We study a nonlinear initial value Cauchy problem depending upon a complex perturbation parameter with vanishing initial data at complex time t = 0 and whose coefficients depend analytically on ( , t) near the origin in C 2 and are bounded holomorphic on some horizontal strip in C w.r.t. the space variable. This problem is assumed to be non-Kowalevskian in time t, therefore analytic solutions at t = 0 cannot be expected in general. Nevertheless, we are able to construct a family of actual holomorphic solutions defined on a common bounded open sector with vertex at 0 in time and on the given strip above in space, when the complex parameter belongs to a suitably chosen set of open bounded sectors whose union form a covering of some neighborhood of 0 in C * . These solutions are achieved by means of Laplace and Fourier inverse transforms of some common -depending function on C × R, analytic near the origin and with exponential growth on some unbounded sectors with appropriate bisecting directions in the first variable and exponential decay in the second, when the perturbation parameter belongs to . Moreover, these solutions satisfy the remarkable property that the difference between any two of them is exponentially flat for some integer order w.r.t. . With the help of the classical Ramis-Sibuya theorem, we obtain the existence of a formal series (generally divergent) in which is the common Gevrey asymptotic expansion of the built up actual solutions considered above.

A definition of summability is put forward in the framework of general Carleman ultraholomorphic classes in sectors, so generalizing k−summability theory as developed by J.-P. Ramis. Departing from a strongly regular sequence of positive numbers, we construct an associated analytic proximate order and corresponding kernels, which allow us to consider suitable Laplace and Borel-type transforms, both formal and analytic, whose behavior closely resembles that of the classical ones in the Gevrey case. An application to the study of the summability properties of the formal solutions to some moment-partial differential equations is included.

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 2 and are bounded holomorphic on some horizontal strip in C w.r.t. the space variable. In our previous contribution (Lastra and Malek in Parametric Gevrey asymptotics for some nonlinear initial value Cauchy problems, arXiv:1403.2350), 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 stemming 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. MSC: 35C10; 35C20

Abstract. We study an inhomogeneous linear q-di¤erence di¤erential Cauchy problem, with a complex perturbation parameter e, whose coe‰cients depend holomorphically on e and on time in the vicinity of the origin in C 2 and are bounded analytic on some horizontal strip in C w.r.t the space variable. This problem is seen as a q-analog of an initial value problem recently investigated by the author and A. Lastra in [9]. Here a comparable result with the one in [9] is achieved, namely we construct a finite set of holomorphic solutions on a common bounded open sector in time at the origin, on the given strip above in space, when e belongs to a well selected set of open bounded sectors whose union covers a neighborhood of 0 in C Ã . These solutions are constructed through a continuous version of a q-Laplace transform of some order k b 1 introduced newly in [6] and Fourier inverse map of some function with q-exponential growth of order k on adequate unbounded sectors in C and with exponential decay in the Fourier variable. Moreover, by means of a q-analog of the classical Ramis-Sibuya theorem, we prove that they share a common formal power series (that generally diverge) in e as q-Gevrey asymptotic expansion of order 1=k.

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