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On Singularly Perturbed Linear Initial Value Problems with Mixed Irregular and Fuchsian Time Singularities

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Abstract

We consider a family of linear singularly perturbed PDE depending on a complex perturbation parameter \(\epsilon \). As in the former study (Lastra and Malek in J Differ Equ 259(10):5220–5270, 2015) of the authors, our problem possesses an irregular singularity in time located at the origin but, in the present work, it also entangles differential operators of Fuchsian type acting on the time variable. As a new feature, a set of sectorial holomorphic solutions are built up through iterated Laplace transforms and Fourier inverse integrals following a classical multisummability procedure introduced by Balser. This construction has a direct consequence on the Gevrey bounds of their asymptotic expansions w.r.t \(\epsilon \) which are shown to increase the order of the leading term which combines both irregular and Fuchsian types operators.

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Acknowledgements

A. Lastra and S. Malek are supported by the Spanish Ministerio de Economía, Industria y Competitividad under the Project MTM2016-77642-C2-1-P.

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Appendix: On Laplace Transforms and Fourier Inverse Maps. Gevrey Asymptotic Expansions and Summability

Appendix: On Laplace Transforms and Fourier Inverse Maps. Gevrey Asymptotic Expansions and Summability

Laplace Transforms of Order k and Fourier Inverse Maps

Let \(k \ge 1\) be an integer. We recall the definition of the Laplace transform of order k as introduced in [16].

Definition 5

Let \(S_{d,\delta } = \{ \tau \in \mathbb {C}^{*} : |d - \mathrm {arg}(\tau )| < \delta \}\) be an unbounded sector with bisecting direction \(d \in \mathbb {R}\) and aperture \(2\delta > 0\), and let \(D(0,\rho )\) be a disc centered at 0 with radius \(\rho >0\). Consider a holomorphic function \(w : S_{d,\delta } \cup D(0,\rho ) \rightarrow \mathbb {C}\) that vanishes at 0 and satisfies the bounds : there exist \(C>0\) and \(K>0\) such that

$$\begin{aligned} |w(\tau )| \le C |\tau | \exp ( K |\tau |^{k} ) \end{aligned}$$
(117)

for all \(\tau \in S_{d,\delta }\). We define the Laplace transform of w of order k in the direction d as the integral transform

$$\begin{aligned} \mathcal {L}_{k}^{d}(w)(T) = k \int \nolimits _{L_{\gamma }} w(u) \exp \left( -\left( \frac{u}{T}\right) ^{k} \right) \frac{\mathrm{d}u}{u} \end{aligned}$$

along a half-line \(L_{\gamma } = \mathbb {R}_{+}\mathrm{e}^{\sqrt{-1}\gamma } \subset S_{d,\delta } \cup \{ 0 \}\), where \(\gamma \) depends on T and is chosen in such a way that \(\cos (k(\gamma - \mathrm {arg}(T))) \ge \delta _{1}\), for some fixed real number \(\delta _{1}>0\). The function \(\mathcal {L}^{d}_{k}(w)(T)\) is well defined, holomorphic and bounded on any sector

$$\begin{aligned} S_{d,\theta ,R^{1/k}} = \{ T \in \mathbb {C}^{*} : |T|< R^{1/k} , \ \ |d - \mathrm {arg}(T) | < \theta /2 \}, \end{aligned}$$

where \(0< \theta < \frac{\pi }{k} + 2\delta \) and \(0< R < \delta _{1}/K\).

If one sets \(w(\tau ) = \sum _{n \ge 1} w_{n} \tau ^n\), the Taylor expansion of w, which converges on the disc \(D(0,\rho /2)\), the Laplace transform \(\mathcal {L}_{k}^{d}(w)(T)\) has the formal series

$$\begin{aligned} \hat{X}(T) = \sum _{n \ge 1} w_{n} \Gamma \left( \frac{n}{k}\right) T^{n} \end{aligned}$$

as Gevrey asymptotic expansion of order 1 / k. This means that for all \(0< \theta _{1} < \theta \), two constants \(C,M>0\) can be chosen such that

$$\begin{aligned} \left| \mathcal {L}_{k}^{d}(w)(T) - \sum _{p=1}^{n-1} w_{p} \Gamma \left( \frac{p}{k} \right) T^{p} \right| \le CM^{n}\Gamma \left( 1 + \frac{n}{k}\right) |T|^{n} \end{aligned}$$

for all \(n \ge 2\), all \(T \in S_{d,\theta _{1},R^{1/k}}\).

In particular, if \(w(\tau )\) represents an entire function w.r.t \(\tau \in \mathbb {C}\) with the bounds (117), its Laplace transform \(\mathcal {L}_{k}^{d}(w)(T)\) does not depend on the direction d in \(\mathbb {R}\) and represents a bounded holomorphic function on \(D(0,R^{1/k})\) whose Taylor expansion is represented by the convergent series \(X(T) = \sum _{n \ge 1} w_{n}\Gamma ( \frac{n}{k} ) T^{n}\) on \(D(0,R^{1/k})\).

We recall the definition of some family of Banach spaces mentioned in [16].

Definition 6

Let \(\beta , \mu \in \mathbb {R}\). \(E_{(\beta ,\mu )}\) stands for the vector space of continuous functions \(h : \mathbb {R} \rightarrow \mathbb {C}\) such that

$$\begin{aligned} ||h(m)||_{(\beta ,\mu )} = \sup _{m \in \mathbb {R}} (1+|m|)^{\mu } \exp ( \beta |m|) |h(m)| \end{aligned}$$

is finite. The space \(E_{(\beta ,\mu )}\) endowed with the norm \(||\cdot ||_{(\beta ,\mu )}\) becomes a Banach space.

Finally, we remind the reader the definition of the inverse Fourier transform acting on the latter Banach spaces and some of its handy formulas relative to derivation and convolution product as stated in [16].

Definition 7

Let \(f \in E_{(\beta ,\mu )}\) with \(\beta > 0\), \(\mu > 1\). The inverse Fourier transform of f is given by

$$\begin{aligned} \mathcal {F}^{-1}(f)(x) = \frac{1}{ (2\pi )^{1/2} } \int \nolimits _{-\infty }^{+\infty } f(m) \exp ( ixm ) \mathrm{d}m \end{aligned}$$

for all \(x \in \mathbb {R}\). The function \(\mathcal {F}^{-1}(f)\) extends to an analytic bounded function on the strips

$$\begin{aligned} H_{\beta '} = \{ z \in \mathbb {C} / |\mathrm {Im}(z)| < \beta ' \}. \end{aligned}$$
(118)

for all given \(0< \beta ' < \beta \).

  1. (a)

    Define the function \(m \mapsto \phi (m) = imf(m)\) which belongs to the space \(E_{(\beta ,\mu -1)}\). Then, it holds that

    $$\begin{aligned} \partial _{z} \mathcal {F}^{-1}(f)(z) = \mathcal {F}^{-1}(\phi )(z). \end{aligned}$$
    (119)
  2. (b)

    Take \(g \in E_{(\beta ,\mu )}\) and construct the convolution product of f and g

    $$\begin{aligned} \psi (m) = \frac{1}{(2\pi )^{1/2}} \int \nolimits _{-\infty }^{+\infty } f(m-m_{1})g(m_{1}) \mathrm{d}m_{1}. \end{aligned}$$

    Then, \(\psi \) belongs to \(E_{(\beta ,\mu )}\) and

    $$\begin{aligned} \mathcal {F}^{-1}(f)(z) \mathcal {F}^{-1}(g)(z) = \mathcal {F}^{-1}(\psi )(z) \end{aligned}$$
    (120)

    for all \(z \in H_{\beta }\).

Gevrey Asymptotic Expansions of Order \(1/\kappa \), \(\kappa \)-Summable Formal Series and a Ramis–Sibuya Theorem

We first recall the definition of \(\kappa \)-summability of formal series with coefficients in a Banach space as introduced in classical textbooks such as [1].

Definition 8

Let \((\mathbb {F},||\cdot ||_{\mathbb {F}})\) be a complex Banach space, and let \(\kappa >1/2\) be a real number. A formal series

$$\begin{aligned} \hat{a}(\epsilon ) = \sum _{j=0}^{\infty } a_{j} \epsilon ^{j} \in \mathbb {F}[[\epsilon ]] \end{aligned}$$

with coefficients taken in \(( \mathbb {F}, ||\cdot ||_{\mathbb {F}} )\) is said to be \(\kappa \)-summable with respect to \(\epsilon \) in the direction \(d \in \mathbb {R}\) if

  1. (i)

    a radius \(\rho \in \mathbb {R}_{+}\) can be chosen in a way that the formal series, called formal Borel transform of order \(\kappa \) of \(\hat{a}\),

    $$\begin{aligned} B_{\kappa }(\hat{a})(\tau ) = \sum _{j=0}^{\infty } \frac{ a_{j} \tau ^{j} }{ \Gamma \left( 1 + \frac{j}{\kappa }\right) } \in \mathbb {F}[[\tau ]], \end{aligned}$$

    converge absolutely for \(|\tau | < \rho \).

  2. (ii)

    One can find an aperture \(2\delta > 0\) in order that the series \(B_{\kappa }(\hat{a})(\tau )\) can be analytically continued with respect to \(\tau \) on the unbounded sector \(S_{d,\delta } = \{ \tau \in \mathbb {C}^{*} : |d - \mathrm {arg}(\tau ) | < \delta \} \). Moreover, there exist suitable \(C >0\) and \(K >0\) with the bounds

    $$\begin{aligned} ||B_{\kappa }(\hat{a})(\tau )||_{\mathbb {F}} \le C \mathrm{e}^{ K|\tau |^{\kappa }} \end{aligned}$$

    whenever \(\tau \in S_{d, \delta }\).

If the constraints above are fulfilled, the vector valued Laplace transform of order \(\kappa \) of \(B_{\kappa }(\hat{a})(\tau )\) in the direction d is set as

$$\begin{aligned} L^{d}_{\kappa }(B_{\kappa }(\hat{a}))(\epsilon ) = \epsilon ^{-\kappa } \int \nolimits _{L_{\gamma }} B_{\kappa }(\hat{a})(u) \mathrm{e}^{ - ( u/\epsilon )^{\kappa } } \kappa u^{\kappa -1}\mathrm{d}u, \end{aligned}$$

along a half-line \(L_{\gamma } = \mathbb {R}_{+}\mathrm{e}^{\sqrt{-1}\gamma } \subset S_{d,\delta } \cup \{ 0 \}\), where \(\gamma \) relies on \(\epsilon \) and is sort in such a way to satisfy \(\cos (k(\gamma - \mathrm {arg}(\epsilon ))) \ge \delta _{1} > 0\), for some fixed \(\delta _{1}\), for all \(\epsilon \) in a sector

$$\begin{aligned} S_{d,\theta ,R^{1/\kappa }} = \{ \epsilon \in \mathbb {C}^{*} : |\epsilon |< R^{1/\kappa } , \ \ |d - \mathrm {arg}(\epsilon ) | < \theta /2 \}, \end{aligned}$$

where the angle \(\theta \) and radius R satisfies \(0< \theta < \frac{\pi }{\kappa } + 2\delta \) and \(0< R < \delta _{1}/K\).

It is worth noticing that this Laplace transform of order \(\kappa \) differs slightly from the one displayed in Definition 5 which appears to be more suitable for the computations related to the problems under study in this work.

The function \(L^{d}_{\kappa }(B_{\kappa }(\hat{a}))(\epsilon )\) is called the \(\kappa \)-sum of the formal series \(\hat{a}(\epsilon )\) in the direction d. It represents a bounded and holomorphic function on the sector \(S_{d,\theta ,R^{1/\kappa }}\) and turns out to be the unique such function that possesses the formal series \(\hat{a}(\epsilon )\) as Gevrey asymptotic expansion of order \(1/\kappa \) with respect to \(\epsilon \) on \(S_{d,\theta ,R^{1/\kappa }}\). It means that for all \(0< \theta _{1} < \theta \), there exist \(C,M > 0\) such that

$$\begin{aligned} \left\| L^{d}_{\kappa }(B_{\kappa }(\hat{a}))(\epsilon ) - \sum _{p=0}^{n-1} a_{p} \epsilon ^{p}\right\| _{\mathbb {F}} \le CM^{n}\Gamma \left( 1+ \frac{n}{\kappa }\right) |\epsilon |^{n} \end{aligned}$$

for all \(n \ge 1\), all \(\epsilon \in S_{d,\theta _{1},R^{1/\kappa }}\).

From Appendix B of [3], we recall the Beta integral formula

$$\begin{aligned} B(\alpha ,\beta ) = \int \nolimits _{0}^{1} (1-t)^{\alpha - 1} t^{\beta - 1} dt = \frac{\Gamma (\alpha ) \Gamma (\beta )}{\Gamma ( \alpha + \beta )} \end{aligned}$$
(121)

which is valid for all positive real numbers \(\alpha ,\beta > 0\). In particular, when \(\alpha ,\beta \ge 1\), we observe that

$$\begin{aligned} \Gamma (\alpha ) / \Gamma ( \alpha + \beta ) \le 1/\Gamma (\beta ) \end{aligned}$$
(122)

This yields the identity

$$\begin{aligned} \int \nolimits _{0}^{x} (x-h)^{\alpha - 1} h^{\beta -1} \mathrm{d}h = x^{\alpha + \beta - 1}\frac{\Gamma (\alpha )\Gamma (\beta )}{\Gamma (\alpha +\beta )} \end{aligned}$$
(123)

which holds for any real number \(x > 0\) whenever \(\alpha ,\beta >0\). We also recall that

$$\begin{aligned} \Gamma (x)/\Gamma (x+a) \sim 1/x^{a} \end{aligned}$$
(124)

as x tends to \(+\infty \), for any real number \(a>0\).

Mittag–Leffler’s function is defined by \(E_{\alpha }(z) = \sum _{n \ge 0} z^{n}/\Gamma (1 + \alpha n)\) for \(\alpha \in (0,2)\), which turns out to be an entire function of exponential order \(k=1/\alpha \) and type \(\tau =1\). Also, Wiman function is defined by \(E_{\alpha ,\beta }(z) = \sum _{n \ge 0} z^{n}/\Gamma (\beta + \alpha n)\), for any \(\alpha \in (0,2)\), \(\beta >0\). Expansion (22) p. 210 in [9] gives rise to the existence of a constant \(C_{\alpha ,\beta }>0\) (depending on \(\alpha ,\beta \)) with

$$\begin{aligned} E_{\alpha ,\beta }(z) \le C_{\alpha ,\beta } z^{\frac{1 - \beta }{\alpha }} \exp ( z^{1/\alpha } ) \end{aligned}$$
(125)

for all \(z \ge 1\).

Lemma 7

(Watson’s Lemma. Exercise 4, page 16 in [1]) Let \(b>0\) and \(f:[0,b] \rightarrow \mathbb {C} \) be a continuous function having the formal expansion \(\sum _{n \ge 0}a_{n}t^n \in \mathbb {C}[[t]]\) as its asymptotic expansion of Gevrey order \(1/\kappa >0\) at 0, meaning there exist \(C,M>0\) such that

$$\begin{aligned} \left| f(t)-\sum _{n=0}^{N-1}a_{n}t^n \right| \le CM^{N}N!^{1/\kappa }|t|^{N}, \end{aligned}$$

for every \(N \ge 1\) and \(t\in [0,\delta ]\), for some \(0<\delta <b\). Then, the function

$$\begin{aligned} I(x)=\int \nolimits _{0}^{b}f(s)\mathrm{e}^{-\frac{s}{x}}\mathrm{d}s \end{aligned}$$

admits the formal power series \(\sum _{n \ge 0} a_{n}n!x^{n+1} \in \mathbb {C}[[x]]\) as its asymptotic expansion of Gevrey order \((1/\kappa )+1\) at 0, it is to say, there exist \(\tilde{C},\tilde{K}>0\) such that

$$\begin{aligned} \left| I(x)-\sum _{n=0}^{N-1}a_{n}n!x^{n+1}\right| \le \tilde{C}\tilde{K}^{N+1}(N+1)!^{1+\frac{1}{\kappa }}|x|^{N+1}, \end{aligned}$$

for every \(N \ge 0\) and \(x \in [0,\delta ']\) for some \(0<\delta '<b\).

Lemma 8

(Exercise 3, page 18 in [1]) Let \(\delta ,q>0\), and \(\psi :[0,\delta ] \rightarrow \mathbb {C} \) be a continuous function. The following assertions are equivalent:

  1. 1.

    There exist \(C,M>0\) such that \(|\psi (x)|\le CM^{n}n!^{q}|x|^{n},\) for every \(n \in \mathbb {N}\), \(n\ge 0\) and \(x \in [0,\delta ]\).

  2. 2.

    There exist \(C',M'>0\) such that \(|\psi (x)|\le C'\mathrm{e}^{-M'/x^{\frac{1}{q}}}\), for every \(x \in (0,\delta ]\).

In the sequel, we state a cohomological criterion for the existence of Gevrey asymptotics of order \(1/\kappa \) for proper families of sectorial holomorphic functions and k-summability of formal series with coefficients in Banach spaces (see [3], p. 121 or [13], Lemma XI-2-6) which is known as the Ramis–Sibuya theorem. This result plays a central role in the proof of our second main statement (Theorem 2).

Theorem (RS)We consider a Banach space\((\mathbb {F},||\cdot ||_{\mathbb {F}})\)over\(\mathbb {C}\)and a good covering\(\{ \mathcal {E}_{p} \}_{0 \le p \le \varsigma -1}\)in\(\mathbb {C}^{*}\)(as explained in Definition 3). For all\(0 \le p \le \varsigma - 1\), let\(G_{p}\)be a holomorphic function from\(\mathcal {E}_{p}\)into the Banach space\((\mathbb {F},||\cdot ||_{\mathbb {F}})\). We denote the cocycle\(\Theta _{p}(\epsilon ) = G_{p+1}(\epsilon ) - G_{p}(\epsilon )\), \(0 \le p \le \varsigma -1\), which represents a holomorphic function from the sector\(Z_{p} = \mathcal {E}_{p+1} \cap \mathcal {E}_{p}\)into\(\mathbb {F}\)(with the convention that\(\mathcal {E}_{\varsigma } = \mathcal {E}_{0}\)and\(G_{\varsigma } = G_{0}\)). We ask for the following requirements.

  1. (1)

    The functions\(G_{p}(\epsilon )\)remain bounded as\(\epsilon \in \mathcal {E}_{p}\)comes close to the origin in\(\mathbb {C}\), for all\(0 \le p \le \varsigma - 1\).

  2. (2)

    The functions\(\Theta _{p}(\epsilon )\)are exponentially flat of order\(\kappa \)on\(Z_{p}\), for all\(0 \le p \le \varsigma -1\), for some real number\(\kappa > 1/2\). In other words, there exist constants\(C_{p},A_{p}>0\)such that

    $$\begin{aligned} ||\Theta _{p}(\epsilon )||_{\mathbb {F}} \le C_{p}\mathrm{e}^{-A_{p}/|\epsilon |^{\kappa }} \end{aligned}$$

    for all\(\epsilon \in Z_{p}\), all\(0 \le p \le \varsigma -1\). Then, for all\(0 \le p \le \varsigma - 1\), the functions\(G_{p}(\epsilon )\)share a common formal power series\(\hat{G}(\epsilon ) \in \mathbb {F}[[\epsilon ]]\)as Gevrey asymptotic expansion of order\(1/\kappa \)on\(\mathcal {E}_{p}\). Moreover, for the special configuration where the aperture of one sector\(\mathcal {E}_{p_0}\)can be chosen slightly larger than\(\pi /\kappa \), the function\(G_{p_0}(\epsilon )\)turns out to be the\(\kappa \)-sum of\(\hat{G}(\epsilon )\)on\(\mathcal {E}_{p_0}\).

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Lastra, A., Malek, S. On Singularly Perturbed Linear Initial Value Problems with Mixed Irregular and Fuchsian Time Singularities. J Geom Anal (2019). https://doi.org/10.1007/s12220-019-00221-3

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Keywords

  • Asymptotic expansion
  • Borel–Laplace transform
  • Fourier transform
  • Initial value problem
  • Formal power series
  • Linear integro-differential equation
  • Partial differential equation
  • Singular perturbation

Mathematics Subject Classification

  • 35R10
  • 35C10
  • 35C15
  • 35C20