Akademik Veri Yönetim Sistemi
Sixth International Conference on Analysis and Applied Mathematics, Antalya, Türkiye, 31 Ekim - 06 Kasım 2022, ss.123
The wave equation is one of the fundamental partial differential equation which arises in different fields such as electromagnetic, traffic flows, fluid dynamics, general relativity, acoustics, atmosphere and ocean dynamics, chemical reactions and biological sciences. By adding a noise term to deterministic equations one can incorporate neglected degrees of freedom, or can involve fluctuations of exterior fields that describes the media. By taking this effects with a space-time white noise into account, the following equation
\begin{equation}
du_{t}+\left[\alpha \left(t\right) u_{t}-\Delta u\right]dt =f \left( u\right)dt
+\sigma\left( u,u_{t},\nabla u\right) d W\left( x,t\right), \quad x \in \Omega,t>0 \label{Eq:stoc._KG}
\end{equation}
with initial and boundary conditions
\begin{align}
u\left( x,0\right) &=u_{0}\left( x\right) ,\quad u_{t}\left(
x,0\right) =u_{1}\left( x\right), \quad x \in \Omega, \label{Eq:init._cond.}\\
u\left( x,t\right) &=0, \quad x \in \partial \Omega, \quad t>0.
\label{Eq:bound._cond.}
\end{align}
is studied in this work. Here $\Omega \subset \mathbb{R}^{n}$, $d\geq 1$ is a bounded domain with smooth boundary, $W\left( x,t\right) $ is a Wiener process, $\alpha \left(t\right):\left[0,\infty\right)\rightarrow \left(0,\infty\right)$ is a nondecreasing, bounded differentiable function. A random exponential attractor was constructed for initial-boundary value problem of (1)-(3) with an additional $g\left(x,t\right)$ term in [1].
In this work, we obtain a local existence result, and then we give the conditions that ensure the blow-up of solutions.