On parametric Gevrey asymptotics for some nonlinear initial value Cauchy problems

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:259

Article

Lastra, A.^1,2  Malek, S.^1,2 

[1] Univ Alcala de Henares, Dept Fis & Matemat, E-28871 Alcala De Henares, Madrid, Spain

[2] Univ Lille 1, Lab Paul Painleve, F-59655 Villeneuve Dascq, France

关键词: Asymptotic expansion;    Borel-Laplace transform;    Fourier transform;    Nonlineai integro-differential equation;    Nonlinear partial differential equation;    Singular perturbation;   

DOI  :  10.1016/j.jde.2015.06.020

来源: Elsevier

【 摘 要 】

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 (e, t) near the origin in C-2 and are bounded holomorplaic 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 c belongs to a suitably chosen set of open bounded sectors whose union form a covering of some neighborhood SZ of 0 in 0'. These solutions are achieved by means of Laplace and Fourier inverse transforms of some common c-depending function on C x 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. (C) 2015 Elsevier Inc. All rights reserved.

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