Admissible inertial manifolds for second order in time evolution equations

Abstract

We prove the existence of admissible inertial manifolds<br /> for the second order in time evolution equations of the form<br /> $$ ddot{x}+2varepsilon dot{x}+Ax=f(t,x)$$<br /> when $A$ is positive definite and self-adjoint with a discrete spectrum<br /> and the nonlinear term $f$ satisfies the $varphi$-Lipschitz condition, that is,<br /> $|f(t,x)-f(t,y)|leqslantvarphi(t)left |A^{beta}(x-y)right |$<br /> for $varphi$ belonging to one of the admissible Banach 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.