Speaker
Description
This work investigates the well-posedness and asymptotic stability of a coupled system of Lam´e wave equations with logarithmic nonlinearity and time delay.
\begin{equation}
\left{
\begin{array}{ll}
\mathbf{Y}{tt}-\gamma \Delta \mathbf{Y}-(\alpha +\gamma )\nabla (\operatorname{div} \mathbf{Y})+\alpha
{1}\mathbf{Y}{t}+\alpha {2}\mathbf{Y}{t}(t-\tau {1})=F_{1}(Y,Z), &
x\in \Omega ,\,t>0, \
\mathbf{Z}{tt}-\gamma \Delta \mathbf{Z}-(\alpha +\gamma )\nabla (\operatorname{div} \mathbf{Z})+\beta
{1}\mathbf{Z}{t}+\beta {2}\mathbf{Z}{t}(t-\tau {2})=F_{1}(Y,Z),&
x\in \Omega ,\,t>0, \
\mathbf{Y}(x,t)=\mathbf{Z}(x,t)=0, & x\in \partial \Omega ,\,t>0, \
\mathbf{Y}(x,0)=\mathbf{Y}{0},\,\mathbf{Y}{t}(x,0)=\mathbf{Y}{1}, & x\in \Omega , \
\mathbf{Z}(x,0)=\mathbf{Z}{0},\,\mathbf{Z}{t}(x,0)=\mathbf{Z}{1}, & x\in \Omega , \
\mathbf{Y}{t}(x,t-\tau {1})=\mathbf{f}{0}(x,t-\tau {1}), & \left( x,t\right) \in \Omega
\times (0,\tau {1}), \
\mathbf{Z}{t}(x,t-\tau {2})=\mathbf{g}{0}(x,t-\tau {2}), & \left( x,t\right) \in \Omega
\times (0,\tau {2}).%
\end{array}%
\right.
\end{equation}
Here:
\begin{equation}
F_{1}(Y,Z)=Y |Y|^{p-2}|Z|^{p}\ln (|Y|^{k}|Z|^{k}), \qquad
F_{2}(Y,Z)=Z |Z|^{p-2}|Y|^{p}\ln (|Y|^{k}|Z|^{k}),
\end{equation}
where $|Y| = \sqrt{\sum_{i=1}^3 Y_i^2}$ denotes the Euclidean norm in $\mathbb{R}^3$. $\Omega \subset \mathbb{R}^3$ is a bounded domain with smooth boundary $\partial \Omega$, $t>0$, $k$ and $p\geq 2$ are positive constants. Here $\mathbf{Y}, \mathbf{Z}: \Omega \times (0,T) \to \mathbb{R}^3$ are vector-valued functions. Our paper features a truly coupled logarithmic term depending on the product $|\mathbf{Y}|^k|\mathbf{Z}|^k$, which is more challenging to analyze due to the mixed dependence. Using semigroup theory and potential well methods, we establish local and global existence of weak solutions, and under enhanced damping conditions, we prove exponential decay of the total energy. This work significantly improves upon existing results: Boularars [4] analyzed a coupled Lam´e system with separate logarithmic sources and without delay, while Yazid et al. [14] studied a scalar Lam´e equation with delay but without full vector coupling. Our contribution unifies and extends these by treating a fully coupled vector-valued Lam´e system incorporating both logarithmic nonlinearity and time delay simultaneously, providing a comprehensive framework for such multi-component elastic models