On parametric Gevrey asymptotics for some nonlinear initial value Cauchy problems

Abstract: We study a nonlinear initial value Cauchy problem depending upon a complex perturbation parameter $\epsilon$ with vanishing initial data at complex time $t=0$ and whose coefficients depend analytically on $(\epsilon,t)$ near the origin in $\mathbb{C}^{2}$ and are bounded holomorphic on some horizontal strip in $\mathbb{C}$ 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 $\epsilon$ belongs to a suitably chosen set of open bounded sectors whose union form a covering of some neighborhood $\Omega$ of 0 in $\mathbb{C}^{\ast}$. These solutions are achieved by means of Laplace and Fourier inverse transforms of some common $\epsilon-$depending function on $\mathbb{C} \times \mathbb{R}$, 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 $\Omega$. 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 $\epsilon$. With the help of the classical Ramis-Sibuya theorem, we obtain the existence of a formal series (generally divergent) in $\epsilon$ which is the common Gevrey asymptotic expansion of the built up actual solutions considered above.