The asymptotic behavior of a family of singularly perturbed PDEs in two time variables in the complex domain is studied. The appearance of a multilevel Gevrey asymptotics phenomenon in the perturbation parameter is observed. We construct a family of analytic sectorial solutions in ϵ which share a common asymptotic expansion at the origin, in different Gevrey levels. Such orders are produced by the action of the two independent time variables.
This work is devoted to the study of a family of nonlinear initial value problems of the form
with initial null data .
The terms Q, P, , are polynomials in all their variables except from the perturbation parameter ϵ in which they turn out to be holomorphic in a neighborhood of the origin. Moreover, we assume that the polynomial
where involves leading terms from the differential operator P which can be factorized in such a way that each of the factors only depend on one of the times variables, i.e.
where the factors will be concreted later in the introduction.
The present work is a natural continuation of that in [6].
In the previous study [6], we focused on families of equations of the form
for given vanishing initial data , where , , H are polynomials in their corresponding variables, and the forcing term turns out to be an analytic function near the origin for , holomorphic with respect to z on a horizontal strip of the form
for some . We constructed a set of genuine bounded holomorphic solutions , for , defined on domains , for well chosen bounded sectors with vertex at the origin, and represents a set of bounded sectors whose union contains a full neighborhood of the origin in , which is called a good covering in . Such functions were built as Laplace transform of order k and inverse Fourier transform
along halflines , where is an unbounded sector with bisecting direction , and represents a holomorphic function with exponential growth w.r.t. u on , continuous w.r.t. with exponential decay and holomorphic w.r.t. ϵ on , for some . The highest order term of H in (4) is of irregular type
for some positive integer k, some integer and a polynomial . We proved that the functions share a common asymptotic expansion w.r.t. ϵ, say , a formal series with bounded holomorphic coefficients on . This asymptotic expansion is (at most) of Gevrey order , meaning that
for all , all . In case the aperture of can be taken slightly larger than , the function is the k-sum of on .
In view of the shape of equation (4), we assume that the factors in (3) are presented as a product of factors of the form
and
The importance of the present work with respect to that previous one is mainly due to the appearance of a multilevel Gevrey asymptotics phenomenon in the perturbation parameter, when dealing with a multivariable approach in time.
A recent overview on summability and multisummability techniques under different points of view is displayed in [10].
In recent years, an increasing interest on complex singularly perturbed PDEs has been observed in the area. Parametric Borel summability has been described in semilinear systems of PDEs of Fuchsian type by H. Yamazawa and M. Yoshino in [14]
where and , for , and . η is a small complex perturbation parameter and f stands for a holomorphic vector function in a neighborhood of the origin in . Also, in partial differential equations of irregular singular type by M. Yoshino in [15]:
where for , and is a holomorphic vector function in some neighborhood of the origin in .
Recently, S.A. Carrillo and J. Mozo-Fernández have studied properties on monomial summability and the extension of Borel–Laplace methods to this theory in [3,4]. In the last section of the second work, a further development on multisummability with respect to several monomials is proposed. Novel Gevrey asymptotic expansions and summability with respect to an analytic germ are described in [11] and applied to different families of ODEs and PDEs such as
where P is an homogeneous polynomial, and h, A, B are convergent power series, and α, β satisfy certain conditions.
We now give some words on the main results obtained in our present study.
In the first principal outcome (Theorem 1), we construct a double indexed family of bounded holomorphic solutions to our main problem (1), presented as a double Laplace transform and Fourier integral
defined on , for some bounded sectors , with vertex at 0, some , and where is a horizontal strip. Here, the set is a good covering of of finite sectors with vertex at the origin (see Definition 4).
In the second main result, Theorem 2, we show that all the functions share a common asymptotic expansion , with bounded holomorphic coefficients on . This formal power series can be split as a sum of two formal power series, and each of the holomorphic solutions is decomposed accordingly in such a way that different Gevrey asymptotic behavior can be observed in each term of the sum. This phenomenon is the key point to multisummability, as described in [1] and also Section 7.5 in [10].
We have labelled these equations as symmetric for the following reason: the function , lying in the Borel plane, is defined on a domain which has the same shape with respect to and , namely a product of unions of discs and unbounded sectors. In this respect, it can be seen as a first step in the generalization of our previous work [6]. In comparison to this work, the result obtained in [9] is a step further in which we study a particular case that we call asymmetric since it deals with a situation where the Borel map presents an asymmetry in its domain of holomorphy. These two geometric configurations give rise to different parametric Gevrey asymptotic behavior of the true solutions.
Throughout the whole paper, we denote , , .
The structure of the paper is as follows. Section 2 states and gives details on the main problem under study (9), and the elements involved in it. It is also explained how to reduce the study of the main problem to that of two auxiliary equations, when searching for solutions under the form of a double Laplace and Fourier transform. In Section 3 we state the definition and some properties of the Banach spaces involved in the solution of the two auxiliary problems. Subsequently, a fixed point argument gives rise to the solutions of these auxiliary problems which lead to an analytic solution of the main problem under study in Section 4, Theorem 1. Also, upper estimates on the difference of two consecutive solutions are described in order to apply a novel version of the multilevel Ramis–Sibuya theorem and provide the existence of a formal solution of the main problem in Theorem 2. The analytic and formal solutions are related by means of a multilevel Gevrey asymptotic representation.
Description of the main problem
Let and be integers. For and , let , , , , be non negative integers. We assume that
for all and .
Let , and for , .
Let , be polynomials belonging to , with coefficients in the space of bounded holomorphic functions on a disc , for some . We assume
for all , all and .
We first introduce the following Banach space of functions.
Let . We denote by the vector space of continuous functions such that
is finite. The space equipped with the norm is a Banach space.
The forcing term is constructed as follows. For every , let , belonging to the Banach space for some and , depending holomorphically on , for some positive . We assume the existence of such that
for all and . We put
which is a convergent series on with values in . We define the forcing term as a time rescaled version of the inverse Fourier transform of F, namely
We make the additional assumption that there exist unbounded sectors
with direction , aperture for some radius such that
for all , and for .
We search for the solutions of our main problem in the form
for some expression .
The next result has already been mentioned in [6].
Letwith,. The inverse Fourier transform of f, defined byextends to an analytic function on the strip. Let. Then, it holds,.
Letand put, the convolution product of f and g, for all. We haveand,.
From the classical properties of the inverse Fourier transform recalled in Proposition 1, it holds that satisfies the auxiliary equation
for given initial data .
We seek for solutions of the auxiliary problem above in the form of a double Laplace transform
where , for , where is an infinite sector to be precised later. The function belongs to a Banach space of functions such that the integrals in the previous expression make sense. The explicit form of such Banach spaces will be described in Section 3.
This is motivated by the fact that the forcing term can be expressed in such a form. Indeed, we will show in the next section that
defines an entire function with values in that belongs to that Banach space. Direct computations show that
Our next goal is to provide a functional equation satisfied by . For that purpose, we need first to expand the equation (12) by means of the following relations (see [12], p. 3630):
for some real numbers , and , . We write (resp. ) for (resp. ) for the sake of simplicity. Let satisfying
for all and . Multiplying the equation (12) by and taking into account (14,15), we rewrite (12) in the form
The proof of the following result, needed in the sequel, can be found in detail in Lemma 1 [9].
Letbe the function constructed in (
13
). Then, it holds that:
We define the operators (resp. ) in the form
Finally, we arrive at the next convolution equation
Banach spaces of functions and solutions of the auxiliary problems
Let be an open disc centered at 0 with radius in , and by its closure. Let be open unbounded sectors with bisecting directions for , and be an open sector with finite radius , all centered at 0 in .
The definition of the following norm heavily rests on that considered in [6]. Here, the exponential growth is held with respect to the two time variables which are involved.
Let and be positive real numbers. Let be integer numbers and let . We put , , , and denote the vector space of continuous functions on the set , which are holomorphic with respect to on and such that
is finite. One can check that the normed space is a Banach space.
Throughout the whole section, we assume , are fixed numbers. We also fix for some positive numbers , , and is a couple of positive integer numbers. Additionally, we take , and write for . The next results are stated without proofs, which are analogous to those in Section 2 of [6]. The integrals appearing in these results can be split accordingly, in order to apply the proof therein.
Letbe a bounded continuous function on, holomorphic with respect toon. Then,for all.
Letbe real numbers. Assume thatare such that, for. Then, a constant(depending on,,,) exists withfor all.
Proposition 2 in [6] is adapted to the Banach space considered in this work.
Letandbe real numbers. Letbe integer numbers. We write. We consider, continuous on, such that
Assume that forand, one of the following holds
and, or
, for someand.
Then, there exists a constant(depending, eventually, on,,,,,,,,) such thatfor all.
The previous result can also be particularized to each of the variables in time in the following manner. We write the result which corresponds to the first time variable, but one can reproduce the same arguments symmetrically on the second variable in time, .
Letandbe real numbers, andbe an integer number. We consider, continuous on, such thatAssume that
and, or
, for someand.
Then, there exists a constant(depending, eventually, on,,,,,) such thatfor all.
Letsuch thatAssume that. Letbe a continuous function onsuch thatThen, there exists a constant(depending on,, R, μ,,) such thatfor all.
We consider the polynomial and assume that and are the complex roots of each polynomial, for . Following an analogous manner as in the construction of [6], one can choose unbounded sectors and , with vertex at 0 and such that
for all , and ; and
for all , and . From now on, we write
for a more compact writing.
Let . The next result guarantees the existence of an element in which turns out to be a fixed point for certain operator to be described, solution of (19). Here β, μ are fixed at the beginning of this section.
We now give some words in order to guarantee that the forcing term belongs to the Banach space .
It holds that
Using classical estimates and Stirling formula we guarantee the existence of depending on , such that, if , then for all . One has
We also refer to [6], Section 4, for further details.
Under the assumption thatfor all,, there exists a constant(depending on,,,, μ,,,,,,,,forand) such that iffor all, there exist,, such that the equation (
19
) has a unique solutionin the space, whereare defined in construction of the forcing term f, which verifies, for all.
Let . We consider the operator , defined by
where
Let and assume that . Assume that for all . We first obtain the existence of such that the operator sends into itself. Here, stands for the closed ball of radius ϖ, centered at 0, in the Banach space .
Using Lemma 2 and Proposition 4, with (27) and (28) we get
In view of (34), (35), (36), (37), (38), and (40), one gets that the operator is such that . The next stage of the proof is to show that, indeed, is a contractive map in that ball. Let with . Then, it holds that
for all .
Analogous estimates as in (34), (35), (36), (38) yield
Finally, put
and . Then, taking into account that
and by using Lemma 2 and analogous estimates as in (37) and (27), we get
Let such that
Then, (48) combined with (42), (43), (44), (45) and (47) yields (41).
We consider the ball constructed above. It turns out to be a complete metric space for the norm . As is a contractive map from into itself, the classical contractive mapping theorem, guarantees the existence of a unique fixed point for . The function depends holomorphically on ϵ in . By construction, defines a solution of the equation (19). □
Regarding the construction of the auxiliary equations, one can obtain the analytic solutions of (12) by means of Laplace transform.
Under the hypotheses of Proposition
6
, choose,and,in such a way that the roots ofandfall appart fromand, respectively, as stated before (
27
).
Notice that they apply for any small enough, provided that (
30
) hold.
Let,be bounded sectors with aperture, for(whereis the opening of), with directionand radiusfor someindependent of ϵ. We choose.
Then, equation (
12
) with initial conditionhas a solutiondefined onfor someand all. Let, then forand all, the functionbelongs to the spaceand for each, the functionis bounded and holomorphic on. Moreover,can be written as a Laplace transform of orderin the directionwith respect toand the Laplace transform of orderin the directionwith respect to,where, forhas bisecting direction which might depend on. The functiondefines a continuous function on, holomorphic with respect toon. Moreover, there exists a constant(independent of ϵ) such thatfor all, all, and.
Let . We take the function constructed in Proposition 6
We follow backwards the details of the construction of Section 2. Since solves (19), we deduce that the function (49) is bounded on with respect to and fulfills the equation (17), in view of Lemma 1. Then, according to (14), (15) it holds that solves (12). □
Construction of a finite set of genuine solutions of the main problem
We recall the definition of a good covering in .
Let be integer numbers. Let be a finite family of open sectors with vertex at 0, radius and opening strictly larger than . We assume that the intersection of three different sectors in the good covering is empty, and , for some neighborhood of 0, . Such set of sectors is called a good covering in .
Let and be a good covering in . Let be open bounded sectors centered at 0 with radius for , and consider two families of open sectors as follows. The first one is given by
with opening , and some , for all . This family is chosen to satisfy that:
There exists a constant such that
for all , , and , for all , and every root of the polynomial .
There exists a constant such that
for some root of , all , , for all .
The second family is chosen in an analogous manner. It is given by
with opening , and some , for all . This family is chosen to satisfy analogous conditions with respect to the roots of the polynomial .
In addition to the previous assumptions, we consider and such that for all , , , and , one has
We say that the family is associated to the good covering .
The first main result of the present work is devoted to the construction of a family of actual holomorphic solutions to the equation (9) for null initial data. Each of the elements in the family of solutions is associated to an element of a good covering with respect to the complex parameter ϵ. The strategy leans on the control of the difference of two solutions defined in domains with nonempty intersection with respect to the perturbation parameter ϵ. The construction of each analytic solution in terms of two Laplace transforms in different time variables requires to distinguish different cases, depending on the coincidence of the integration paths or not.
We consider the equation (
9
) and we assume that (
6
–
10
) and (
11
) hold. We also make the additional assumption thatfor alland.
Letbe a good covering inwithbeing a family associated to this good covering can be considered.
Then, there exist,, such that for everyand, one can construct a solutionof (
9
) withwhich defines a bounded holomorphic function on the domainfor any givenand for some.
Moreover, there exist constants,(independent of ϵ), and setssuch that for every, one of the following holds:
.
andfor all. In this situation, we say thatbelong to.
andfor all. In this situation, we say thatbelong to.
Regarding Proposition 7, one can choose , , and in a way that for each pair , we fix the multidirection with and construct such that and is a solution of
where
is a convergent series in with values in , for all . The function is well defined on where , for all . Moreover, can be written as the iterated Laplace transform of order in the direction , and the Laplace transform of order in the direction
along which might depend on . Here, defines a continuous function on , holomorphic with respect to on for all . Moreover, there exists a constant (independent of ϵ) such that
for all , all and . The function
turns out to be holomorphic on , for all and . For all , let
By construction (see Definition 4), the function defines a bounded holomorphic function on . Moreover, . Moreover, the properties of inverse Fourier transform described in Proposition 1 guarantee that is a solution of the main problem under study (9) on .
It is worth mentioning that all the functions provide the analytic continuation of a common function to .
Different digressions are considered, due to the presence of two time variables. Let for , and assume that . Then, three different cases should be considered:
Case 1: Assume that the path coincides with , and does not coincide with . Then, using that is holomorphic on for all , and every , one can deform one of the integration paths to write
in the form
where , and is an arc of circle connecting and with the adequate orientation.
The estimates for the previous expression can be found in detail in the proof of Theorem 1, [6]. Namely, we get the existence of constants such that
for and and . We have
The last integral is estimated via the reparametrization and the change of variable by
for some , whenever .
From the fact that , we get that belong to .
Path deformation in Case 1.
Case 2: The path coincides with , and does not coincide with . It can be handled analogously as Case 1. We get that the set belongs to . More precisely, we arrive at the expression
Case 3: Assume that neither coincides with , nor coincides with .
We deform the integration paths with respect to the first time variable and write
where
where is such that θ is an argument between and . The path (resp. ) consists of the concatenation of the arc of circle connecting with (resp. with ) and the half line (resp. ).
Path deformation in Case 3.
We first give estimates for . We have
for some , and . Using the parametrization and the change of variable . Using analogous estimations as in the Case 1, we arrive at
for some , for all , where and , .
Analogous calculations yield to
for some , for all , where and , .
In order to give upper bounds for , we consider
We choose a deformation path in the form of that considered in Case 1. We get the previous expression is upper estimated by
for , , . We finally get
We conclude that
uniformly for for some , and for any fixed , where , are positive constants. □
Asymptotics of the problem in the perturbation parameter
k-Summable formal series and Ramis–Sibuya theorem
For the sake of completeness, we recall the definition of k-Borel summability of formal series with coefficients in a Banach space, and Ramis–Sibuya Theorem. A reference for the details on the first part is [1], whilst the second part of this section can be found in [2], p. 121, and [5], Lemma XI-2-6.
Let be an integer. A formal series
with coefficients in a Banach space is said to be k-summable with respect to ϵ in the direction if
there exists such that the following formal series, called formal Borel transform of of order k
is absolutely convergent for ,
there exists such that the series can be analytically continued with respect to τ in a sector . Moreover, there exist , and such that
for all .
If this is so, the vector valued Laplace transform of order k of in the direction d is defined by
along a half-line , where γ depends on ϵ and is chosen in such a way that , for some fixed , for all ϵ in a sector
where and . The function is called the k-sum of the formal series in the direction d. It is bounded and holomorphic on the sector and has the formal series as Gevrey asymptotic expansion of order with respect to ϵ on . This means that for all , there exist such that
for all , all .
Multisummability of a formal power series is a recursive process that allows to compute the sum of a formal power series in different Gevrey orders. One of the approaches to multisummability is that stated by W. Balser, which can be found in [1], Theorem 1, p. 57. Roughly speaking, given a formal power series which can be decomposed into a sum such that each of the terms is -summable, with sum given by , then, turns out to be multisummable, and its multisum is given by . More precisely, one has the following definition.
Let be a complex Banach space and let . Let be a bounded open sector with vertex at 0, and opening for some , and let be a bounded open sector with vertex at the origin in , with opening , for some and such that holds. A formal power series is said to be -summable on if there exist which is -summable on , with -sum given by , and which is -summable on , with -sum given by , such that . Furthermore, the holomorphic function on is called the -sum of on . In that situation, can be obtained from the analytic continuation of the -Borel transform of by the successive application of accelerator operators and Laplace transform of order , see Section 6.1 in [1].
A novel version of Ramis–Sibuya Theorem has been developed in [13], and has provided successful results in previous works by the authors, [7,8]. A version of the result in two different levels which fits our needs is now given without proof, which can be found in [7,8].
Letbe a Banach space overandbe a good covering in. Assume that. For alland, letbe a holomorphic function frominto the Banach spaceand for everysuch thatwe definebe a holomorphic function from the sectorinto. We make the following assumptions.
The functionsare bounded astends to the origin in, for alland.
, where
iff,
iffandfor all.
iffandfor all.
Then, there exists a convergent power seriesand two formal power seriessuch thatcan be split in the formwhere, and admitsas its asymptotic expansion of Gevrey orderon, for.
Moreover, assume thatis a subset of, for some positive integer y, andfor some sectorwith opening larger than. Then, the formal power seriesis-summable onand its-sum ison.
Existence of formal power series solutions in the complex parameter and asymptotic behavior
The second main result of our work states the existence of a formal power series in the perturbation parameter ϵ, with coefficients in the Banach space of holomorphic and bounded functions on , with the norm of the supremum. Here , , are determined in Theorem 1.
The importance of this result compared to the main one in [6] lies on the fact that a multisummability phenomenon can be observed here, in contrast to [6]. This situation is attained due to the appearance of different Gevrey levels coming from the different variables in time.
Under the assumptions of Theorem
1
, a formal power seriesexists, with the following properties.is a formal solution of (
9
). In addition to that,can be split in the formwhere, and. Moreover, for everyand, the functioncan be written aswhereis an-valued function which admitsas its-Gevrey asymptotic expansion on, for.
Moreover, assume thatis a subset of, for some positive integer y, andfor some sectorwith opening larger than. Then,is-summable onand its-sum ison.
Let be the family constructed in Theorem 1. We recall that is a good covering in .
The function belongs to . We consider such that and belong to , and and are consecutive sectors in the good covering, so their intersection is not empty. In view of (54) and (55), one has that satisfies exponentially flat bounds of certain Gevrey order, which is in the case that and if . Multilevel-RS Theorem guarantees the existence of formal power series such that
and the splitting
for some , such that for every , one has that admits as its Gevrey asymptotic expansion of order , and admits as its Gevrey asymptotic expansion of order . We define
It only rests to prove that is a formal solution of (9). For every , and , the existence of an asymptotic expansion concerning and implies that
for every . By construction, the function is a solution of (9). Taking derivatives of order with respect to ϵ on that equation yield
for every and . Tending in (62) together with (61), we obtain a recursion formula for the coefficients of the formal solution.
for every , and . From the analyticity of f with respect to ϵ in a vicinity of the origin we get
for every and as above. On the other hand, a direct inspection from the recursion formula (63) and (64) allow us to affirm that the formal power series solves the equation (9). □
Footnotes
Acknowledgements
The authors want to express their gratitude to the anonymous referee for his/her careful reading of our work and the helpful comments and suggestions that helped to improve the quality of the manuscript.
The authors are partially supported by the project MTM2016-77642-C2-1-P of Ministerio de Economía y Competitividad, Spain.
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