|We consider the diffusion phenomena
for the linear wave equation with space-dependent damping.
It is well known that if the damping term has a constant coefficient,
then the solution behaves like that of the corresponding heat equation
as time goes to infinity.
On the other hand, it is also known that
when the coefficient of the damping term decays
sufficiently fast near infinity,
the solution behaves like that of the free wave equation.
In this talk, we consider the intermediate case,
that is, the case where the coefficient of the damping term
decays slowly near infinity,
and we show that the asymptotic profile of the solution is given by
a solution of the corresponding heat equation.
This result corresponds to that of J. Wirth,
who treated the time-dependent damping cases.
The key point of the proof is weighted energy estimates
for higher order derivatives,
which are based on some recent results by
G. Todorova, B. Yordanov, R. Ikehata and K. Nishihara.