Admissible inertial manifolds for second order in time evolution equations

Document Type : Original Article


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



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.


Main Subjects