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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References
 1.
Balser, W.: From divergent power series to analytic functions. Theory and application of multisummable power series. Lecture Notes in Mathematics, 1582, p. x+108. SpringerVerlag, Berlin (1994)
 2.
Balser, W.: Multisummability of complete formal solutions for nonlinear systems of meromorphic ordinary differential equations. Complex Var. Theory Appl. 34(1–2), 19–24 (1997)
 3.
Balser, W.: Formal power series and linear systems of meromorphic ordinary differential equations. Universitext, p. xviii+299. SpringerVerlag, New York (2000)
 4.
Balser, W.: Multisummability of formal power series solutions of partial differential equations with constant coefficients. J. Differ. Equ. 201(1), 63–74 (2004)
 5.
Balser, W., Braaksma, B., Ramis, J.P., Sibuya, Y.: Multisummability of formal power series solutions of linear ordinary differential equations. Asymptotic Anal. 5(1), 27–45 (1991)
 6.
Braaksma, B.: Multisummability of formal power series solutions of nonlinear meromorphic differential equations. Ann. Inst. Fourier (Grenoble) 42(3), 517–540 (1992)
 7.
Chen, H., Luo, Z., Zhang, C.: On the summability of divergent power series satisfying singular PDEs. C. R. Math. Acad. Sci. Paris 357(3), 258–262 (2019)
 8.
Chen, H., Tahara, H.: On totally characteristic type nonlinear partial differential equations in the complex domain. Publ. Res. Inst. Math. Sci. 35(4), 621–636 (1999)
 9.
Erdelyi, A.: Higher transcendental functions, vol. III. McGrawHill, NewYork (1953)
 10.
Costin, O., Tanveer, S.: Existence and uniqueness for a class of nonlinear higherorder partial differential equations in the complex plane. Commun. Pure Appl. Math. 53(9), 1092–1117 (2000)
 11.
Costin, O., Tanveer, S.: Short time existence and Borel summability in the NavierStokes equation in \(\mathbb{R}^{3}\). Commun. Partial Differ. Equ. 34(7–9), 785–817 (2009)
 12.
Gérard, R., Tahara, H.: Singular nonlinear partial differential equations. Aspects of mathematics, p. viii+269. Friedr. Vieweg and Sohn, Braunschweig (1996)
 13.
Hsieh, P., Sibuya, Y.: Basic theory of ordinary differential equations. Universitext. Springer, New York (1999)
 14.
Ichinobe, K.: On ksummability of formal solutions for certain higher order partial differential operators with polynomial coefficients. Analytic, algebraic and geometric aspects of differential equations, Trends Math. Springer, Cham (2017)
 15.
Ichinobe, K.: On ksummability of formal solutions for a class of partial differential operators with time dependent coefficients. J. Differ. Equ. 257(8), 3048–3070 (2014)
 16.
Lastra, A., Malek, S.: Parametric Gevrey asymptotics for some nonlinear initial value Cauchy problems. J. Differ. Equ. 259(10), 5220–5270 (2015)
 17.
Lastra, A., Malek, S.: On parametric multisummable formal solutions to some nonlinear initial value Cauchy problems. Adv. Differ. Equ. 2015, 200 (2015)
 18.
Lastra, A., Malek, S.: Parametric Gevrey asymptotics for initial value problems with infinite order irregular singularity and linear fractional transforms. Adv. Differ. Equ. 2018, 386 (2018)
 19.
Lastra, A., Malek, S.: On parametric Gevrey asymptotics for some initial value problems in two asymmetric complex time variables. Results Math. 73(4), 46 (2018)
 20.
Lastra, A., Malek, S., Sanz, J.: On Gevrey solutions of threefold singular nonlinear partial differential equations. J. Differ. Equ. 255(10), 3205–3232 (2013)
 21.
LodayRichaud, M.: Divergent series, summability and resurgence. II. Simple and multiple summability. With prefaces by JeanPierre Ramis, Éric Delabaere, Claude Mitschi and David Sauzin. Lecture Notes in Mathematics, 2154, p. xxiii+272. Springer, Cham (2016)
 22.
LodayRichaud, M.: Stokes phenomenon, multisummability and differential Galois groups. Ann. Inst. Fourier (Grenoble) 44(3), 849–906 (1994)
 23.
Lope, J.E., Ona, M.P.: Solvability of a system of totally characteristic equations related to Kähler metrics. Electron. J. Differ. Equ. 51, 15 (2017)
 24.
Luo, Z., Chen, H., Zhang, C.: Exponentialtype Nagumo norms and summability of formal solutions of singular partial differential equations. Ann. Inst. Fourier 62(2), 571–618 (2012)
 25.
Malek, S.: On Gevrey asymptotics for some nonlinear integrodifferential equations. J. Dyn. Control Syst. 16(3), 377–406 (2010)
 26.
Malgrange, B., Ramis, J.P.: Fonctions multisommables. (French) [Multisummable functions] Ann. Inst. Fourier (Grenoble) 42(1–2), 353–368 (1992)
 27.
Mandai, T.: Existence and nonexistence of nullsolutions for some nonFuchsian partial differential operators with \(T\)dependent coefficients. Nagoya Math. J. 122, 115–137 (1991)
 28.
Michalik, S.: On the multisummability of divergent solutions of linear partial differential equations with constant coefficients. J. Differ. Equ. 249(3), 551–570 (2010)
 29.
Michalik, S.: Multisummability of formal solutions of inhomogeneous linear partial differential equations with constant coefficients. J. Dyn. Control Syst. 18(1), 103–133 (2012)
 30.
Ramis, J.P., Sibuya, Y.: A new proof of multisummability of formal solutions of nonlinear meromorphic differential equations. Ann. Inst. Fourier (Grenoble) 44(3), 811–848 (1994)
 31.
Tahara, H., Yamazawa, H.: Multisummability of formal solutions to the Cauchy problem for some linear partial differential equations. J. Differ. Equ. 255(10), 3592–3637 (2013)
 32.
Yamazawa, H., Yoshino, M.: Parametric Borel summability for some semilinear system of partial differential equations. Opuscula Math. 35(5), 825–845 (2015)
 33.
Yoshino, M.: Parametric Borel summability of partial differential equations of irregular singular type. Analytic, algebraic and geometric aspects of differential equations, pp. 455–471. Springer, Cham (2017)
Acknowledgements
A. Lastra and S. Malek are supported by the Spanish Ministerio de Economía, Industria y Competitividad under the Project MTM201677642C21P.
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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
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
along a halfline \(L_{\gamma } = \mathbb {R}_{+}\mathrm{e}^{\sqrt{1}\gamma } \subset S_{d,\delta } \cup \{ 0 \}\), 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
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
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
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
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
for all \(x \in \mathbb {R}\). The function \(\mathcal {F}^{1}(f)\) extends to an analytic bounded function on the strips
for all given \(0< \beta ' < \beta \).

(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) 
(b)
Take \(g \in E_{(\beta ,\mu )}\) and construct the convolution product of f and g
$$\begin{aligned} \psi (m) = \frac{1}{(2\pi )^{1/2}} \int \nolimits _{\infty }^{+\infty } f(mm_{1})g(m_{1}) \mathrm{d}m_{1}. \end{aligned}$$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
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

(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 \(\tau  < \rho \).

(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
along a halfline \(L_{\gamma } = \mathbb {R}_{+}\mathrm{e}^{\sqrt{1}\gamma } \subset S_{d,\delta } \cup \{ 0 \}\), 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
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
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
which is valid for all positive real numbers \(\alpha ,\beta > 0\). In particular, when \(\alpha ,\beta \ge 1\), we observe that
This yields the identity
which holds for any real number \(x > 0\) whenever \(\alpha ,\beta >0\). We also recall that
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
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
for every \(N \ge 1\) and \(t\in [0,\delta ]\), for some \(0<\delta <b\). Then, the function
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
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.
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.
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 ksummability of formal series with coefficients in Banach spaces (see [3], p. 121 or [13], Lemma XI26) 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)
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)
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/s12220019002213
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Keywords
 Asymptotic expansion
 Borel–Laplace transform
 Fourier transform
 Initial value problem
 Formal power series
 Linear integrodifferential equation
 Partial differential equation
 Singular perturbation
Mathematics Subject Classification
 35R10
 35C10
 35C15
 35C20