# Admissible inertial manifolds for second order in time evolution equations

Document Type : Original Article

Author

Department of Mathematical Analysis, Faculty of Natural Sciences, Hongduc University, Thanh Hoa, Vietnam

10.22034/kjm.2020.109813

Abstract

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

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