Smoothness of inertial manifolds for semilinear evolution equations in complex Banach spaces

Abstract

We study inertial manifolds for a semilinear evolution equation $du/dt+Au=F(t,u)$ in a complex Banach space. It is known that various conditions ensure existence of inertial manifolds for the equation, however, Miklavčič gave a sharp but simple condition so as to show the existence of inertial manifolds. In this paper, we show smoothness of inertial manifolds using the sharp condition with additional assumptions on $F$, and also apply to a scalar reaction diffusion equation $u_t-u_{xx}=f(t,x,u,u_x)$ with the Dirichlet boundary conditions.