
956 HIGHER-ORDER PAR TIAL DIFFERENTIAL EQUATIONS
2.
∂w
∂t
− a∆w − b
∂
∂t
∆w = Φ(x, t).
Equation of filtration of a fluid in a cracked porous medium.
Domain: x ∈ R
n
. Cauchy problem.
An initial condition is prescribed:
w = f (x) at t = 0.
Solution:
w =
Z
t
0
Z
R
n
E
e
(x − y, t −τ)Φ(y, τ ) dV
y
dτ +
Z
R
n
E
e
(x − y, t)
f(y) − b∆
y
f(y)
dV
y
,
where ∆
y
is the Laplace operator in the integration variables y
1
, . . . , y
n
.
For the function E
e
(x, t) with n = 2 and n = 3, see Items 1
◦
and 2
◦
in Eq. 11.1.5.1.
3.
∂
2
w
∂t
2
− a∆w − b
∂
∂t
∆w = 0.
This equation arises when decomposing unsteady equations describing 3D motions of vis-
cous compressible barotropic fluids (see Section 12.12.2).
Consider a fi