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Global Gevrey solvability for a class of perturbations of involutive systems. (English) Zbl 1454.35050
In this article, the author is interested in the global solvability, in the Gevrey meaning, on the $$n$$-dimensional torus $$\mathbb T^n$$ of the system $(L_j=\partial_{t_j}+a_j(t)\partial_x+b_j(t))_{1\leq j\leq n}\in\mathbb T_t^n\times S_x^1,$ where $$a_j\in G^s(\mathbb T^n,\mathbb R)$$ and $$b_j\in G^s(\mathbb T^n)$$ are both $$s$$-Gevrey on the torus $$\mathbb T^n$$, and where $$\displaystyle\sum_{j=1}^n a_jdt_j$$ and $$\displaystyle\sum_{j=1}^n b_jdt_j$$ are both closed. More precisely, he focuses in the following question: supposing that the system $(\partial_{t_j}+a_j(t)\partial_x)_{1\leq j\leq n}$ is globally $$s$$-solvable, when the system $$(L_j)_{1\leq j\leq n}$$ is also globally $$s$$-solvable?
To do that, the author first reduces the study of the system $$(L_j)_{1\leq j\leq n}$$ to the study of a system whose the principal part has constant coefficients. Then, he generalizes the results of [G. Petronilho and S. L. Zani, J. Differ. Equations 244, No. 9, 2372–2403 (2008; Zbl 1155.35010)] in order to characterize the global $$s$$-solvability of the latter system.
##### MSC:
 35F05 Linear first-order PDEs 35N10 Overdetermined systems of PDEs with variable coefficients
##### Keywords:
involutive system; global Gevrey solvability
Full Text:
##### References:
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