) + N-alpha(N phi + g(phi)) + sigma phi = 0, alpha = 0, 1, (H-epsilon) and {phi(t) + N-alpha(N phi + g(phi) - u) + sigma phi = 0, epsilon u(t) + phi(t) + N-u = 0, alpha = 0,1, (S-epsilon) respectively, where sigma > 0 and the Laplace operator is defined as N = -Delta : D(N) = {psi is an element of H-2(Omega), psi subject to the BC} -> L-2(Omega) or L-2(Omega). We assume that, for a given real number c(1) > 0, there exists a positive integer n = n(c(1)) such that lambda(n+1) - lambda(n) > c(1), where {lambda(k)}(k is an element of N*) are the eigenvalues of N. There exists a real number M > 0 such that the nonlinear function g : V-j -> V-j satisfies the conditions parallel to g(psi)parallel to(j) <= M and parallel to g(psi) - g(phi)parallel to V-j <= M parallel to psi -phi parallel to V-j , V psi, phi is an element of V-j, where V-j = D(N-j/2), j = 1 for Problems (P-epsilon) and (S-epsilon) and j = 0, 2 alpha for Problem (H-epsilon). We further require g is an element of C-1(V-j, V-j), parallel to g'(psi)phi parallel to(j) <= M parallel to phi parallel to(j) for Problems (H-epsilon) and (S-epsilon).