Sufficient conditions for the continuity of inertial manifolds for singularly perturbed problems

We consider a nonlinear evolution equation in the form

$ {{\rm{U}}_t} + {{\rm{A}}_\varepsilon }{\rm{U}} + {{\rm{N}}_\varepsilon }{{\rm{G}}_\varepsilon }({\rm{U}}) = 0,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left( {{{\rm{E}}_\varepsilon }} \right)$

together with its singular limit problem as $ \varepsilon\to 0 $

$ \begin{align*} U_t+{\rm A} U+ {\rm N}{\rm G}(U) = 0, \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left( {{{\rm{E}} }} \right)\end{align*} $

where $ \varepsilon\in (0,1] $ (possibly $ \varepsilon = 0 $), $ {\rm A}_\varepsilon $ and $ {\rm A} $ are linear closed (possibly) unbounded operators, $ {\rm N}_\varepsilon $ and $ {\rm N} $ are linear (possibly) unbounded operators, $ {\rm G}_\varepsilon $ and $ {\rm G} $ are nonlinear functions. We give sufficient conditions on $ {\rm A}_\varepsilon, $ $ {\rm N}_\varepsilon $ and $ {\rm G}_\varepsilon $ (and also $ {\rm A} $, $ {\rm N} $ and $ {\rm G} $) that guarantee not only the existence of an inertial manifold of dimension independent of $ \varepsilon $ for (E_ε) on a Banach space $ {\mathcal H} $, but also the Hölder continuity, lower and upper semicontinuity at $ \varepsilon = 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{\mathbb R}^d $ is a bounded domain with smooth boundary) or periodic BC (in which case $ \Omega = \Pi_{i = 1}^d(0,L_i), $ $ L_i>0 $), $ d = 1,2\; {\rm or} \;3$, are considered. These three classes of dissipative equations read

$ \begin{align*} \phi_{t}+N(\varepsilon \phi_t+N^{\alpha+1} \phi +N\phi + g(\phi))+\sigma\phi = 0,\quad\alpha\in\mathbb N, \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left( {{{\rm{P}}_\varepsilon }} \right)\end{align*} $

$ \begin{align*} \varepsilon \phi_{tt}+\phi_{t}+N^\alpha(N \phi + g(\phi))+ \sigma\phi = 0,\quad\alpha = 0, 1,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left( {{{\rm{H}}_\varepsilon }} \right) \end{align*} $

and

$ \begin{align*} \left\{\begin{aligned} & \phi_{t}+N^\alpha (N \phi + g(\phi)-u)+\sigma\phi = 0,\\&\varepsilon u_t+\phi_t+N u = 0,\end{aligned}\right.\quad\alpha = 0, 1,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left( {{{\rm{S}}_\varepsilon }} \right) \end{align*} $

respectively, where $ \sigma\ge 0 $ and the Laplace operator is defined as

$ N = -\Delta:{\mathscr D}(N) = \{\psi\in H^2(\Omega),\,\psi\,{\rm subject \,\,to \,\,the\,\, BC}\}\to \dot L^2(\Omega)\,\,{\rm or}\,\,L^2(\Omega). $

We assume that, for a given real number $ {\frak c}_1>0, $ there exists a positive integer $ n = n({\frak c}_1) $ such that $ \lambda_{n+1}-\lambda_n>{\frak c}_1 $, where $ \{\lambda_k\}_{k\in\mathbb N^*} $ are the eigenvalues of $ N $. There exists a real number $ {\mathscr M}>0 $ such that the nonlinear function $ g: V_j\to V_j $ satisfies the conditions $ \|g(\psi)\|_j\le {\mathscr M} $ and $ \|g(\psi)-g(\varphi)\|_{V_j}\le {\mathscr M}\|\psi-\varphi\|_{V_j} $, $ \forall\psi,\varphi\in V_j $, where $ V_j = {\mathscr D}(N^{j/2}) $, $ j = 1 $ for Problems (P_ε) and (S_ε) and $ j = 0, 2\alpha $ for Problem (H_ε). We further require $ g\in{\mathcal C}^1( V_j, V_j) $, $ \|g'(\psi)\varphi\|_j\le {\mathscr M}\|\varphi\|_j $ for Problems (H_ε) and (S_ε).