Exponential Stabilization of the Kirchhoff-Viscoelastic Equations
Resumo
In this work we study the existence, uniqueness and decay of solutions to a
class of viscoelastic equations in a separable Hilbert space $H$ given by
\begin{align}
\partial^2_tu(t)&+M([u(t)]Au(t)\nonumber\\
& -\int_0^tg(t-\tau )Au(\tau )d\tau =0, \label{e.glo1}\\
u(0)=u_{_{0}},&\qquad \partial_tu(0)=u_{_{1}}
\end{align}
We show that for a class of initial data in $\mathcal{D}(A)$, there exists global solutions for large data.
Such nonlinear model describing a homogeneous and isotropic viscoelastic solid. In recent years have been subject of study of researchers.
The particular case $A=-\Delta $ and $[u]= \|A^{1/2}u\|^2$, was studied by Torrejon and Young \cite{tor}. The authors showed the existence of global solution, for analytical data and the asymptotic stability when $t\rightarrow \infty$.
class of viscoelastic equations in a separable Hilbert space $H$ given by
\begin{align}
\partial^2_tu(t)&+M([u(t)]Au(t)\nonumber\\
& -\int_0^tg(t-\tau )Au(\tau )d\tau =0, \label{e.glo1}\\
u(0)=u_{_{0}},&\qquad \partial_tu(0)=u_{_{1}}
\end{align}
We show that for a class of initial data in $\mathcal{D}(A)$, there exists global solutions for large data.
Such nonlinear model describing a homogeneous and isotropic viscoelastic solid. In recent years have been subject of study of researchers.
The particular case $A=-\Delta $ and $[u]= \|A^{1/2}u\|^2$, was studied by Torrejon and Young \cite{tor}. The authors showed the existence of global solution, for analytical data and the asymptotic stability when $t\rightarrow \infty$.