The well-posedness of the cauchy problem for partial differential equations with multiple characteristics

Regularity Properties, Representation of Solutions, and Spectral Asymptotics of Systems with Multiplicities

“…Example 1. In scalar equations with Levi conditions studied by Chazarain [3], Mizohata-Ohya [18], Zeman [28], one assumed that {a j , a k } = C jk (a j − a k ). It is clear that in this situation a j (x, ξ) = a k (x, ξ) implies {a j , a k }(x, ξ) = 0.…”

“…Example 1. In scalar equations with Levi conditions studied by Chazarain [3], Mizohata-Ohya [18], Zeman [28], one assumed that {a j , a…”

Hyperbolic Differential Equations and Wave Propagation

“…Thus the difficulties of proving an existence theorem are centred about the occurrence of non-real or multiple roots. The most extensive recent result related to multiple roots is a theorem by Zeman (22) which may be described briefly, as follows. We assume the most general hyperbolic condition ImA^x, r,^)>0, for f>0 and |£| = 1, tjeR n , thus excluding any exponential solutions that could increase arbitrarily rapidly for large times f>0.…”

The Cauchy problems for non-effectively hyperbolic differential operators and their product

“…W2 m _ i a module generated by L, = 1, ..., 2m.Hence we can apply the argument of[19] to this operator: Since Y 1we have, mod R (u) the greatest multiplicity of characteristic roots (not greater than m) and x e L '(X ) with support in outside of E.Using a partition of unity a n d taking the form o f [L, x] into consideration, we have for u e K c c X . Since t L also satisfies the same hypothesis, repeating the argument of [5], we have finished the proof of theorem 6.…”