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

and Applied Analysis 3 PDEs of this latter result. Furthermore, in our study we are also able to describe the behaviour of our specific solutions near (t, ε) = (0, 0). For more recent and advanced works related to WKB analysis and local/global studies of solutions to linear ODEs near turning points, we refer to contributions related to the 1D Schrödinger equation with simple poles [11], with merging pairs of simple poles and turning points [12], and with merging triplet of poles and turning points [13, 14] and for analytic continuation properties of the Borel transform (resurgence) of WKB expansions in the problem of confluence of two simple turning points we quote [15]. Concerning the structure of singular formal solutions to singularly perturbed linear systems of ODEs with turning points we point out [16] solving an old question of Wasow. We mention also preeminent studies on WKB analysis for higher order differential equations which reveal new Stokes phenomena giving rise to so-called virtual turning points [17, 18]. In the framework of linear PDEs, normal forms for completely integrable systems near a degenerate point where two turning points coalesce have been obtained in [19], which is a first step toward the so-called Dubrovin conjecture which concerns the question of universal behaviour of generic solutions near gradient catastrophe of singularlyHamiltonian perturbations of first-order hyperbolic equations; see [20]. We mention also that sectorial analytic transformations to normal forms have been obtained for systems of singularly perturbed ODEs near a turning point with multiplicity using the recent approach of composite asymptotic expansions developed in [2]; see [21]. The paper is organized as follows. In Section 2, we recall the definition introduced in the work [3] of some weighted Banach spaces of continuous functions with exponential growth on unbounded sectors in C and with exponential decay on R. We analyze the continuity of specific multiplication and linear/nonlinear convolution operators acting on these spaces. In Section 3, we remind the reader of basic statements concerning mk-Borel-Laplace transforms, a version of the classical Borel-Laplace maps already used in previous works [3, 22, 23] and Fourier transforms acting on exponentially flat functions. In Section 4, we display our main problems and explain the leading strategy in order to solve them. It consists in four operations. In a first step, we restrict our inquiry for the sets of solutions to time rescaled function spaces; see (63). Then, we consider candidates for solutions to the resulting auxiliary problem (64) that are small perturbations of a so-called slow curve which solves a second-order algebraic equation and which may be singular at the origin in C. In a third step, we search again for time rescaled functions solutions for the associated problem (84) solved by the small perturbation of the slow curve; see (85). In the last step, we write down the convolution problem (95) solved by a suitable mκ-Borel transform of a formal solution to the attached problem (87). In Section 5, we solve the main convolution problem (95) within the Banach spaces described in Section 2 using some fixed point theorem argument. In Section 6, we provide a set of actualmeromorphic solutions to our initial equation (61) by executing backwards the operations described in Section 4. In particular, we show that our singular functions actually solve problem (164) which is a factorized part of (61) with a more restrictive forcing term. Furthermore, the difference of any two neighboring solutions tends to 0 as ε tends to 0 faster than a function with exponential decay of order (χ + α)κ. In Section 7, we show the existence of a common asymptotic expansion of Gevrey order 1/(χ + α)κ for the nonsingular parts of these solutions of (61) and (164) based on the flatness estimates obtained in Section 6 using a theorem by Ramis and Sibuya. 2. Banach Spaces with Exponential Growth and Exponential Decay We denote byD(0, ρ) the open disc centered at 0 with radius ρ > 0 in C and by D(0, ρ) its closure. Let Sd be an open unbounded sector in directiond ∈ R andE be an open sector with finite radius rE, both centered at 0 in C. By convention, these sectors do not contain the origin in C. We first give definitions of Banach spaces which already appear in our previous work [3]. Definition 1. Let β > 0 and μ > 1 be real numbers. We denote by E(β,μ) the vector space of functions h : R→ C such that ‖h (m)‖(β,μ) = sup m∈R (1 + |m|)μ exp (β |m|) |h (m)| (8) is finite. The space E(β,μ) endowed with the norm ‖ ⋅ ‖(β,μ) becomes a Banach space. As a direct consequence of Proposition 5 from [3], we notice the following. Proposition 2. The Banach space (E(β,μ), ‖ ⋅ ‖(β,μ)) is a Banach algebra for the convolution product (f ⋆ g) (m) = ∫ −∞ f (m − m1) g (m1) dm1. (9) Namely, there exists a constant C0 > 0 (depending on μ) such that 󵄩󵄩󵄩󵄩(f ⋆ g) (m)󵄩󵄩󵄩󵄩(β,μ) ≤ C0 󵄩󵄩󵄩󵄩f (m)󵄩󵄩󵄩󵄩(β,μ) 󵄩󵄩󵄩󵄩g (m)󵄩󵄩󵄩󵄩(β,μ) (10) for all f, g ∈ E(β,μ). Definition 3. Let ], ρ > 0 and β > 0, μ > 1 be real numbers. Let κ ≥ 1 and χ, α ≥ 0 be integers. Let ε ∈ E. We denote by Fd (],β,μ,χ,α,κ,ε) the vector space of continuous 4 Abstract and Applied Analysis functions (τ, m) 󳨃→ h(τ,m) on (D(0, ρ) ∪ Sd) × R, which are holomorphic with respect to τ onD(0, ρ) ∪ Sd and such that ‖h (τ,m)‖(],β,μ,χ,α,κ,ε) = sup τ∈D(0,ρ)∪Sd ,m∈R (1 + |m|)μ exp (β |m|) ⋅ 1 + 󵄨󵄨󵄨󵄨τ/εχ+α󵄨󵄨󵄨󵄨2κ |τ/εχ+α| exp(−] 󵄨󵄨󵄨󵄨󵄨󵄨󵄨 τ εχ+α 󵄨󵄨󵄨󵄨󵄨󵄨󵄨 ) |h (τ,m)| (11) is finite. One can check that the normed space (Fd (],β,μ,χ,α,κ,ε), ‖ ⋅ ‖(],β,μ,χ,α,κ,ε)) is a Banach space. Throughout the whole section, we keep the notations of Definitions 1 and 3. In the next lemma, we check that some parameter depending functions with polynomial growth with respect to the variable τ and exponential decay with respect to the variablem, whichwill appear later on in our study (Section 5), belong to the Banach spaces described above. Lemma 4. Let γ1 ≥ 0, γ2 ≥ 1 be integers. Let ?̃?(X) be a polynomial that belongs to C[X] such that ?̃?(im) ̸ = 0 for all m ∈ R. We take a function B(m) located in E(β,μ) and we consider a continuous function aγ1,κ(τ, m) on (D(0, ρ)∪Sd)×R, holomorphic with respect to τ onD(0, ρ) ∪ Sd, such that 󵄨󵄨󵄨󵄨󵄨aγ1,κ (τ, m)󵄨󵄨󵄨󵄨󵄨 ≤ 1 (1 + |τ|κ)γ1 󵄨󵄨󵄨󵄨󵄨?̃? (im)󵄨󵄨󵄨󵄨󵄨 (12) for all τ ∈ D(0, ρ) ∪ Sd, allm ∈ R. Then, the function ε−χγ2τγ2B(m)aγ1 ,κ(τ, m) belongs to Fd (],β,μ,χ,α,κ,ε). Moreover, there exists a constant C1 > 0 (depending on κ and γ2) such that 󵄩󵄩󵄩󵄩󵄩ε−χγ2τγ2B (m) aγ1,κ (τ, m)󵄩󵄩󵄩󵄩󵄩(],β,μ,χ,α,κ,ε) ≤ C1 󵄩󵄩󵄩󵄩󵄩B (m)󵄩󵄩󵄩󵄩󵄩(β,μ) infm∈R 󵄨󵄨󵄨󵄨󵄨?̃? (im)󵄨󵄨󵄨󵄨󵄨 |ε| γ2α (13)

Here we consider the case when  1 (0, ) vanishes identically near 0. The point  = 0 is known to be called a turning point in that situation; see [1,2] for a more detailed description of this terminology in the linear and nonlinear settings.Let us recall the definition of the valuation val  () of an analytic function near  = 0 as the smallest integer  ≥ 0 with the factorization () =   f() for an analytic function f near  = 0 with f(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:  (  )    (, , ) = ( 1 (  )  (, , )) ( 2 (  )  (, , )) +  (, ,   ,   )  (, , ) +  (, , ) (2) for given vanishing initial data (0, , ) ≡ 0, where ,  1 ,  2 , and  are polynomials with complex coefficients and (, , ) is a forcing term constructed as above.Under appropriate assumptions on the shape of (2), we established the existence of a family of actual bounded holomorphic solutions   (, , ), 0 ≤  ≤  − 1, for some integer  ≥ 2, defined on domains T ×   × E  , for some fixed bounded sector T with vertex at 0 and E = {E  } 0≤≤−1 , a set of bounded sectors whose union covers a full neighborhood of 0 in C * .These solutions are obtained by means of Laplace and inverse Fourier transforms.On each sector E  , they share with respect to  a common asymptotic expansion ŷ(, , ) = ∑ ≥0   (, )  which defines a formal series with bounded holomorphic coefficients on T ×   .Moreover, this asymptotic expansion is shown to be of Gevrey order (at most) 1/ that appears in the highest order term of the operator  which is of irregular type in the sense of [4] outlined as  (  −1)  (  −1)(+1)       (  ), for some integer   ≥ 2 and a polynomial   with complex coefficients.Conjointly, since the aperture of the sectors E  can be chosen slightly larger than /, the functions   →   (, , ) can be viewed as -sums of the formal series ŷ as defined in [5].
In this work, our goal is to achieve a similar statement, namely, the existence of sectorial holomorphic solutions and asymptotic expansions as  tends to 0. However, the main contrast with problem (2) is that, due to the presence of the turning point, our solutions are no longer bounded in the vicinity of the origin, being meromorphic in both time variable  and parameter .Namely, we build a set of actual meromorphic solutions  d  (, , ) to problem (1) of the form  d  (, , ) =   ( 0 (  ) + (  )  V d  (, , )) , where  > 1,  are some rational numbers,  is an integer, and  0 () is a nonidentically vanishing root of a secondorder algebraic equation with polynomial coefficients related to the polynomials  1 ,  2 , see (70), and where V d  (, , ) is a bounded holomorphic function on products T ×   × E  similar to the ones mentioned above, which can be expressed as a Laplace transform of some order  ≥ 1 and Fourier inverse transform along some half line  d  = R +  d  , for some positive rational number  > 0, where  d   (, , ) represents a function with at most exponential growth of order  on a sector containing  d  with respect to , with exponential decay with respect to  on R and with analytic dependence on  near 0 (see Theorem 19).Furthermore, we show that these functions V d  (, , ) own with respect to  a common asymptotic expansion V(, , ) = ∑ ≥0 V  (, )  which represents a formal series with bounded holomorphic coefficients on T ×   .We specify also the nature of this asymptotic expansion which turns out to be of Gevrey order (at most) 1/( + ).Besides, since the aperture of the sectors E  may be selected slightly larger than /( + ), the functions V d  can be identified as (+)-sums of the formal series V (Theorem 21).By construction, the integer  shows up in the highest order term of the operator  3 which is of irregular type of the form  Δ     (+1)+ 0       (  ), with  0 = val  ( 1 ), for some integers Δ  ≥ 0,   ≥ 2 and a polynomial   with complex coefficients.The rational number  is built with the help of the integers Δ  ,   ,  0 , and  and the rational numbers , ; see (86).According to the fact that ,  are mainly related to constraints assumed on the polynomials  1 and  2 (see (66), (67)), we observe that the Gevrey order 1/( + ) of the asymptotic expansion involves information coming both from the highest irregular term and from the two polynomials  1 and  2 that shape the turning point at  = 0, whereas, in our previous contribution [3], the Gevrey order was exclusively stemming from the irregular singularity at  = 0.
The kind of equations with quadratic nonlinearity we investigate in this work is strongly related to singularly perturbed ODEs which are nonsingular at the origin of the form   / = (, , , ) for some analytic functions , small complex parameter , and a complex additional parameter , described in the seminal joint paper by Canalis-Durand et al., see [6], where they study asymptotic properties of actual overstable solutions near a slow curve  0 () (meaning that (,  0 (), , 0) ≡ 0) in the case when the Jacobian   (,  0 (), , 0) is not invertible at  = 0.The main notable difference is that we assume the origin to be at the same time a turning point and an irregular singularity.More precisely, with the rescaling map (, )  → ( = , ) the transformed equation (64) possesses a rational slow curve  0 () and  = 0 remains a turning point and an irregular singularity for this new equation.
The construction of the distinguished solution performed in Section 4 and the parametric Borel/Laplace summable character of these solutions shown in Section 7 are also intimately linked to recent developments of exact WKB analysis of formal and analytic solutions to second-order linear ODEs of Schrödinger type.Namely, let  2   (, ) =  ()  (, ) be a singularly perturbed ODE where  is a small complex parameter and () is some polynomial with complex coefficients.WKB solutions of (5) are known as special solutions that are described as an exponential ψ(, ) = exp(∫   0 Ŝ(, )) where the expression Ŝ(, ) satisfies a socalled Riccati equation  2 Ŝ (, ) +  2 Ŝ2 (, ) =  () .

𝜖 T󸀠
± (, ) + 2 −1 () T± (, ) +  T2 ± (, ) +   −1 () = 0 (7) with turning points at the roots of ().Our main PDE (1) resembles this last one provided that  −1 () is a polynomial and with the significant distinction that our equation only involves differential operators with irregular singularity at  = 0.An essential feature of the theory is that the formal series T± (, ) are 1-summable in suitable directions  ∈ R with respect to  (that are related to the function ∫   0  −1 ()) for any fixed  ∈ .Different proofs of this fact can be found in [7][8][9][10].Our second main statement, Theorem 21, can be considered as a similar contribution for some higher order PDEs of this latter result.Furthermore, in our study we are also able to describe the behaviour of our specific solutions near (, ) = (0, 0).
For more recent and advanced works related to WKB analysis and local/global studies of solutions to linear ODEs near turning points, we refer to contributions related to the 1D Schrödinger equation with simple poles [11], with merging pairs of simple poles and turning points [12], and with merging triplet of poles and turning points [13,14] and for analytic continuation properties of the Borel transform (resurgence) of WKB expansions in the problem of confluence of two simple turning points we quote [15].Concerning the structure of singular formal solutions to singularly perturbed linear systems of ODEs with turning points we point out [16] solving an old question of Wasow.We mention also preeminent studies on WKB analysis for higher order differential equations which reveal new Stokes phenomena giving rise to so-called virtual turning points [17,18].
In the framework of linear PDEs, normal forms for completely integrable systems near a degenerate point where two turning points coalesce have been obtained in [19], which is a first step toward the so-called Dubrovin conjecture which concerns the question of universal behaviour of generic solutions near gradient catastrophe of singularly Hamiltonian perturbations of first-order hyperbolic equations; see [20].We mention also that sectorial analytic transformations to normal forms have been obtained for systems of singularly perturbed ODEs near a turning point with multiplicity using the recent approach of composite asymptotic expansions developed in [2]; see [21].
The paper is organized as follows.In Section 2, we recall the definition introduced in the work [3] of some weighted Banach spaces of continuous functions with exponential growth on unbounded sectors in C and with exponential decay on R. We analyze the continuity of specific multiplication and linear/nonlinear convolution operators acting on these spaces.
In Section 3, we remind the reader of basic statements concerning   -Borel-Laplace transforms, a version of the classical Borel-Laplace maps already used in previous works [3,22,23] and Fourier transforms acting on exponentially flat functions.
In Section 4, we display our main problems and explain the leading strategy in order to solve them.It consists in four operations.In a first step, we restrict our inquiry for the sets of solutions to time rescaled function spaces; see (63).Then, we consider candidates for solutions to the resulting auxiliary problem (64) that are small perturbations of a so-called slow curve which solves a second-order algebraic equation and which may be singular at the origin in C. In a third step, we search again for time rescaled functions solutions for the associated problem (84) solved by the small perturbation of the slow curve; see (85).In the last step, we write down the convolution problem (95) solved by a suitable   -Borel transform of a formal solution to the attached problem (87).
In Section 5, we solve the main convolution problem (95) within the Banach spaces described in Section 2 using some fixed point theorem argument.
In Section 6, we provide a set of actual meromorphic solutions to our initial equation (61) by executing backwards the operations described in Section 4. In particular, we show that our singular functions actually solve problem (164) which is a factorized part of (61) with a more restrictive forcing term.Furthermore, the difference of any two neighboring solutions tends to 0 as  tends to 0 faster than a function with exponential decay of order ( + ).
In Section 7, we show the existence of a common asymptotic expansion of Gevrey order 1/( + ) for the nonsingular parts of these solutions of (61) and (164) based on the flatness estimates obtained in Section 6 using a theorem by Ramis and Sibuya.

Banach Spaces with Exponential Growth and Exponential Decay
We denote by (0, ) the open disc centered at 0 with radius  > 0 in C and by (0, ) its closure.Let   be an open unbounded sector in direction  ∈ R and E be an open sector with finite radius  E , both centered at 0 in C. By convention, these sectors do not contain the origin in C. We first give definitions of Banach spaces which already appear in our previous work [3].
Definition 1.Let  > 0 and  > 1 be real numbers.We denote by  (,) the vector space of functions ℎ : R → C such that is finite.The space  (,) endowed with the norm ‖ ⋅ ‖ (,) becomes a Banach space.
Throughout the whole section, we keep the notations of Definitions 1 and 3.
In the next lemma, we check that some parameter depending functions with polynomial growth with respect to the variable  and exponential decay with respect to the variable , which will appear later on in our study (Section 5), belong to the Banach spaces described above.
Then, the function  − which yields the lemma since an exponential grows faster than any polynomial.
The next proposition provides norm estimates for some linear convolution operators acting on the Banach spaces introduced above.These bounds are more accurate than the one supplied in Proposition 2 from [3].These new estimates will be essential in Section 5 in order to solve problem (95).The improvements are due to the use of thorough upper bounds estimates of a generalized Mittag-Leffler function described in the proofs of Propositions 1 and 5 from [23].
Proposition 5. Let   , 0 ≤  ≤ 3, be real numbers with  1 ≥ 0. Let R() and R () be polynomials with complex coefficients such that deg( R) ≤ deg( R ) and with R () ̸ = 0 for all  ∈ R. We consider a continuous function   1 , (, ) on ((0, ) ∪   ) × R, holomorphic with respect to  on (0, ) ∪   , such that for all  ∈ (0, ) ∪   , all  ∈ R. We make the following assumptions: (1) If 1 +  3 ≤ 0, then there exists a constant  2 > 0 (depending on ], ,  2 ,  3 and R(), R ()) such that where Again by the definition of the norm of  and by the constraints on the polynomials ,   , we deduce that  ≤ We perform the change of variable ℎ = || (+) ℎ  inside the integral which is a part of As a result, we obtain the bounds where We now proceed as in Proposition 1 of [23].We split the function () into two pieces and study them separately.Namely, we decompose () =  1 () +  2 (), where We first provide estimates for  1 ().
In a second step, we study  2 ().
(1) We consider the first case when 1 +  3 ≤ 0. Bearing in mind (31) and (33), we deduce that sup which is finite.On the other hand, when 0 ≤  < 1, we make the change of variable ℎ  =   inside  2.1 () and, taking (31) into account, we get sup which is finite provided that the constraints ( 16) are fulfilled.
The forthcoming proposition presents norm estimates for some bilinear convolution operators acting on the aforementioned Banach spaces.Proposition 6.There exists a constant  3 > 0 (depending on  and ) such that We provide upper bounds that can be split in two parts, where is finite under the condition that  > 1 according to Lemma 4 of [24] and We carry out the change of variable ℎ  = || (+) ℎ inside the integral piece of  3.2 () which yields the bounds where A change of variable ℎ =  in this last expression followed by a partial fraction decomposition allow us to write which acquaints us with the fact that () is finite provided that  ≥ 1 and bounded on R + with respect to .At last, collecting (38), ( 40), ( 42), ( 43), (45), and (47) leads to the statement of Proposition 6.

Borel-Laplace and Fourier Transforms
In this section, we review some basic statements concerning a -Borel summability method of formal power series which is a slightly modified version of the more classical procedure (see [5], Section 3.2).This novel version has already been used in works such as [3,22] when studying Cauchy problems under the presence of a small perturbation parameter.We remind also the reader of the definition of Fourier inverse transform acting on functions with exponential decay.Definition 7. Let  ≥ 1 be an integer.Let (  ()) ≥1 be the sequence Let (E, ‖ ⋅ ‖ E ) be a complex Banach space.We say a formal power series is   -summable with respect to  in the direction  ∈ [0, 2) if the following assertions hold: (1) There exists  > 0 such that the   -Borel transform of X, B   ( X), is absolutely convergent for || < , where (2) The series B   ( X) can be analytically continued in a sector  = { ∈ C ⋆ : | − arg()| < } for some  > 0. In addition to this, the extension is of exponential growth at most  in , meaning that there exist ,  > 0 such that Under these assumptions, the vector valued Laplace transform of B   ( X) along direction  is defined by where   is the path parametrized by  ∈ [0, ∞)  →   , for some appropriate direction  depending on , such that   ⊆  and cos(( − arg())) ≥ Δ > 0 for some Δ > 0.
The function L    (B   ( X)) is well defined and turns out to be a holomorphic and bounded function in any sector of the form  ,, 1/ = { ∈ C ⋆ : || <  1/ , | − arg()| < /2}, for some / <  < /+2 and 0 <  < Δ/.This function is known as the   -sum of the formal power series X() in the direction .
The following are some elementary properties concerning the   -sums of formal power series which will be crucial in our procedure.
(1) The function L    (B   ( X))() admits X() as its Gevrey asymptotic expansion of order 1/ with respect to  in  ,, 1/ .More precisely, for every / <  1 < , there exist ,  > 0 such that for every  ≥ 2 and  ∈  , 1 , 1/ .Watson's lemma (see Proposition 11 p. 75 in [25]) allows us to affirm that (2) Whenever E is a Banach algebra, the set of holomorphic functions having Gevrey asymptotic expansion of order 1/ on a sector with values in E turns out to be a differential algebra (see Theorems 18,19,and 20 in [25]).This and the uniqueness provided by Watson's lemma allow us to obtain some properties on   -summable formal power series in direction .
By ⋆ we denote the product in the Banach algebra and also the Cauchy product of formal power series with coefficients in for every  ∈  ,, 1/ .The next proposition is written without proof for it can be found in [3], Proposition 6.
Let ,  ≥ 1 be integers.The following formal identities hold.
In the last part of the section, we recall without proofs some properties of the inverse Fourier transform acting on continuous functions with exponential decay on R; see [3], Proposition 7 for more details.

Proposition 9.
(1) Let  : R → R be a continuous function with a constant  > 0 such that |()| ≤  exp(−||) for all  ∈ R, for some  > 0. The inverse Fourier transform of  is defined by the integral representation for all  ∈ R. It turns out that the function F −1 () extends to an analytic function on the horizontal strip Let () = ().Then, we have the commuting relation for all  ∈   .

Abstract and Applied Analysis
We consider the following nonlinear singularly perturbed PDE: The coefficients   () are constructed as follows.For all 0 ≤  ≤ , we consider functions   → B () that belong to the Banach space  (,) for some  > 1 and  > 0. We define   () = () V B () where V is the integer introduced above, for 0 ≤  ≤ .We set where F −1 denotes the Fourier inverse transform defined in Proposition 9. From (58), it turns out by construction that one can write   () =  V  b (), where b () is the inverse Fourier transform of B ().
Remark 10.The reason why we make these factorizations hypotheses on the polynomials (),   (), and the functions   () will be explained later on in Remark 14 of next section and is related to the construction of the Banach spaces in Section 2 and their Fourier inverse transforms.

Construction of a Distinguished Solution.
We make the additional assumption that ,  set above can be chosen in such a way that the following inequalities for all 1 ≤  ≤ , 0 ≤  ≤  and for all 0 ≤  ≤  and all  + 1 ≤  ≤ , for some integer 0 ≤  ≤  − 1, together with for all 0 ≤  ≤   and all   + 1 ≤  ≤ , for some integer 0 ≤   ≤  − 1, hold.
Remark 11.In the case  = 1,  0 ,  1 ≥ 1, the roots of the polynomial (in ) (, ) =  0   0   0 +  1 except the trivial root 0. The constraints (66) imply in particular that  1 −  0 > ( 1 −  0 ).As a result, all the nonvanishing roots of (, ) tend to ∞ as  tends to 0 and 0 is therefore the only root (with order  0 ) of (, ) in the vicinity of 0 as  stays near the origin.Let us assume that the expression (, , ) is allowed to be written as a perturbation series with respect to : where the constant term  0 () is taken independently of  and is not identically equal to 0. The coefficient  0 () is called the slow curve of (64) in the terminology of [6].
In the following, we make the assumption that  0 () solves the following second-order algebraic equation: As  0 () is not identically vanishing, it must be equal to Bearing in mind that  0 ,  0 ̸ = 0, we get its asymptotic behaviour as  tends to 0.
Remark 12.Under the hypotheses (60) and (62), we observe, by factoring out the operator  V  from (64), that (, , ) must solve the related PDE Abstract and Applied Analysis 11 where the forcing term (, , ) is a polynomial in  of degree less than V − 1.According to the assumptions (65), (66), and (67) and using the fact that Q(0) ̸ = 0, by taking  = 0 into (72), we see that the constraint (70) is equivalent to the fact that (, , 0) ≡ 0. The precise shape of the term (, , ) will be given later in Section 6; see (183).
We set  = 2,  = 1,  = 6,  = 1 and we choose the powers of  and  in the coefficients of (61) as follows: For these data, we can check that the constraints (65), ( 66 We can divide this last equation by , but not by , and the resulting equation still possesses a turning point and an irregular singularity at  = 0. (2) For the case ℎ 0 >  0 , we may take  < 0 and hence one can choose some   <  0 for some 0 ≤  ≤ .Let, for instance,  = 1,  = 1,  = 0, and  = 2.We choose  = 2,  = −1,  = −2, and  = 1 and we select the powers of  and  in the coefficients of (61) as follows: For these data, we can figure out that the constraints (65), (66), ( 67), ( 77), (78), and (79) above are satisfied.Moreover, all the forthcoming requirements (88), ( 105), ( 106), (107), and (184) stated in Theorem 19 are also verified.In this particular case, the main equation (61) writes We can divide this latter equation by  3 and by .The corresponding equation still suffers the presence of a turning point and an irregular singularity at  = 0.
In a second step, we divide the left-and right-hand sides of (76) by the monomial   0 + .We obtain the following equation:  Notice that the additional constraints (65), (66), ( 67) and (77), ( 78), (79) ensure that the coefficients of the PDE (84) are analytic with respect to  and  on a neighborhood of the origin in C 2 .Moreover, the coefficient of (  )(, , ) is invertible at  = 0 since  0 ̸ = 0. We will see later that this fact is essential in order to solve this equation within some function space of analytic functions.
We look for solutions which are rescaled in time of the form where As a result, provided that T =    holds, the expression V(T, , ) is supposed to solve the following equation: We make further assumptions on the coefficients   and   for 1 ≤  ≤  which are stronger than the constraint (79).Assume the existence of integers  ≥ 1 and  ,0 ≥ 1 such that for all 1 ≤  ≤  − 1.Then, for all 1 ≤  ≤ , and all integers  1 ≥ 0,  2 ≥ 0 with  1 +  2 =   we deduce the existence of a nonnegative integer  , 1 , 2 which is larger than 1 except  ,0,  = 0 such that Indeed, if one puts  ,0 = 0, from (88), we can write According to (88) and (89), with the help of formula (8.7) from [26], p. 3630, we can expand the following pieces appearing in (87) satisfied by V(T, , ): for all 1 ≤  ≤  and all integers  1 ≥ 0 and  2 ≥ 2 such that  1 +  2 =   , for some real constants    , , 1 ≤  ≤   − 1, and   2 , , 1 ≤  ≤  2 − 1.
In a third step, let us assume that the expression V(T, , ) has a formal power series expansion where each coefficient V  (, ) is defined as an inverse Fourier transform for some function   →   (, ) belonging to the Banach space  (,) and depending holomorphically on  on some punctured disc (0,  0 ) \ {0} centered at 0 with radius  0 > 0.
Remark 14.The hypotheses (60) and (62) ensure that (87) does not contain terms that involve isolated polynomials in T which are not inverse Fourier transformable.

Analytic Solutions of a Convolution Problem with Complex Parameters
Our main goal in this section is the construction of a unique solution of problem (95) within the Banach spaces introduced in Section 2. We make the following further assumptions.The conditions below are very similar to the ones proposed in Section 4 of [3].Namely, we demand that there exists an unbounded sector with direction  Q, R ∈ R and aperture  Q, R > 0 for some radius  Q, R > 0 such that for all  ∈ R. The polynomial P () = − Q() 0 − R ()       can be factorized in the form where for all 0 ≤  ≤    − 1 and all  ∈ R.
We select an unbounded sector   centered at 0 and a small closed disc (0, ) and we require the sector  Q, R to fulfill the next conditions.
(1) There exists a constant  1 > 0 such that for all 0 ≤  ≤    − 1, all  ∈ R, and all  ∈   ∪ (0, ).Indeed, from (99) and the explicit expression (101) of   (), we first observe that |  ()| > 2 for every  ∈ R, all 0 ≤  ≤    − 1 for an appropriate choice of  Q, R and of  > 0. We also see that, for all  ∈ R and all 0 ≤  ≤    − 1, the roots   () remain in a union U of unbounded sectors centered at 0 that do not cover a full neighborhood of the origin in C * provided that  Q, R is small enough.Therefore, one can choose an adequate sector   such that   ∩ U = 0 with the property that for all 0 ≤  ≤    − 1 the quotients   ()/ lay outside some small disc centered at 1 in C for all  ∈   and all  ∈ R.This yields (102) for some small constant  1 > 0.
By construction of the roots (101) in the factorization (100) and using the lower bound estimates (102) and ( 103 for all  ∈ (0,  0 ) \ {0}, where the direction  ∈ R can be chosen for any sector   that fulfills the constraints ( 102) and ( 103) above.
Proof.We undertake the proof with a lemma that studies some shrinking map on the Banach spaces mentioned above and reduces the main convolution problem (95) to the existence of a unique fixed point for this map.
At the very end of the proof, we now take for granted that all conditions (124), ( 133), ( 139), ( 143), (152), and (158) hold for the radii  Q, R and .Then both (113) and (114) hold at the same time and Lemma 16 is shown.

Singular Analytic Solutions on Sectors to the Main Problem
We go back to the sequence of formal constructions performed in Section 4 under the new light shed in Section 5 on problem (95).We first recall the definitions of a good covering and associated sets of sectors as introduced in [3].Definition 17.Let  ≥ 2 be an integer.For all 0 ≤  ≤  − 1, we consider open sectors E  centered at 0, with radius  0 > 0 and opening /( + ) +   with   > 0 small enough such that E  ∩ E +1 ̸ = 0, for all 0 ≤  ≤  − 1 (with the convention that E  = E 0 ).Moreover, we assume that the intersection of any three different elements in {E  } 0≤≤−1 is empty and that for some  0 ∈ {0, . . .,    − 1}; all  ∈ R; all  ∈  d  ∪ (0, ), for all 0 ≤  ≤  − 1.
In the next main first outcome, we construct a family of actual holomorphic solutions to the principal equation (61) which may be meromorphic at (, ) = (0, 0) and defined on the sectors E  with respect to the complex parameter .Furthermore, we can also control the difference between any two neighboring solutions on the intersections E  ∩E +1 and state that it is exponentially flat of order at most ( + ) with respect to .Theorem 19.One considers the nonlinear singularly perturbed PDE (61) and takes for granted that all the assumptions (60), ( 62), ( 65), ( 66), ( 67), ( 77), ( 78), ( 79), ( 88), ( 99), ( 105), (106), and (107) hold for some rational numbers  > 1,  ∈ Q and integers  ∈ Z,  ≥ 1.Let {E  } 0≤≤−1 a good covering in C * be given, for which a family of open sectors {( d  ,, 0  T ) 0≤≤−1 , T} associated with this good covering can be singled out.
Then, there exist a radius  Q, R > 0 large enough and  0 > 0 small enough, for which a family { d  (, , )} 0≤≤−1 of actual solutions of (61) can be built up.More exactly, the functions  d  (, , ) solve the following singularly perturbed PDE: where () is holomorphic on some disc (0,   ),   > 0, and V Bearing in mind the identities of Proposition 8 and using the properties for the   -sum with respect to derivatives and products (within the Banach algebra E =  (,) equipped with the convolution product ⋆ as described in Proposition 2), we check that the functions Ω We examine now the function which defines a bounded holomorphic function with respect to T on  d  ,,ℎ  || + , with respect to  on    for any 0 <   < , and for all  on (0,  0 ) \ {0}.Using the properties of the Fourier inverse transform described in Proposition 9 and watching out the expansions (91), we extract from equality (172) the next equation satisfied by V d  (T, , ); namely, ⋅  −(  − 0 − 1 ) T   − 0 −  ( Observe that (, ) is bounded holomorphic with respect to  and is analytic in  near 0 provided that the following additional conditions hold: +  0 − ℎ 0 ≥ 0, ℎ  + 2 ( 0 − ℎ 0 ) ≥ 0,   +  0 − ℎ 0 −   ≥ 0 with additional forcing term (  , ).As a spin-off, by applying the operator  V  on the left-and right-hand side of this last equation, we see that  d  (, , ) is also an actual solution of problem (61) disclosed at the beginning of Section 4.
In the last part of the proof, we proceed to justify estimates (166).The steps of the verification are similar to the arguments displayed in Theorem 1 of [3], but we choose to present them for the sake of completeness.Let  ∈ {0, . . .,  − 1}.By the sequence of constructions performed above, we see that the function V d  (, , ) can be written as a   -Laplace and Fourier transform where    = R +    ⊂  d  .Using the fact that the function   →   (, , )exp(−(/ + )  )/ is holomorphic on (0, ) for all (, ) ∈ R × ((0,  0 ) \ {0}), its integral along the union of a segment starting from 0 to (/2)  +1 , an arc of circle with radius /2 which connects (/2)  +1 and (/2)   , and a segment starting from (/2)   to 0 are vanishing.Therefore, we can write the difference V d +1 − V d  as a sum of three integrals: (189) By construction, the direction  +1 (which depends on  + ) is chosen in such a way that cos(( +1 − arg( + ))) ≥  1 , for all  ∈ E  ∩ E +1 , for all  ∈ T, and for some fixed  1 > 0.
From the estimates (167), we get that