Inertial manifolds for a singularly non-autonomous semi-linear parabolic equations

Abstract:

This paper devotes to the existence of an $N$-dimensional inertial manifold for a class of singularly, i.e. $A(t)$ may degenerate to $0$ at some time $t$, non-autonomous parabolic equations \begin{equation*} \partial _{t}u+A(t)u=F(t,u)+g(x,t),\;t>\tau ;\; \; u|_{t=\tau }=u_{\tau }(x),\;x\in \Omega , \end{equation*} where $A(t)\geq 0$ for any $t\geq \tau$, and $\Omega \subset \mathbb {R}^{d}$ is a bounded domain with smooth boundary. Since the operator $A(t)$ may degenerate, a compatibility condition for the operator $A(t)$ and the nonlinear term $F(t,u)$ was proposed to construct the inertial manifolds.