| Abstract: |
| In this work we study the stochastic (Stratonovich) superconducting model with an infinite dimensional multiplicative noise in a bounded domain {%
\ $\mathcal{O}$ in $\mathbb{R}^{2}$}. The evolution system on a time
interval $(0,T)$ reads for a.s. in $\Omega $ as
\begin{equation*}
\left\{
\begin{array}{l}
du= \mathrm{div}(g(u)\ \nabla h)\,dt+\sigma (u)\circ d{\mathcal{W}}_{t}, \
\
-\Delta h+h=u,%
\end{array}%
\right.
\quad\quad \text{for}\quad {(t,\mathbf{x})\in }(0,T)\times
\mathcal{O}
\end{equation*}%
where {$u=u(t,\mathbf{x})$} denotes the cell density and $g(u)=|u|;$ ${%
\mathcal{W}}_{t}$ is a Wiener process given on the probability space $(\Omega
,\mathcal{F},P)$ and $\sigma =\sigma (u)$ is the diffusion coefficient. \ {%
This system is closed by the boundary condition
\begin{equation*}
h=a \quad \quad \quad \text{on}\quad (0,T)\times \partial {%
\mathcal{O}}
\end{equation*}%
on the boundary }$\partial $$\mathcal{O}$\ of the domain $\mathcal{O},$ {\,
the influx boundary condition }%
\begin{equation*}
u=b \quad \text{on}\quad (0,T)\times \partial {\mathcal{O}}^{%
\mathbf{-}}
\end{equation*}%
and the initial condition%
\begin{equation*}
u=u_{0}\quad \quad \quad \text{in}\quad \mathcal{O} \quad \text{at}\quad t=0,
\end{equation*}%
where
\begin{equation*}
\partial {\mathcal{O}}^{\mathbf{-}}=\left\{ \mathbf{x}\in \partial {\mathcal{%
O}}:\quad g^{\prime }(u)\left( \nabla h\cdot \mathbf{n}\right) (t,\mathbf{x}%
) |
|