15-53 Alberto Lastra, Stephane Malek

On parametric Gevrey asymptotics for some nonlinear initial value Cauchy problems (618K, pdf) Jun 17, 15

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Abstract. We study a nonlinear initial value Cauchy problem depending upon a complex perturbation parameter with vanishing initial data at the origin in time and whose coefficients are analytic in time and parameter in the vicinity of zero and are bounded holomorphic on some horizontal strip w.r.t the space variable. This problem is assumed to be non-Kowalevskian in time, therefore analytic solutions at the origin cannot be expected in general. Nevertheless, we can 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 the complex domain. 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 the parameter. With the help of the classical Ramis-Sibuya theorem, we obtain the existence of a formal series (generally divergent) in the parameter which is the common Gevrey asymptotic expansion of the built up actual solutions considered above.

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