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 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.

On Parametric Gevrey Asymptotics for a <i>q</i>-Analog of Some Linear Initial Value Problem

“…On the other hand, from the functional equation (35) In the following we recall some well known properties of the Fourier transform already given in [9]. …”

“…The equation (1) under study can be seen as a q-analog of the initial value problem investigated in our previous work [9] in its linear version (1) is a discretized version of (2) in the sense that the derivative q t is replaced by the operator ð f ðqtÞ À f ðtÞÞ=ðqt À tÞ for q > 1 (which formally tends to q t as q tends to 1). In [9], for a given suitable set of open bounded sectors fE p g 0a pavÀ1 whose union covers a full neighborhood of 0 in C à , for some integer v b 2 and for well selected directions m p A R, one constructs a family of holomorphic bounded functions y p ðt; z; eÞ, solutions of (2) on products T  H b  E p , where T stands for a fixed bounded sector centered at 0 with small aperture, that can be written as Laplace transforms of some adequate order k b 1 in direction m p and Fourier inverse transform y p ðt; z; eÞ ¼ k where the inner integration is made along some halfline L m p ¼ R þ e ffiffiffiffi ffi À1 p m p and where o p ðu; m; eÞ denotes a function with at most exponential growth of order k in u=e and with exponential decay in m A R. Moreover, all these functions y p ðt; z; eÞ turn out to share a common formal power seriesŷ yðt; z; eÞ ¼ P mb0 g m ðt; zÞe m A G½½e, where G is the Banach space of bounded holomorphic functions on T  H b endowed with the L y norm, as asymptotic expansion of Gevrey order 1=k on E p .…”

“…Gathering the estimates (7), (8), we see that (6) holds. r We recall that ðE ð b; mÞ ; k:k ð b; mÞ Þ can be equipped as a Banach algebra for some noncommutative product introduced below (see Proposition 5 of Section 2 in [9]). In the next proposition, we study the continuity property of some q-analog of a convolution operator acting on the Banach spaces mentioned above.…”

“…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 .…”

On Singular Solutions to PDEs with Turning Point Involving a Quadratic Nonlinearity

“…Let us recall the definition of the valuation val ( ) of an analytic function near = 0 as the smallest integer ≥ 0 with the factorization ( ) =̃( ) for an analytic functioñnear = 0 with̃(0) ̸ = 0. The most interesting case examined in this work is when the valuation val ( 1 ) of 1 ( , ) with respect to is larger than the valuation val ( 2 ) or val ( ( , , )) since the problem cannot be reduced to the case 1 (0, 0) ̸ = 0 by dividing (1) by a suitable power of and ; see Remark 13. In our previous study [3], we already have considered a similar problem which corresponds to the situation when 1 (0, 0) ̸ = 0 for our equation (1). Namely, we focused on the following problem:…”