
1026 HIGHER-ORDER PARTIAL DIFFERENTIAL EQUATIONS
where p
k
(z) is a polynomial and i
2
= −1. Let r = r(σ) be the number of roots (taking into
account their multiplicities) of the characteristic equation
λ
m
−
m−1
X
k=0
p
k
(σ)λ
k
= 0 (2)
whose real parts are nonpositive (or bounded above) for a given σ. If r is the same (up to a
set of measure zero) for all σ ∈ (−∞, ∞), then E q. (1) is said to be regular with regularity
index r.
Classical equations such as the heat, wave, and L aplace equations are regular.
2
◦
. In the Cauchy problem for a regular equation (1), one should set r initial conditions of
the form
w
t=0
= f
0
(x),
∂w
∂t
t=0
= f
1
(x), . . . ,
∂
r−1
w
∂t
r−1