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Interpolation spaces and weighted pseudo almost automorphic solutions to parabolic equations and applications to fluid dynamics

Thieu Huy Nguyen;  Thi Ngoc Ha Vu;  The Sac Le;  Truong Xuan Pham — 2022

Czechoslovak Mathematical Journal

We investigate the existence, uniqueness and polynomial stability of the weighted pseudo almost automorphic solutions to a class of linear and semilinear parabolic evolution equations. The necessary tools here are interpolation spaces and interpolation theorems which help to prove the boundedness of solution operators in appropriate spaces for linear equations. Then for the semilinear equations the fixed point arguments are used to obtain the existence and stability of the weighted pseudo almost...

Admissibly integral manifolds for semilinear evolution equations

Nguyen Thieu Huy;  Vu Thi Ngoc Ha — 2014

Annales Polonici Mathematici

We prove the existence of integral (stable, unstable, center) manifolds of admissible classes for the solutions to the semilinear integral equation u ( t ) = U ( t , s ) u ( s ) + ∫ s t U ( t , ξ ) f ( ξ , u ( ξ ) ) d ξ when the evolution family ( U ( t , s ) ) t ≥ s has an exponential trichotomy on a half-line or on the whole line, and the nonlinear forcing term f satisfies the (local or global) φ-Lipschitz conditions, i.e., ||f(t,x)-f(t,y)|| ≤ φ(t)||x-y|| where φ(t) belongs to some classes of admissible function spaces. These manifolds are formed by trajectories of the solutions belonging...

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