Cauchy Problem for Differential Operators with Double Characteristics by Tatsuo Nishitani

Author:Tatsuo Nishitani
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

5.1 Main Results on Well-Posedness

We study the Cauchy problem for with principal symbol p(x, ξ);

where A j (x, ξ′) ∼ A j0 + A j1 + ⋯ and A 0 = 1.

Lemma 5.1

Let Γ be a conic neighborhood of , and I be an open interval containing 0. Assume that p has a factorization p(x, ξ) = p 1(x, ξ)p 2(x, ξ) when (x 0, x′, ξ′) ∈ I ×Γ where

with a j ν (x, ξ′) ∈ C ∞ (I ×Γ) which are homogeneous of degree j in ξ′. Assume that p 1 and p 2 have no common zero ξ 0 for (x 0, x′, ξ′) ∈ I ×Γ. Then one can find , i = 1, 2 such that P ≡ P 1 P 2 at (0, ρ′).

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