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Well-posedness in the Gevrey classes of the Cauchy problem for a non- strictly hyperbolic equation with coefficients depending on time. (English) Zbl 0543.35056

The authors consider the Cauchy problem \((1)\quad u_{tt}- \sum^{n}_{i,j=1}a_{ij}(t)u_{x_ ix_ j}=0, u(x,0)=\phi(x),\quad u_ t(x,0)=\psi(x)\) on \({\mathbb{R}}^ n\times [0,T]\), provided the non- strict hyperbolicity condition \(\sum_{i,j}a_{ij}(t)\xi_ i\xi_ j\geq 0,\) for all \(\xi \in {\mathbb{R}}^ n\) is fulfilled. They prove that if the coefficients \(a_{ij}\in C^{k,\alpha}([0,T])\) with \(k\geq 0\) integer and 0\(\leq \alpha \leq 1\), then the problem (1) is well posed in the Gevrey class \({\mathcal E}^ s\) if \(1\leq s<1+(k+\alpha)/2.\) If the coefficients \(a_{ij}\) are analytic on [0,T], then (1) is \(C^{\infty}\) well-posed. The proof is based on the Fourier-Laplace transform as well as on approximate energy estimates. Some counter-examples are considered showing that the principal result cannot be improved.
Reviewer: V.Petkov

MSC:

35L15 Initial value problems for second-order hyperbolic equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35A22 Transform methods (e.g., integral transforms) applied to PDEs
35B45 A priori estimates in context of PDEs
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References:

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