Alberto Lastra, Stephane Malek
On parametric multisummable formal solutions to some nonlinear initial value Cauchy problems
(872K, pdf)
ABSTRACT. We study a nonlinear initial value Cauchy problem depending upon a complex perturbation parameter whose coefficients depend holomorphically on time near the origin
and are bounded holomorphic on some horizontal strip in the complex domain w.r.t the space variable. In a previous
contribution, we assumed the forcing term of the Cauchy problem to be analytic near 0. Presently, we consider a family of forcing terms that are holomorphic on a common sector in time and on sectors w.r.t the
parameter whose union form a covering of some neighborhood of 0, which are asked to share a common formal power series asymptotic expansion of some Gevrey order as the parameter tends
to 0. We construct a family of actual holomorphic solutions to our Cauchy problem defined on the
sector in time and on the sectors in the parameter mentioned above. These solutions are achieved by means
of a version of the so-called accelero-summation method in the time variable and by Fourier inverse transform in
space. It appears that these functions share a common formal asymptotic expansion in the perturbation parameter.
Furthermore, this formal series expansion can be written as a sum of two formal series with a corresponding
decomposition for the actual solutions which possess two different asymptotic Gevrey orders, one steming
from the shape of the equation and the other originating from the forcing terms. The special case of parametric multisummability in is also analyzed thoroughly. Finally, we give an application to the study of parametric multi-level Gevrey solutions
for some nonlinear initial value Cauchy problems with holomorphic coefficients and forcing term in time near 0 and
bounded holomorphic on a strip in the complex space variable.