Sufficient conditions for the continuity of inertial manifolds for singularly perturbed problems
Authors: Bonfoh Ahmed 
Published: September 2021

<p style='text-indent:20px;'>We consider a nonlinear evolution equation in the form</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1a"> \begin{document}$ {{\rm{U}}_t} + {{\rm{A}}_\varepsilon }{\rm{U}} + {{\rm{N}}_\varepsilon }{{\rm{G}}_\varepsilon }({\rm{U}}) = 0,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left( {{{\rm{E}}_\varepsilon }} \right)$\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>together with its singular limit problem as <inline-formula><tex-math id="M1">\begin{document}$ \varepsilon\to 0 $\end{document}</tex-math></inline-formula></p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1E"> \begin{document}$ \begin{align*} U_t+{\rm A} U+ {\rm N}{\rm G}(U) = 0, \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left( {{{\rm{E}} }} \right)\end{align*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M2">\begin{document}$ \varepsilon\in (0,1] $\end{document}</tex-math></inline-formula> (possibly <inline-formula><tex-math id="M3">\begin{document}$ \varepsilon = 0 $\end{document}</tex-math></inline-formula>), <inline-formula><tex-math id="M4">\begin{document}$ {\rm A}_\varepsilon $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M5">\begin{document}$ {\rm A} $\end{document}</tex-math></inline-formula> are linear closed (possibly) unbounded operators, <inline-formula><tex-math id="M6">\begin{document}$ {\rm N}_\varepsilon $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M7">\begin{document}$ {\rm N} $\end{document}</tex-math></inline-formula> are linear (possibly) unbounded operators, <inline-formula><tex-math id="M8">\begin{document}$ {\rm G}_\varepsilon $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M9">\begin{document}$ {\rm G} $\end{document}</tex-math></inline-formula> are nonlinear functions. We give sufficient conditions on <inline-formula><tex-math id="M10">\begin{document}$ {\rm A}_\varepsilon, $\end{document}</tex-math></inline-formula> <inline-formula><tex-math id="M11">\begin{document}$ {\rm N}_\varepsilon $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M12">\begin{document}$ {\rm G}_\varepsilon $\end{document}</tex-math></inline-formula> (and also <inline-formula><tex-math id="M13">\begin{document}$ {\rm A} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M14">\begin{document}$ {\rm N} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M15">\begin{document}$ {\rm G} $\end{document}</tex-math></inline-formula>) that guarantee not only the existence of an inertial manifold of dimension independent of <inline-formula><tex-math id="M16">\begin{document}$ \varepsilon $\end{document}</tex-math></inline-formula> for (E<sub><i>ε</i></sub>) on a Banach space <inline-formula><tex-math id="M17">\begin{document}$ {\mathcal H} $\end{document}</tex-math></inline-formula>, but also the Hölder continuity, lower and upper semicontinuity at <inline-formula><tex-math id="M18">\begin{document}$ \varepsilon = 0 $\end{document}</tex-math></inline-formula> of the intersection of the inertial manifold with a bounded absorbing set. Applications to higher order viscous Cahn-Hilliard-Oono equations, the hyperbolic type equations and the phase-field systems, subject to either Neumann or Dirichlet boundary conditions (BC) (in which case <inline-formula><tex-math id="M19">\begin{document}$ \Omega\subset{\mathbb R}^d $\end{document}</tex-math></inline-formula> is a bounded domain with smooth boundary) or periodic BC (in which case <inline-formula><tex-math id="M20">\begin{document}$ \Omega = \Pi_{i = 1}^d(0,L_i), $\end{document}</tex-math></inline-formula> <inline-formula><tex-math id="M21">\begin{document}$ L_i&gt;0 $\end{document}</tex-math></inline-formula>), <inline-formula><tex-math id="M22">\begin{document}$ d = 1,2\; {\rm or} \;3$\end{document}</tex-math></inline-formula>, are considered. These three classes of dissipative equations read</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE2a"> \begin{document}$ \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*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE3a"> \begin{document}$ \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*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>and</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE4a"> \begin{document}$ \begin{align*} \left\{\begin{aligned} &amp; \phi_{t}+N^\alpha (N \phi + g(\phi)-u)+\sigma\phi = 0,\\&amp;\varepsilon u_t+\phi_t+N u = 0,\end{aligned}\right.\quad\alpha = 0, 1,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left( {{{\rm{S}}_\varepsilon }} \right) \end{align*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>respectively, where <inline-formula><tex-math id="M23">\begin{document}$ \sigma\ge 0 $\end{document}</tex-math></inline-formula> and the Laplace operator is defined as</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ 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). $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>We assume that, for a given real number <inline-formula><tex-math id="M24">\begin{document}$ {\frak c}_1&gt;0, $\end{document}</tex-math></inline-formula> there exists a positive integer <inline-formula><tex-math id="M25">\begin{document}$ n = n({\frak c}_1) $\end{document}</tex-math></inline-formula> such that <inline-formula><tex-math id="M26">\begin{document}$ \lambda_{n+1}-\lambda_n&gt;{\frak c}_1 $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M27">\begin{document}$ \{\lambda_k\}_{k\in\mathbb N^*} $\end{document}</tex-math></inline-formula> are the eigenvalues of <inline-formula><tex-math id="M28">\begin{document}$ N $\end{document}</tex-math></inline-formula>. There exists a real number <inline-formula><tex-math id="M29">\begin{document}$ {\mathscr M}&gt;0 $\end{document}</tex-math></inline-formula> such that the nonlinear function <inline-formula><tex-math id="M30">\begin{document}$ g: V_j\to V_j $\end{document}</tex-math></inline-formula> satisfies the conditions <inline-formula><tex-math id="M31">\begin{document}$ \|g(\psi)\|_j\le {\mathscr M} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M32">\begin{document}$ \|g(\psi)-g(\varphi)\|_{V_j}\le {\mathscr M}\|\psi-\varphi\|_{V_j} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M33">\begin{document}$ \forall\psi,\varphi\in V_j $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M34">\begin{document}$ V_j = {\mathscr D}(N^{j/2}) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M35">\begin{document}$ j = 1 $\end{document}</tex-math></inline-formula> for Problems (P<sub><i>ε</i></sub>) and (S<sub><i>ε</i></sub>) and <inline-formula><tex-math id="M36">\begin{document}$ j = 0, 2\alpha $\end{document}</tex-math></inline-formula> for Problem (H<sub><i>ε</i></sub>). We further require <inline-formula><tex-math id="M37">\begin{document}$ g\in{\mathcal C}^1( V_j, V_j) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M38">\begin{document}$ \|g'(\psi)\varphi\|_j\le {\mathscr M}\|\varphi\|_j $\end{document}</tex-math></inline-formula> for Problems (H<sub><i>ε</i></sub>) and (S<sub><i>ε</i></sub>).</p>