CPAA
Exponential boundary stabilization for nonlinear wave equations with localized damping and nonlinear boundary condition
Takeshi Taniguchi
Communications on Pure & Applied Analysis 2017, 16(5): 1571-1585 doi: 10.3934/cpaa.2017075
Let
$ D\subset R^{d}$
be a bounded domain in the
$d- $
dimensional Euclidian space
$R^{d} $
with smooth boundary $Γ=\partial D.$ In this paper we consider exponential boundary stabilization for weak solutions to the wave equation with nonlinear boundary condition:
$\left\{ \begin{gathered}u_{tt}(t)-ρ(t)Δ u(t)+b(x)u_{t}(t)=f(u(t)), \\ u(t)=0\ \ \text{on }Γ_{0}×(0,T), \\ \dfrac{\partial u(t)}{\partialν}+γ(u_{t}(t))=0\ \ \text{on }Γ _{1}×(0,T), \\ u(0)=u_{0},u_{t}(0)=u_{1},\end{gathered} \right.$
where
$\left\| {{u_0}} \right\| < {\lambda _\beta }, $
$ E(0) < d_{β},$
where
$λ_{β}, $
$d_{β} $
are defined in (21), (22) and
$Γ=Γ_{0}\cupΓ_{1} $
and
$\bar{Γ}_{0}\cap\bar{Γ}_{1}=φ. $
keywords: Wave equation exponential behavior of solutions nonlinear boundary condition
DCDS
The exponential behavior of Navier-Stokes equations with time delay external force
Takeshi Taniguchi
Discrete & Continuous Dynamical Systems - A 2005, 12(5): 997-1018 doi: 10.3934/dcds.2005.12.997
In this paper we discuss the existence and the exponential behaviour of the solutions to a 2D-Navier-Stokes equation with time delay external force $f(t-\tau(t),u(t-\tau (t))),$ where $f(t,u)$ is a locally Lipschitz function in $u$ and $|f(t,u)|^2\leq a|u|^k+b_f,$ $a>0,b_f\geq 0,k\geq 2.$ $\tau (t)$ is a differentiable function with $0\leq \tau (t)\leq r, r>0,\frac{d}{dt}\tau (t)\leq M<1,$ $M$ a constant. We show the relations between the kinematic viscosity $\nu ,$ time delay $r>0$ and $\lambda_1, a, b_{f}, k, M$ play an important role. Furthermore, we consider the exponential behaviour of the strong solutions to a 3D-Navier-Stokes equation with time delay external force $f(t-\tau(t),u(t-\tau (t))),$ where $f(t,u)$ is a locally Lipschitz function in $u$ and $|f(t,u)|^2\leq a|u|^2+b_f,$ $a>0,b_f\geq 0.$ We extend Corollary 64.5[11]. Furthermore we discuss the existence of a periodic solution.
keywords: Navier-Stokes equations with delay external force strong solutions weak solutions periodic solutions
DCDS
The existence and decay estimates of the solutions to $3$D stochastic Navier-Stokes equations with additive noise in an exterior domain
Takeshi Taniguchi
Discrete & Continuous Dynamical Systems - A 2014, 34(10): 4323-4341 doi: 10.3934/dcds.2014.34.4323
In this paper we consider the local existence and global existence with probability $1-\sigma $ $(0<\sigma <1)$ of pathwise solutions to the three-dimensional stochastic Navier-Stokes equation perturbed by a cylindrical Wiener processe $W(t)$ in an exteriour domain: \begin{equation*} dX(t)=[-AX(t)+B\left( X(t)\right) +f_{\ast }(t)]dt+\Phi (t)dW(t), \end{equation*} where $A=-P\Delta $ is the Stokes operator, and $f_{\ast }(t)$ and $\Phi (t)$ satisfy some conditions. We also consider the decay of pathwise solutions.
keywords: strong solutions Stochastic Navier-Stokes equations exterior domain additive noise asymptotic behavior of solutions.

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