On parametric multisummable formal solutions to some nonlinear initial value Cauchy problems

We study a nonlinear initial value Cauchy problem depending upon a complex perturbation parameter whose coefficients depend holomorphically on ( , t) near the origin in C and are bounded holomorphic on some horizontal strip in C w.r.t the space variable. In our previous contribution [14], 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 t and on sectors w.r.t the parameter whose union form a covering of some neighborhood of 0 in C∗, which are asked to share a common formal power series asymptotic expansion of some Gevrey order as 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 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 multisummability in is also analyzed thoroughly. The proof leans on a version of the so-called Ramis-Sibuya theorem which entails two distinct Gevrey orders. 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 ( , t) near 0 and bounded holomorphic on a strip in the complex space variable.