Dichotomy and periodic solutions to partial functional differential equations
Nguyen Thieu Huy Ngo Quy Dang
Discrete & Continuous Dynamical Systems - B 2017, 22(8): 3127-3144 doi: 10.3934/dcdsb.2017167

We establish a sufficient condition for existence and uniqueness of periodic solutions to partial functional differential equations of the form $\dot{u}=A(t)u+F(t)(u_t)+g(t,u_t)$ on a Banach space $X$ where the operator-valued functions $t\mapsto A(t)$ and $t\mapsto F(t)$ are $1$-periodic, the nonlinear operator $g(t,φ)$ is $1$-periodic with respect to $t$ for each fixed $φ∈ {\mathcal{C}}:=C([-r,0],X)$, and satisfying $\|g(t,φ_1)-g(t,φ_2)\|≤\varphi(t)\|φ_1-φ_2\|_C$ for $φ_1, φ_2∈ {\mathcal{C}}$ with $\varphi$ being a positive function such that $\sup_{t≥0}∈t_{t}^{t+1}\varphi(τ)dτ < ∞$. We then apply the results to study the existence, uniqueness, and conditional stability of periodic solutions to the above equation in the case that the family $(A(t))_{t≥ 0}$ generates an evolution family having an exponential dichotomy. We also prove the existence of a local stable manifold near the periodic solution in that case.

keywords: Partial functional differential equations periodic solutions exponential dichotomy conditional stability local stable manifolds
Inertial manifolds for a class of non-autonomous semilinear parabolic equations with finite delay
Cung The Anh Le Van Hieu Nguyen Thieu Huy
Discrete & Continuous Dynamical Systems - A 2013, 33(2): 483-503 doi: 10.3934/dcds.2013.33.483
Using the Lyapunov-Perron method, we prove the existence of an inertial manifold for the process associated to a class of non-autonomous semilinear parabolic equations with finite delay, where the linear principal part is positive definite with a discrete spectrum having a sufficiently large distance between some two successive point spectra, and the Lipschitz coefficient of the nonlinear term may depend on time and belongs to some admissible function spaces.
keywords: semilinear parabolic equations non-autonomous dynamical systems spectral gap condition Inertial manifolds finite delay admissible function spaces.
Boundedness and stability of solutions to semi-linear equations and applications to fluid dynamics
Nguyen Thieu Huy Vu Thi Ngoc Ha Pham Truong Xuan
Communications on Pure & Applied Analysis 2016, 15(6): 2103-2116 doi: 10.3934/cpaa.2016029
For an exterior domain $\Omega\subset R^d$ with smooth boundary, we study the existence and stability of bounded mild solutions in time $t$ to the abstract semi-linear evolution equation $u_t + Au = Pdiv (G(u)+F(t))$ where $-A$ generates a $C_0$-semigroup on the solenoidal space $L^d_{\sigma,w}(\Omega)$ (known as weak-$L^d$), $P$ is Helmholtz projection; $G$ is a nonlinear operator acting from $L^d_{\sigma,w}(\Omega)$ into $L^{d/2}_{\sigma,w}(\Omega)^{d^2}$, and $F(t)$ is a second-order tensor in $L^{d/2}_{\sigma,w}(\Omega)^{d^2}$. Our obtained abstract results can be applied not only to reestablish the known results on Navier-Stokes flows on exterior domains and/or around rotating obstacles, but also to obtain a new result on existence and polynomial stability of bounded solutions to Navier-Stokes-Oseen equations on exterior domains.
keywords: exterior domains. fluid dynamics boundedness and polynomial stability of solutions Navier-Stokes-Oseen equations Semi-linear evolution equations

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