Gevrey regularity and summability of the formal solutions in time of some inhomogeneous linear systems of partial differential equations

In this article, we investigate the Gevrey regularity and the summability of the formal solutions in time of some inhomogeneous linear systems of partial differential equations with analytic coefficients at the origin of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>ℂ</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> . Given such a system, we first prove that the Gevrey regularity of its formal solutions follows a noteworthy dichotomy with respect to the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>s</mml:mi> </mml:math> -Gevrey regularity of the inhomogeneity: for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>s</mml:mi> <mml:mo>≥</mml:mo> <mml:msub> <mml:mi>s</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:mrow> </mml:math> with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>s</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> a nonnegative rational number entirely determined by a convenient Newton polygon attached to the given system, the solutions inherit the Gevrey regularity of the inhomogeneity; for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>s</mml:mi> <mml:mo>&lt;</mml:mo> <mml:msub> <mml:mi>s</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:mrow> </mml:math> , if any exists, the solutions keep the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>s</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> -Gevrey regularity defined by the structure of the initial system. Then, assuming <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>s</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> and some convenient additional assumptions on the system, we give a necessary and sufficient condition for the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:msub> <mml:mi>s</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:mrow> </mml:math> -summability of the formal solutions in a given direction. Several examples illustrate these results.