Vol. 52, No. 1, 1974

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Phragmén-Lindelöf type theorems for a system of nonhomogeneous equations

Kuang-Ho Chen

Vol. 52 (1974), No. 1, 17–28

Hyperbolic systems of N equations are considered

∂u ∕∂t = P(D )u+ f(x,t), with u in RN and (x,t) in Rn × R1,

where D = i∂∕∂x. For suitable source functions f(x,t) there are solutions satisfying the boundedness condition

                 γ     𝜃
|u(x,t)| ≦ Cexp{a|x| + b|t|},0 ≦ 𝜃 < 1,0 ≦ γ < p,

where p is the conjugate of 2p0, with p0 the reduced order of the matrix P(ξ). Furthermore, the solutions are polynomials in t if the initial states u(x,0) grow at infinity like polynomials. However, these solutions are not unique; a requirement of a certain type of u(x,0) at infinity is needed. The one-dimensional classical Phragmén-Lindelöf theorem and some results of Shilov for homogeneous systems are instances of this. It is the purpose here to supply a general (necessary and for some cases sufficient) condition for uniqueness. Preliminary to that a necessary condition is found on f(x,t) so that (1) admits solutions that are polynomials in t.

Mathematical Subject Classification 2000
Primary: 35L40
Received: 28 September 1973
Published: 1 May 1974
Kuang-Ho Chen