Abstract and Applied Analysis

Volume 2013 (2013), Article ID 394564, 30 pages

http://dx.doi.org/10.1155/2013/394564

Research Article

On Complex Singularity Analysis for Some Linear Partial Differential Equations in

A. Lastra,¹ S. Malek,² and C. Stenger³

¹Facultad de Ciencias, University of Alcalá, Apartado de Correos 20, 28871 Alcalá de Henares (Madrid), Spain

²UFR de Mathématiques, University of Lille 1, Cité Scientifique M2, 59655 Villeneuve d'Ascq Cedex, France

³Laboratoire Mathématiques, Images et Applications, University of La Rochelle, Avenue Michel Crépeau, 17042 La Rochelle Cedex, France

Received 1 April 2013; Accepted 4 June 2013

Academic Editor: Graziano Crasta

Copyright © 2013 A. Lastra et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We investigate the existence of local holomorphic solutions Y of linear partial differential equations in three complex variables whose coefficients are holomorphic on some polydisc in outside some singular set . The coefficients are written as linear combinations of powers of a solution X of some first-order nonlinear partial differential equation following an idea, we have initiated in a previous work (Malek and Stenger 2011). The solutions Y are shown to develop singularities along with estimates of exponential type depending on the growth's rate of X near the singular set. We construct these solutions with the help of series of functions with infinitely many variables which involve derivatives of all orders of X in one variable. Convergence and bounds estimates of these series are studied using a majorant series method which leads to an auxiliary functional equation that contains differential operators in infinitely many variables. Using a fixed point argument, we show that these functional equations actually have solutions in some Banach spaces of formal power series.

1. Introduction

In this paper, we study a family of linear partial differential equations of the form

for given initial data , , where is a subset of and is an integer which satisfies the constraints (175). The coefficients are holomorphic functions on some domain where is some singular set of (where denotes the disc centered at 0 in with radius ) and the initial data are assumed to be holomorphic functions on the polydisc .

In order to avoid cumbersome statements and tedious computations, the authors have chosen to restrict their study to (1) that involves at most first-order derivatives with respect to and but the method proposed in this work can also be extended to higher order derivatives too.

In this work, we plan to construct holomorphic solutions of the problem (1) on and we will give precise growth rate for these solutions near the singular set of the coefficients (Theorem 21).

There exists a huge literature on the study of complex singularities and analytic continuation of solutions to linear partial differential equations starting from the fundamental contributions of Leray in [1]. Many important results are known for singular initial data and concern equations with bounded holomorphic coefficients. In that context, the singularities of the solution are generally contained in characteristic hypersurfaces issued from the singular locus of the initial conditions. For meromorphic initial data, we may refer to [2–5] and for more general ramified multivalued initial data, we may cite [6–9]. In our framework, the initial data are assumed to be nonsingular and the coefficients of the equation now carry the singularities. To the best knowledge of the authors, few results have been worked out in that case. For instance, the research of the so-called Fuchsian singularities in the context of partial differential equations is widely developed; we provide [10–13] as examples of references in this direction. It turns out that the situation we consider is actually close to a singular perturbation problem since the nature of the equation changes nearby the singular locus of it coefficients.

This work is a continuation of our previous study [14]. In [14], the authors focused on linear partial differential equations in . They have constructed local holomorphic solutions with a careful study of their asymptotic behaviour near the singular locus of the initial data. These initial data were chosen to be polynomial in , and a function satisfying some nonlinear differential equation of first order on some punctured disc and owning an isolated singularity at which is either a pole or an algebraic branch point according to a result of Painlevé. Inspired by the classical  tanh method introduced in [15], they have considered formal series solutions of the form

where are holomorphic functions on where is a small disc centered at 0. They have given suitable conditions for these series to be well defined and holomorphic for in a sector with vertex and moreover as tends to the solutions are shown to carry at most exponential bounds estimates of the form for some constants .

In this work, the coefficients are constructed as polynomials in some function with holomorphic coefficients in , where is now assumed to solve some nonlinear partial differential equation of first order and is asked to be holomorphic on a domain and to be singular along the set . The class of functions in which one can choose the coefficients is quite large since it contains meromorphic and multivalued holomorphic functions in (see the example of Section 2.1).

In our setting, one cannot achieve the goal only dealing with formal expansions involving the function like (2) since the derivatives of with respect to or cannot be expressed only in terms of . In order to get suitable recursion formulas, it turns out that we need to deal with series expansions that take into account all the derivatives of with respect to . For this reason, the construction of the solutions will follow the one introduced in a recent work of Tahara and will involve Banach spaces of holomorphic functions with infinitely many variables.

In [16], Tahara introduced a new equivalence problem connecting two given nonlinear partial differential equations of first order in the complex domain. He showed that the equivalence maps have to satisfy the so-called coupling equations which are nonlinear partial differential equations of first order but with infinitely many variables. It is worthwhile saying that within the framework of mathematical physics, spaces of functions of infinitely many variables play a fundamental role in the study of nonlinear integrable partial differential equations known as solitons equations as described in the theory of Sato. See [17] for an introduction.

The layout of the paper is a follows. In a first step described in Section 2.2, we construct formal series of the form

solutions of some auxiliary nonhomogeneous integrodifferential equation (17) with polynomial coefficients in . The coefficients , , are holomorphic functions on some polydisc in that satisfy some differential recursion (Proposition 2).

In Section 2.3, we establish a sequence of inequalities for the modulus of the differentials of arbitrary order of the functions denoted by for all nonnegative integers with (Proposition 3). In the next section, we construct a sequence of coefficients which is larger than the latter sequence

for any nonnegative integers with and whose generating formal series satisfies some integrodifferential functional equation (51) that involves differential operators with infinitely many variables (Propositions 5 and 6). The idea of considering recursions over the complete family of derivatives and the use of majorant series which lead to auxiliary Cauchy problems were already applied in former papers by the authors of this work; see [14, 18–21].

In Section 3, we solve the functional equation (51) by applying a fixed point argument in some Banach space of formal series with infinitely many variables (Proposition 19). The definition of these Banach spaces (Definition 7) is inspired from formal series spaces introduced in our previous work [14]. The core of the proof is based on continuity properties of linear integrodifferential operators in infinitely many variables explained in Section 3.1 and constitutes the most technical part of the paper.

Finally, in Section 4, we prove the main result of our work. Namely, we construct analytic functions , solutions of (1) for the prescribed initial data, defined on sets for any compact set with precise bounds of exponential type in terms of the maximum value of over (Theorem 21). The proof puts together all the constructions performed in the previous sections. More precisely, for some specific choice of the nonhomogeneous term in (17), a formal solution (3) of (17) gives rise to a formal solution of (1) with the given initial data that can be written as the sum of the integral and a polynomial in having the initial data as coefficients. Owing to the fact that the generating series of the sequence , solution of (51), belongs to the Banach spaces mentioned above, we get estimates for the holomorphic functions with precise bounds of exponential type in terms of the radii of the polydiscs where they are defined; see (196). As a result, the formal solution is actually convergent for near the origin and for belonging to any compact set of . Moreover, exponential bounds are achieved; see (197). The same properties then hold for .

2. Formal Series Solutions of Linear Integrodifferential Equations

2.1. Some Nonlinear Partial Differential Equation

We consider the following nonlinear partial differential equation:

where is some integer and the coefficients , are holomorphic functions on some polydisc such that is not identically equal to zero on .

Notice that (5) can be solved by using the classical method of characteristics which is described in some classical textbooks like [22, page 118] or [23, page 100]. However, the solutions of (5) cannot in general be expressed in closed form. Nevertheless, we can mention some general results concerning qualitative properties of holomorphic solutions to (5) and even to more general first-order partial differential equations of the form

for where is some holomorphic function and an integer. For the construction of holomorphic functions to (6) with singularities located on some specific hypersurfaces (like ), see [24, 25]. For the existence of local multivalued holomorphic solutions ramified around some singular sets, we may refer to [26, 27]. Concerning the study of the analytic continuation of singular solutions bounded on some hypersurface, we cite [28] and with prescribed upper estimates, we quote [29, 30].

In this work, we make the assumption that (5) has a holomorphic solution on where is some set of ( will be called a singular set in the sequel).

In the next example, we show that a large class of functions can be obtained as solutions of equations of the form (5).

Example 1. Let be an integer and let be a holomorphic function which is not identically equal to zero. We consider

which defines a holomorphic function on . Then, the function is a holomorphic solution of the equation

on where is the singular set defined by and is some half-line with depending on the choice of the determination of the logarithm.

2.2. Composition Series

Let be as in the previous subsection. In the following, we choose a compact subset with nonempty interior of for some and we consider a real number such that

Let be a compact set with nonempty interior . From the Cauchy formula, there exists a real number such that

for all integers . For all integers , we denote . We consider a sequence of functions which are holomorphic and bounded on the polydisc , for all .

We define the formal series in the variable as

For all , we consider a holomorphic and bounded function on the product . We define the formal series

Let be a finite subset of and let be an integer which satisfies the property that

for all . For all , , and all integers , we define a function which is holomorphic on and satisfies estimates of the following form. There exist two constants , and an integer such that

for all , with all . In particular, each function is a polynomial of degree at most for all . Finally, for all , , we consider the series

which define holomorphic functions on , for any .

Proposition 2. Assume that the sequence of functions satisfies the following recursion:

for all , all , all , for . Then, the formal series satisfies the following integrodifferential equation:

for all , where denotes the -iterate of the usual integration operator .

Proof . We have that

and we also see that

with

for all . We also get that

with

for all . Now, from (5) and the classical Schwarz’s result on equality of mixed partial derivatives, we get that

and from the Leibniz formula, we can write

for all . Finally, gathering all the equalities above and using the recursion (16), one gets the integrodifferential equation (17).

2.3. Recursion for the Derivatives of the Functions ,

We consider a sequence of functions , , which are holomorphic and bounded on some polydisc for some real numbers and and which satisfy the equalities (16). We introduce the sequence

for all , all , , for all . We define also the following sequences:

for and . We put

for all , and , . We define the sequences

for all , all , all , , for all . We also recall the definition of the Kronecker symbol which is equal to 0 if and equal to 1 if .

Proposition 3. The sequence satisfies the following inequality:

for all , all for .

Proof. In order to get the inequality (30), we apply the differential operator on the left and right hand side of the recursion (16) and we use the expansions that are computed below.

From the Leibniz formula, we deduce that

Moreover, we can write

with

with

By construction, we have

for all . Again, by the Leibniz formula, we get that

Inside the formula (37), we can rewrite the relations (34) and

with

In the same way, one gets the following equalities:

with the factorizations

We recall that

for all and we deduce that

Inside the formula (44), we can rewrite the relations (41) and

with the factorization (39).

2.4. Majorant Series and a Functional Equation with Infinitely Many Variables

Definition 4. One denotes by the vector space of formal series in the variables of the form

where for all .

We keep the notations of the previous section and we introduce the following formal series:

for , all , and

for all , all . We also introduce the following linear operators acting on . Let

for all . We stress the fact that although these operators act on their image does not have to belong to this space.

Proposition 5. A formal series

satisfies the following functional equation:

if and only if its coefficients satisfy the following recursion:

for all , all with .

Proof. We proceed by identification of the coefficients in the Taylor expansion with respect to the variables , and for all . By definition, we have that

where the coefficients can be rewritten, using the Kronecker symbol , in the form

Hence,

We also have that

where the coefficients can be rewritten in the form

Therefore,

On the other hand, using similar computations we get

where

We also have that

where

where

Finally, gathering the expansions (55), (58), (60), and (62) with (64) yields the result.

Proposition 6. The sequences and satisfy the following inequalities:

for all , all , all , .

Proof. For , using the recursions (16) and (52), we get that

for all . By induction on and using the inequalities (30) together with the equalities (52), one gets the result.

3. Convergent Series Solutions for a Functional Equation with Infinitely Many Variables

3.1. Banach Spaces of Formal Series

Let and be real numbers. For any given real number , we define the sequences for all and for all .

Definition 7. Let be an integer. One denotes by the vector space of formal series

that belong to such that the series

is convergent. One denotes also by the vector space of formal series

where belong to for all , such that the series

is convergent. One checks that the space equipped with the norm is a Banach space.

In the next two propositions, we study norm estimates for linear operators acting on the Banach spaces constructed above.

Proposition 8. Consider a formal series

which is absolutely convergent on the polydisc . One uses the notation

Let belong to . Then, the following inequality:

holds.

Proof. Let

which belongs to . By definition, we have that

We can give upper bounds for this latter expression

Lemma  9. For all integers , all , all , all , and all for , one has that

Proof . For any integers and , one has

by using the factorization . Therefore, one gets the inequality

Now, from the identity and the binomial formula, we deduce that

for all , , . Therefore, we deduce that

and the lemma follows from the inequalities (79) and (81).

Finally, the inequality (73) follows from (76) and (77).

Proposition 10. Let be integers such that and . Let and . One has that

for all .

Proof . Let that we write in the form

By definition, we get that

We give upper bounds for this latter expression

Lemma . One has

Proof. We notice that

and, with the help of (78), that for all integers ,

The lemma follows.

We get that the inequality (82) follows from (87) together with (88). Finally, using similar arguments, one gets also the inequalities (83) and (84).

In the next two propositions, we study norm estimates for linear operators acting on the Banach space .

Proposition 12. Let a formal series be absolutely convergent on the polydisc . Let belong to . Then, the product belongs to and the following inequality:

holds.

Proof. Let

By definition, we get

Lemma . One has

Proof. We can write

By remembering (73) of Proposition 8, we deduce that

Lemma . One has

Proof. We write

and we use the inequality

for all and all which follows from (78). This yields the lemma.

Using the fact that and gathering the inequalities (96) and (97) yield (94).

Finally, using (93) with (94), one gets

from which the inequality (91) follows.

Proposition 15. Let be integers such that

Then, there exists a constant (which is independent of ) such that

for all .

Let be integers such that

Then, there exists a constant (which is independent of ) such that

for all .

Proof. We show the first inequality (102). We expand

By definition, we have

Now, using Lemma 13, we deduce that

In the next lemma, we give estimates for the coefficients of the series and .

Lemma .   The coefficients of the Taylor series of

satisfy the next estimates. There exist constants , with , ,   such that

for all , all , all ,   where   is defined in (115).

  The coefficients of the Taylor series of

satisfy the following inequalities. There exist constants ,   with

for all , all .

Proof. We first treat the estimates for . From the Cauchy formula in several variables, one can write

for all , , , and where is introduced in Section 2.2. The integration is made along positively oriented circles with radius , , and for . We choose the real number in such a way that where is defined in Section 2.1 and at the beginning of Section 3.1. Now, since the functions and are holomorphic on , the number (see (10)) can be chosen large enough such that there exist real numbers , for , with

for all . We recall also that for any integers , the number of tuples such that is . From these latter statements and the definition of given by (27), we deduce that

(since ), where

is a polynomial of degree in with positive coefficients, for all , , , and . Gathering (112) and (114) yields (109).

Again, from the Cauchy formula in several variables, one can write

for all , , . Again, one chooses the real number in such a way that . By construction of in Section 2.2, we know that there exist two constants such that

for all , all , , . Gathering (116) and (117) yields (111).

From (111), we deduce that

On the other hand, from Proposition 8, we deduce that

From (109), we deduce that

for all . Now, from the definition of , where , we know that there exists such that

for all . From Proposition 10, we have that

Collecting the estimates (120), (121), and (122), we get from (119) that

where

Now, we recall the following classical estimates. Let be positive real numbers, and then

holds. Hence,

Under the assumptions (101), one gets a constant (depending on ,) such that

for all , all . Finally, gathering (107), (118), (123), and (127), one gets that

provided that , , , and , which yields (102).

Now, we turn towards the estimates (104) which will follow from the same arguments as in . Using Lemma 13, we get that

In the next lemma, we give estimates for the coefficients of the series and .

Lemma .  The coefficients of the Taylor series of

satisfy the next estimates. There exist a constant , with   such that

for all , all , all , .

The coefficients of the Taylor series of

satisfy the following inequalities. There exist constants , with

for all , all .

Proof. From the Cauchy formula in several variables, one can check that

for all , , , , and . We choose the real number in such a way that . From the definition given in (28), we get that

for all , , , , and . Gathering (134) and (135) yields (131).

The proof is exactly the same as in Lemma .

From (133), we deduce that

Using Propositions 8 and 10, we deduce that

where

and where is introduced in (121). Using the estimates (125), we get

Under the assumptions (103), one gets a constant (depending on ) such that

for all , all . Finally, gathering (129), (136), (137), and (140), we get (104).

Proposition 18.  Let be integers such that

Then, for , there exists a constant (which is independent of ) such that

for all .

Let be integers such that

Then, there exists a constant (which is independent of ) such that

for all .

Proof. We expand

By definition, we have

Now, using Lemma 13, we deduce that

From Proposition 10, we know that

From (118), (136), (147), and (148), we get that

where

Using the estimates (125), we deduce that

Under the assumption (141), we get a constant (depending on ) such that

for all , all . Finally, collecting (149) and (152), we get

which yields (142).

We expand

By definition, we have

Now, using Lemma 13, we deduce that

From Proposition 10, we know that

On the other hand, the coefficients of the Taylor series of

satisfy the following inequalities. There exist constants , with

for all , all . The proof copies from Lemma 16. From (159), we deduce that

From (160), (156), and (157), we get that

where

Using the estimates (125), we deduce that

Under the assumption (143), we get a constant (depending on ) such that

for all , all . Finally, collecting (161) and (164), we get

which yields (144).

3.2. A Functional Partial Differential Equation in the Banach Spaces of Infinitely Many Variables

In the next proposition, we solve a functional fixed point equation within the Banach spaces of formal series introduced in the previous subsection.

Proposition 19. One makes the following assumptions:

for all . Then, for given , there exists (independent of ) such that, for all , the functional equation

has a unique solution . Moreover, one has that

Proof . We consider the map from the space of formal series (introduced in Definition 4) into itself defined as follows:

for all .

In order to prove the proposition, we need the following lemma.

Lemma . Let be the identity map      into itself. Then, for a well-chosen , the map   defines an invertible map such that   is defined from   into itself. Moreover, one has that

for all .

Proof. Taking care of the constraints (166), we get from Propositions 15 and 18 a constant (depending on the constants introduced above and also on the aforementioned propositions: , , , , , , , , , and but independent of ) such that

for all with . Since for all , we can choose such that

together with . We deduce that

for all . This yields the estimates (170).

Finally, let for chosen as in Lemma 20. We define

By construction, belongs to and solves (167) with the estimates (168).

4. Analytic Solutions with Growth Estimates of Linear Partial Differential Equations in

We are now in position to state the main result of our work.

Theorem 21. Let be the functions defined in (15) for and . Let one assume that there exists such that

for all . For all , one considers functions which are assumed to be holomorphic and bounded on the product .

Then, there exist constants such that the problem

with initial data

has a solution which is holomorphic on and which fulfills the following estimates:

where , for any compact set with nonempty interior Int for some and any which satisfies (10). One stresses that the constants do not depend neither on nor on .

Proof. By convention, we will put for all . On the other hand, we specialize the functions which were introduced in (12) in order that

By construction and using the definition (26), we can write with the help of the Kronecker symbol,

where

Lemma 22. There exist   such that the formal series

belongs to . Moreover, there exists a constant   (independent of ) such that

Proof. Let . Due to the estimates (14) for the functions , we get couples of constants , , and such that

for all , all , , . Moreover, we also get couples of constants , , and such that

for all , all , . From (184) and (185) we deduce

for all , all , , . From the Cauchy formula in several variables, one can write

for all , , . We deduce that

for all , all . Using (180), we get that

From (188), (125), and with the help of the classical estimates

for all , we get a constant (depending on , , , , , , , , for all , , )  such that

We choose

From (191) we deduce the inequality (183).

Under the assumption (175), we get from Proposition 19 four constants , , , and (independent of ) such that the functional equation

has a unique solution belonging to which satisfies moreover the estimates

Now, from Proposition 6, we know that the sequence introduced in (25) satisfies the inequality

for all , all , for . Gathering (194) and (195) and from the definition of the Banach spaces in Section 3.1, we get, in particular, for , for all , all , that

for all and where . From (196), we get that the formal series introduced in (11) actually defines a holomorphic function (denoted again by ) on for which the estimates

hold and which satisfies (17) on .

Finally, we define the function

By construction, defines a holomorphic function on with bounds estimates

and solves the problem (176), (177). This yields the result.

Acknowledgments

A. Lastra is partially supported by Project MTM2012-31439 of Ministerio de Ciencia e Innovacion, Spain. S. Malek is partially supported by the French ANR-10-JCJC 0105 project and the PHC Polonium 2013 Project no. 28217SG.

References

J. Leray, “Problème de Cauchy. I. Uniformisation de la solution du problème linéaire analytique de Cauchy près de la variété qui porte les données de Cauchy,” Bulletin de la Société Mathématique de France, vol. 85, pp. 389–429, 1957.

O. Costin, H. Park, and Y. Takei, “Borel summability of the heat equation with variable coefficients,” Journal of Differential Equations, vol. 252, no. 4, pp. 3076–3092, 2012.

Y. Hamada, “The singularities of the solutions of the Cauchy problem,” Publications of the Research Institute for Mathematical Sciences, vol. 5, pp. 21–40, 1969.

S. Ōuchi, “An integral representation of singular solutions of linear partial differential equations in the complex domain,” Journal of the Faculty of Science, vol. 27, no. 1, pp. 37–85, 1980.

S. Ōuchi, “The behaviour near the characteristic surface of singular solutions of linear partial differential equations in the complex domain,” Japan Academy A, vol. 65, no. 4, pp. 102–105, 1989.

Y. Hamada, J. Leray, and C. Wagschal, “Systèmes d'équations aux dérivées partielles à caractéristiques multiples: problème de Cauchy ramifié; hyperbolicité partielle,” Journal de Mathématiques Pures et Appliquées, vol. 55, no. 3, pp. 297–352, 1976.

K. Igari, “On the branching of singularities in complex domains,” Japan Academy, vol. 70, no. 5, pp. 128–130, 1994.

B. Sternin and V. Shatalov, Differential Equations on Complex Manifolds, vol. 276 of Mathematics and its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1994.

C. Wagschal, “Sur le problème de Cauchy ramifié,” Journal de Mathématiques Pures et Appliquées, vol. 53, pp. 147–163, 1974.

S. Alinhac, “Problèmes de Cauchy pour des opérateurs singuliers,” Bulletin de la Société Mathématique de France, vol. 102, pp. 289–315, 1974.

A. Bove, J. E. Lewis, and C. Parenti, Propagation of Singularities for Fuchsian Operators, vol. 984 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1983.

R. Gérard and H. Tahara, Singular Nonlinear Partial Differential Equations, Aspects of Mathematics, Friedrich Vieweg & Sohn, Braunschweig, Germany, 1996.

T. Mandai, “The method of Frobenius to Fuchsian partial differential equations,” Journal of the Mathematical Society of Japan, vol. 52, no. 3, pp. 645–672, 2000.

S. Malek and C. Stenger, “On complex singularity analysis of holomorphic solutions of linear partial differential equations,” Advances in Dynamical Systems and Applications, vol. 6, no. 2, pp. 209–240, 2011.

W. Malfliet, “Solitary wave solutions of nonlinear wave equations,” American Journal of Physics, vol. 60, no. 7, pp. 650–654, 1992.

H. Tahara, “Coupling of two partial differential equations and its application,” Publications of the Research Institute for Mathematical Sciences, vol. 43, no. 3, pp. 535–583, 2007.

Y. Ohta, J. Satsuma, D. Takahashi, and T. Tokihiro, “An elementary introduction to Sato theory. Recent developments in soliton theory,” Progress of Theoretical Physics, no. 94, pp. 210–241, 1988.

A. Lastra, S. Malek, and J. Sanz, “On -asymptotics for linear -difference-differential equations with Fuchsian and irregular singularities,” Journal of Differential Equations, vol. 252, no. 10, pp. 5185–5216, 2012.

S. Malek, “On the summability of formal solutions of linear partial differential equations,” Journal of Dynamical and Control Systems, vol. 11, no. 3, pp. 389–403, 2005.

S. Malek, “On functional linear partial differential equations in Gevrey spaces of holomorphic functions,” Annales de la Faculté des Sciences de Toulouse, vol. 16, no. 2, pp. 285–302, 2007.

S. Malek, “On Gevrey functional solutions of partial differential equations with Fuchsian and irregular singularities,” Journal of Dynamical and Control Systems, vol. 15, no. 2, pp. 277–305, 2009.

L. Debnath, Nonlinear Partial Differential Equations for Scientists and Engineers, Birkhäuser, Boston, Mass, USA, 1997.

L. C. Evans, Partial Differential Equations, vol. 19 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, USA, 2nd edition, 2010.

H. Tahara, “On the singularities of solutions of nonlinear partial differential equations in the complex domain. II,” in Differential Equations & Asymptotic Theory in Mathematical Physics, vol. 2 of Series in Analysis, pp. 343–354, World Scientific, Hackensack, NJ, USA, 2004.

H. Tahara, “On the singularities of solutions of nonlinear partial differential equations in the complex domain,” in Microlocal Analysis and Complex Fourier Analysis, pp. 273–283, World Scientific, River Edge, NJ, USA, 2002.

M. Kametani, “On multi-valued analytic solutions of first order nonlinear Cauchy problems,” Publications of the Research Institute for Mathematical Sciences, vol. 27, no. 1, pp. 1–131, 1991.

T. Kobayashi, “Propagation of singularities for a first order semilinear system in ,” Annali della Scuola Normale Superiore di Pisa, vol. 12, no. 2, pp. 173–189, 1985.

Y. Tsuno, “On the prolongation of local holomorphic solutions of nonlinear partial differential equations,” Journal of the Mathematical Society of Japan, vol. 27, no. 3, pp. 454–466, 1975.

T. Kobayashi, “Singular solutions and prolongation of holomorphic solutions to nonlinear differential equations,” Publications of the Research Institute for Mathematical Sciences, vol. 34, no. 1, pp. 43–63, 1998.

J. E. C. Lope and H. Tahara, “On the analytic continuation of solutions to nonlinear partial differential equations,” Journal de Mathématiques Pures et Appliquées, vol. 81, no. 9, pp. 811–826, 2002.