Inertial manifolds for semi-linear parabolic equations in admissible spaces

We prove the existence of inertial manifolds for the solutions to the semi-linear parabolic equation when the partial differential operator A is positive definite and self-adjoint with a discrete spectrum having a sufficiently large distance between some two successive points of the spectrum, and the nonlinear forcing term f satisfies the φ-Lipschitz conditions on the domain D(Aθ), 0⩽θ<1, i.e., ‖f(t,x)−f(t,y)‖⩽φ(t)‖Aθ(x−y)‖ and ‖f(t,x)‖⩽φ(t)(1+‖Aθx‖) where φ(t) belongs to one of admissible function spaces containing wide classes of function spaces like Lp-spaces, the Lorentz spaces Lp,q and many other function spaces occurring in interpolation theory.