
11.6. Higher-Order Linear Equations with Constant Coefficients 1021
Here dV
y
= dy
1
. . . dy
n
and the convolution E
e
∗ Φ is assumed to make sense.
⊙ Literature: G. E. Shilov (1965), S. G. Krein (1972), L. H¨ormander (1983), V. S. Vladimirov (1988).
◮ Domain: 0 ≤ t < ∞, −∞ < x
k
< ∞; k = 1, . . . , n. Cauchy problem.
Now let P
∂
∂t
,
∂
∂x
1
, . . . ,
∂
∂x
n
be a constant coefficient linear differential operator of or-
der m with respect to t. Then a distribution E (t, x) = E (t, x
1
, . . . , x
n
) that is a solution of
the homogeneous equation
P
∂
∂t
,
∂
∂x
1
, . . . ,
∂
∂x
n
E (t, x) = 0 (1)
and satisfies the initial conditions
∗
E
t=0
= 0,
∂E
∂t
t=0
= 0, . . . ,
∂
m−2
E
∂t
m−2
t=0