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 d u ( t ) d t + A u ( t ) = f ( t , u ) 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 ( A θ ) , 0 ⩽ θ < 1 , i.e., ‖ f ( t , x ) − f ( t , y ) ‖ ⩽ φ ( t ) ‖ A θ ( x − y ) ‖ and ‖ f ( t , x ) ‖ ⩽ φ ( t ) ( 1 + ‖ A θ x ‖ ) where φ ( t ) belongs to one of admissible function spaces containing wide classes of function spaces like L p -spaces, the Lorentz spaces L p , q and many other function spaces occurring in interpolation theory.