Necessary Conditions for the Cauchy Problem for Non-Strictly Hyperbolic Equations to Be Well-Posed

Geometric transition for a class of hyperbolic operators with double characteristics

“…9 We point out that in obtaining the second term in the right hand side of the above inequality we used an analogue of Lemma 4.3.1 in the opposite direction , i.e. estimating µ0p(*{ f, q}) with Op({ fµ, Q2}).…”

“…Recalling the celebrated necessary conditions of Ivrii-Petkov ( [9], [8]) we see that in order that the local Cauchy problem for P2 (and hence for P2) be well posed we must necessarily have k(x) = x22lk(x). Hence if k1 in Example C is zero, we conclude that Assumption (H4) cannot be improved.…”

Weakly Hyperbolic Equations — A Modern Field in the Theory of Hyperbolic Equations

“…From [Ole70] we know that the Cauchy problem for u tt − t 2l x 2n u xx + t k x m u x = 0 is C ∞ -well-posed if k ≥ l − 1, m ≥ n. The necessity of this conditions was proved in [IP74]. These conditions are called Levi conditions of C ∞ -type which are used to ensure the well-posedness in C ∞ .…”

Continuation of Bicharacteristics for Effectively Hyperbolic Operators

“…(There were many contributions of proving this fact in general, and Iwasaki's factorization theorem in [10] was fundamental in order to reduce the general problem to a special case.) The notion of effective hyperbolicity was first introduced by Ivrii and Petkov in [9]. Its definition requires the existence of (necessarily two) non-vanishing real eigenvalues of the Hamilton map i.e.…”