  The L\'evy walk process with rests is discussed. The jumping time is governed
by an $\alpha$-stable distribution with $\alpha>1$ while a waiting time
distribution is Poissonian and involves a position-dependent rate which
reflects a nonhomogeneous trap distribution. The master equation is derived and
solved in the asymptotic limit for a power-law form of the jumping rate. The
relative density of resting and flying particles appears time-dependent and the
asymptotic form of both distribution obey a stretched-exponential shape at
large time. The diffusion properties are discussed and it is demonstrated that,
due to the heterogeneous trap structure, the enhanced diffusion, observed for
the homogeneous case, may turn to a subdiffusion. The density distributions and
mean squared displacements are also evaluated from Monte Carlo simulations of
individual trajectories.
