## 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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