Regularly hyperbolic systems and Gevrey classes.(English)Zbl 0583.35074

This paper deals with the first order Cauchy problem $(1)\quad \partial U/\partial t=\sum A_ h(t,x) \partial U/\partial x_ h+B(t,x),\quad U(0,x)=g(x),$ $$0\leq t\leq T$$, $$x\in {\mathbb{R}}^ n$$, where $$A_ h$$ (1$$\leq h\leq n)$$ and $$B$$ are $$N\times N$$ real matrices, while U and g are real $$N$$-vectors. System (1) is assumed to be regularly hyperbolic. Suppose that the coefficients $$A_ h(t,x)$$ are Hölder continuous of order $$\alpha$$ in t $$(0<\alpha <1)$$ and belong to the Gevrey class of order s in x and that $$B(t,x)$$ is locally bounded and belongs to the Gevrey class of order s in x. Then the author proves that the Cauchy problem is well posed in the Gevrey class of order s provided that $$1\leq s<1/(1-\alpha)$$. The method of energy inequalities is used.
Reviewer: P.Jeanquartier

MSC:

 35L45 Initial value problems for first-order hyperbolic systems 35F10 Initial value problems for linear first-order PDEs 35R25 Ill-posed problems for PDEs 35L40 First-order hyperbolic systems 35L85 Unilateral problems for linear hyperbolic equations and variational inequalities with linear hyperbolic operators
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References:

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