  We consider a random walk with death in $[-N,N]$ moving in a time dependent
environment. The environment is a system of particles which describes a current
flux from $N$ to $-N$. Its evolution is influenced by the presence of the
random walk and in turn it affects the jump rates of the random walk in a
neighborhood of the endpoints, determining also the rate for the random walk to
die. We prove an upper bound (uniform in $N$) for the survival probability up
to time $t$ which goes as $c\exp\{-bN^{-2}t\}$, with $c$ and $b$ positive
constants.
