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Optimalité du caractère bien posé dans les classes de Gevrey du problème de Cauchy pour une équation faiblement hyperbolique. (French) Zbl 0533.35061
The author studies how the multiplicity of characteristic roots and lower order terms affect the regularity of solutions to the Cauchy problem for a hyperbolic operator \(P=\sum_{j+| \alpha | \leq m}a_{j,\alpha}(t,x)D^ j\!_ tD_ x\!^{\alpha}\) where \(a_{j,\alpha}\in C^{\infty}(V)\) for V, an open set in \({\mathbb{R}}^{n+1}\!\!\!_{t,x}\), \(P_ m(t,x;1,0)\neq 0\) and the characteristic roots are real. First he gives an energy estimate in a sharp form in case P has characteristic roots of multiplicity \(\leq d\) and, conversely, proves if \(\| u\|_{p+m-d}\leq C\| Pu\|_ p, u\in C_ 0\!^{\infty}(U)\), U (open \(set)\subset [0,T]\times {\mathbb{R}}^ n\) holds, then the multiplicity of characteristic roots are at most d. Next, he assumes \[ \int^{t}_{0}\|<{\hat \xi},D>^{m- d}u(\tau)\|_ 0\!^{\ell}d\tau \leq C\int^{t}_{0}(t-\tau)^ p\| Pu(\tau)\|_ 0\!^{\ell}d\tau \quad(\ell =1\quad or\quad 2) \] or \(\| u\|_ p\leq C\| Pu\|_ q\) hold for any \(u\in C_ 0\!^{\infty}(V_{[0,T]})\) where \(V_{[0,T]}=\{(t,x)\in V| 0\leq t\leq T\}.\) Then he derives some relations between r,d,p and q under the hypotheses that the Cauchy problem for P is well posed in a sense of Ivrij-Petkov and P has a finite propagation speed and that P has a characteristic root of multiplicity r.
Finally he discusses the effects of lower order terms to the energy inequalities and derives a similar type theorem to the result of V. Y. Ivrij and V. M. Petkov [Usp. Mat. Nauk 29, No.5(179), 3-70 (1974; Zbl 0312.35049)] for P replaced by the effectively hyperbolic operator.
Reviewer: T.Kakita
35L25 Higher-order hyperbolic equations
35L80 Degenerate hyperbolic equations
35D10 Regularity of generalized solutions of PDE (MSC2000)
35A05 General existence and uniqueness theorems (PDE) (MSC2000)