  We present a simple paradigm for detection of an immobile target by a
space-time coupled random walker with a finite lifetime. The motion of the
walker is characterized by linear displacements at a fixed speed and
exponentially distributed duration, interrupted by random changes in the
direction of motion and resumption of motion in the new direction with the same
speed. We call these walkers "mortal creepers". A mortal creeper may die at any
time during its motion according to an exponential decay law characterized by a
finite mean death rate $\omega_m$. While still alive, the creeper has a finite
mean frequency $\omega$ of change of the direction of motion. In particular, we
consider the efficiency of the target search process, characterized by the
probability that the creeper will eventually detect the target. Analytic
results confirmed by numerical results show that there is an
$\omega_m$-dependent optimal frequency $\omega=\omega_{opt}$ that maximizes the
probability of eventual target detection. We work primarily in one-dimensional
($d=1$) domains and examine the role of initial conditions and of finite domain
sizes. Numerical results in $d=2$ domains confirm the existence of an optimal
frequency of change of direction, thereby suggesting that the observed effects
are robust to changes in dimensionality. In the $d=1$ case, explicit
expressions for the probability of target detection in the long time limit are
given. In the case of an infinite domain, we compute the detection probability
for arbitrary times and study its early- and late-time behavior. We further
consider the survival probability of the target in the presence of many
independent creepers beginning their motion at the same location and at the
same time. We also consider a version of the standard "target problem" in which
many creepers start at random locations at the same time.
