Abstract
We construct formal power series solutions of nonlinear partial integro-differential equations with Fuchsian and irregular singularities at the origin of \( \mathbb{C}^2 \) 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 asymptotic order for the constructed holomorphic solutions which becomes larger than the Gevrey order of the initial conditions.
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Malek, S. On Gevrey functional solutions of partial differential equations with Fuchsian and irregular singularities. J Dyn Control Syst 15, 277–305 (2009). https://doi.org/10.1007/s10883-009-9061-4
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DOI: https://doi.org/10.1007/s10883-009-9061-4
Key words and phrases
- Cauchy problem
- partial integro-differential equation
- Gevrey estimates
- Fuchsian singularity
- irregular singularity
- small divisor