08-135 Stephane MALEK

On Gevrey functions solutions of partial differential equations with fuchsian and irregular singularities (428K, postscript) Jul 7, 08

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Abstract. We construct formal power series solutions of non linear partial integro-differential equations with fuchsian and irregular singularities at the origin in C^2 for given initial conditions being formal power series. We give sufficient conditions under which there exist actual sectorial holomorphic solutions which are Gevrey asymptotic to the given formal series solutions for given 1-summable formal series initial conditions. A phenomenon of small divisors is observed for the appearance of singularities of the Borel transform of the constructed formal series due to the presence of the fuchsian singularity. This property has an effect on the Gevrey order asymptotic for the constructed holomorphic solutions which becomes larger than the Gevrey order of the initial conditions.

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