An Introduction to Nonlinear Partial Differential Equations in the Complex Domain

An Introduction to Nonlinear Partial

Differential Equations in the Complex Domain

Hidetoshi TAHARA

Department of Mathematics, Sophia University

1 Preliminaries

This introductory text surveys some known results in the theory of nonlinear partial differential equations in the complex domain. For simplicity, we will present the results in only two variables (t, x) ∈C². Moreover, we will only deal with equations which are of order one with respect to ∂/∂t.

We will start with the most fundamental theorem on the existence and uniqueness of holomorphic solutions to nonlinear partial differential equa- tions, the Cauchy-Kowalewski Theorem. Then we will proceed by examin- ing whether or not these equations possess singular solutions which admit singularities in some hypersurface. (Such solutions will be termed singular solutions.) This will be done either by examining the possibility of analytic continuation of solutions or by actually constructing solutions that possess singularities on a particular hypersurface. Two theorems on analytic con- tinuation, one due to Tsuno [8] and the other due to Kobayashi [6], will be presented. In the course of constructing singular solutions, we will come across singular partial differential equations. The final section will feature some results by Gerard-Tahara [4],[5] and Chen-Tahara [1],[2] on this type of equations.

We will employ majorant functions to prove the theorems stated in this lecture notes. The proofs will be made significantly simpler (than previ- ously known proofs using majorant series) with the use of a certain family of majorant functions which is a modified version of the one considered by Lax [7].

Notes of lectures at Sophia University in April-July, 2000 (summarized by J.E.C.Lope).

Hereafter, we agree to use the following notations to describe majorant relations: if a(x) =

a_ixⁱ and A(x) =

A_ixⁱ, then we say that a(x) A(x) if and only if |a_i| ≤A_i for each i. Likewise, if g(t, x) =

g_ik(t−ε)ⁱx^k and G(t, x) =

G_ik(t−ε)ⁱx^k, then we say that g(t, x) ε G(t, x) if and only if |g_ik| ≤G_ik for all i and k.

Denote byNthe set of all nonnegative integers and byN^∗ the set N\{0}. LetS = 1 + 1/2²+ 1/3²+. . . =π²/6. We now define the following family of majorant functions: for each i∈N, we set

(1.1) φ⁽ⁱ⁾(z) = 1

4S ∞ n=0

zⁿ (n+ 1)²⁺ⁱ.

The constant 1/4S was introduced by the author to facilitate computation;

without this constant, several other constants (dependent on i) will appear in computations involving φ⁽ⁱ⁾(z) and will thus make the situation appear more complicated.

Note that each φ⁽ⁱ⁾(z) converges for all |z| < 1. Moreover, this family of functions enjoys some interesting majorant relations, as is stated in the following proposition.

Proposition 1.1. The following relations hold:

(i) φ⁽⁰⁾(z)φ⁽⁰⁾(z)φ⁽⁰⁾(z) ;

(ii) φ⁽⁰⁾(z)φ⁽¹⁾(z)φ⁽²⁾(z) · · · ; (iii)

1 2

_2+i

φ⁽ⁱ⁻¹⁾(z) d

dz φ⁽ⁱ⁾(z)φ⁽ⁱ⁻¹⁾(z) ;

(iv) Given any 0< ε < 1, then there exists a positive constant C_i,ε, depen- dent on i and ε, such that

1

1−εz φ⁽ⁱ⁾(z)C_i,εφ⁽ⁱ⁾(z).

The first three relations are easily verified using the definition of φ⁽ⁱ⁾(z).

To prove the fourth relation, it is sufficient to show that

(1.2) 1

1−εz = ∞ n=1

εⁿzⁿ C_i,εφ⁽ⁱ⁾(z),

for some constant C_i,ε > 0. (For one can easily verify that φ⁽ⁱ⁾(z)φ⁽ⁱ⁾(z) 2ⁱφ⁽ⁱ⁾(z) holds.) But this is equivalent to saying that for all n, we have

(1.3) εⁿ ≤ C_i,ε

4S(n+ 1)²⁺ⁱ .

Clearly, such a constant exists sinceεⁿ(n+1)²⁺ⁱis close to zero for sufficiently large values of n.

We remark that if a(x) is holomorphic in a neighborhood of |x| ≤ R₀, then a(x) is majorized by

a(x) M

1−x/R₀ M

1−εx/R ×4S φ⁽ⁱ⁾(x/R) (1.4)

4SM C_i,εφ⁽ⁱ⁾(x/R), for any 0< R < εR₀.

2 The Cauchy-Kowalewski Theorem

LetF(t, x, u, v) be a function which is holomorphic in a neighborhood of the point (a, b, c, d) ∈ C⁴, and let ϕ(x) be a function which is holomorphic in a neighborhood ofx=band which satisfiesϕ(b) =cand ∂ϕ

∂x(b) =d. Consider the initial value problem

(E)

⎧⎪

⎨

⎪⎩

∂u

∂t = F

t, x, u,∂u

∂x

, u

t=a = ϕ(x) in a neighborhood ofx=b.

Then we have the following fundamental unique solvability result.

Theorem 2.1 (Cauchy-Kowalewski Theorem). The initial value prob- lem (E) has one and only one solution u(t, x) which is holomorphic in a neighborhood of (a, b)∈Ct×Cx.

Proof. We will present the proof of this well-known theorem to illustrate the convenience of using the functions φ⁽ⁱ⁾(z). (The reader may compare this proof to other known proofs of the Cauchy-Kowalewski Theorem.)

To simplify things, we translate the setting from the point (a, b) into the origin (0,0). Next, we perform a change of variable by setting w(t, x) = u(t, x)−ϕ(x), where w(t, x) is the new unknown function. Then the initial value problem (E) becomes

(2.1)

⎧⎪

⎨

⎪⎩

∂w

∂t = G

t, x, w,∂w

∂x

,

w t=0 ≡ 0 in a neighborhood of x= 0.

Here, the functionG(t, x, w, v) is holomorphic in a neighborhood of the origin inC⁴. It is thus sufficient to consider the reduced initial value problem (2.1).

We can easily see that the above equation has a unique formal solution of the form w(t, x) = _∞

k=0w_k(x)t^k. We will show that this formal solution converges.

Letr₀ >0,R₀ >0 andρ >0 be small enough and suppose that the func- tion G(t, x, w, v) is holomorphic in a neighborhood of the set {(t, x, w, v) ∈ C⁴; |t| ≤ r₀, |x| ≤ R₀, |w| ≤ ρ and |v| ≤ ρ}; suppose further that G is bounded by M in this domain. Since G is holomorphic, we may expand it into

(2.2) G(t, x, w, v) =

p,q,s

a_p,q,s(x)t^pw^qv^s.

By Cauchy’s inequality and the fact that the coefficient a_p,q,s(x) is holo- morphic in a neighborhood of {x∈C; |x| ≤R₀}, we have

(2.3) a_p,q,s(x) M

r^p₀ρ^q+s 1 1−x/R₀.

This means that if we can find any W(t, x) satisfying the majorant relations

(2.4)

⎧⎪

⎨

⎪⎩

∂W

∂t

(p,q,s)∈N³

M r^p₀ρ^q+s

1

1−x/R₀ t^pW^q ∂W

∂x _s

, W

t=0 0,

then this function W(t, x) majorizes the formal solution w(t, x).

Let 0< r < r₀, 0< R < R₀ and define

(2.5) W(t, x) = Lφ⁽¹⁾

t cr + x

R

.

We shall show that for suitable values of L > 0 and 0< c≤1, the function W(t, x) just defined satisfies (2.4). For brevity, set X =t/cr+x/R. Using property (iii) of the majorant function, we see that

(2.6) ∂W

∂t = L cr

dφ⁽¹⁾

dz (X) L

8crφ⁽⁰⁾(X).

On the other hand, the summation at the right-hand side of (2.4) may be majorized as follows: there exists a large constant C₀ such that

(q,s)∈N²

M ρ^q+s

1 1−x/R0

1 1−t/r0

W^q ∂W

∂x _s (2.7)

(q,s)∈N²

M ρ^q+s

1 1− t

cr₀ − x R₀

Lφ⁽⁰⁾(X)_qLφ⁽⁰⁾(X) R

_s

M C₀

1−L/ρ−L/ρR φ⁽⁰⁾(X), if L ρ + L

ρR <1.

In simplifying the above expressions, we have used properties (i), (iii) and (iv) of the majorant function. Now, comparing this with (2.6), if we could force

(2.8) L

8cr ≥ M C₀

1−L/ρ−L/ρR,

then the majorant relations in (2.4) will be satisfied by W(t, x) defined in (2.5). By choosing a sufficiently small L, the condition L/ρ+L/ρR < 1 is satisfied. Having chosen and fixed this L, we now choose csmall enough so that (2.8) will hold true.

Thus, we were able to prove that W(t, x) in (2.5) majorizes the formal solution w(t, x). This now implies that w(t, x) converges in a domain con- taining

(t, x)∈C²; t/cr+x/R <1 .

3 Analytic Continuation

Let Ω be a neighborhood of the origin. Let F(t, x, u, v) be a holomorphic function in Ω×Cu×Cv and consider the nonlinear partial differential equation

(e) ∂u

∂t =F

t, x, u,∂u

∂x

.

If we impose a boundary condition on the solution, say u(t, x) ≡ 0 on the hypersurfaceS ={t= 0}, then the Cauchy-Kowalewski Theorem asserts that there exists only one holomorphic solution to (e) satisfying the said boundary condition.

But suppose we include into consideration the singular solutions of (e).

For example, we may ask: does (e) admit solutions which possess singularities on the hypersurface S? One method of arguing the non-existence of such solutions is by means of analytic continuation.

If equation (e) is linear, then Zerner’s Theorem (1971) states that any solution which is holomorphic in Ω₊ = {(t, x)∈Ω; t >0} can be analyt- ically extended to some neighborhood of the origin (0,0). In other words, there does not exist a solution with singularity only on S.

If equation (e) is nonlinear, we have the following nonlinear analogue of Zerner’s theorem due to Tsuno.

Theorem 3.1 (Tsuno, 1975). If (e) has a solution u(t, x) which is holo- morphic in Ω₊, and if the solution satisfies

(3.1) sup

x∈ω|u(t, x)|=O(1) (as t→0),

in some neighborhood ω of the origin, then the solution can be analytically continued up to a neighborhood of the origin.

The assumption that u(t, x) be bounded in some neighborhood of the origin seemed too strong to other researchers at that time. Some might have believed that Zerner’s result can be extended to the nonlinear case without any additional assumption. However, this is not possible if the equation is nonlinear, as can be seen in this example: the equation

(3.2) ∂u

∂t =u ∂u

∂x _m

has for a solution the function u(t, x) = (−1/m)^1/mxt^−1/m. Clearly, this has a singularity at t = 0.

We actually have a more precise result on this problem which is due to Kobayashi. To present his result, we make a slight modification in the formulation of the problem. SinceF(t, x, u, v) is holomorphic in Ω×C_u×C_v, we may expand it into the convergent series

(3.3) F(t, x, u, v) =

(j,α)∈N²

a_j,α(t, x)u^jv^α.

Let ∆₀ ={(j, α)∈N²; a_j,α(t, x) ≡0}and ∆ ={(j, α)∈∆₀; j +α ≥2}. We remark that F is linear if and only if ∆ = ∅; it is nonlinear otherwise.

Since we already have Zerner’s result for the linear case, we will assume henceforth that F is nonlinear, that is, ∆ is nonempty. In the following, we will write the coefficients as

(3.4) a_j,α(t, x) =t^kj,αb_j,α(t, x),

where k_j,α is a nonnegative integer and b_j,α(0, x) = 0. Using the above, (e) may now be written as

(e) ∂u

∂t =

(j,α)∈∆0

t^kj,αb_j,α(t, x)u^j ∂u

∂x _α

.

Define for σ∈R the quantity (3.5) δ(σ) ^def= inf

(j,α)∈∆

k_j,α+ 1 +σ(j+α−1) .

Note that if σ = 0, then δ(σ)≥1. We now state the following improvement of Tsuno’s theorem.

Theorem 3.2 (Kobayashi, 1998). Suppose u(t, x) is a solution of (e) which is holomorphic in Ω₊. If for some σ ∈R satisfyingδ(σ)>0, we have

(3.6) sup

x∈ω|u(t, x)|=O(|t|^σ) (as t →0),

then the solution u(t, x) can be extended analytically as a holomorphic solu- tion of (e) up to a neighborhood of the origin.

We remark that if we takeσ= 0, i.e., if we assume thatu(t, x) is bounded near the origin, thenδ(σ) is clearly positive and the above theorem is nothing but Tsuno’s Theorem.

The next proposition identifies a critical value for the indexσ. Set

(3.7) σ_K = sup

(j,α)∈∆

−k_j,α−1 j+α−1. Then it is easy to see that

Proposition 3.3. If σ is greater than σ_K, then δ(σ) is positive.

Recall equation (3.2). For that equation, σ_K may be verified to be equal to −1/m. Hence, by the above proposition, analytic continuation of the solution up to the origin is possible if σ > −1/m. Note further that the counterexample u(t, x) = (−1/m)^1/mxt^−1/m has growth order in t equal to

−1/m and may not be analytically continued up to the origin.

Proof of Theorem 3.2. Let us suppose u(t, x) is a solution of (e) which is holomorphic in Ω₊. Suppose further that expansion (3.3) is valid in D = {(t, x, u, v) ∈ C⁴; |t| ≤ 2r0, |x| ≤2R, |u| ≤ ρ and |v| ≤ ρ}. Here, ρ is any positive number while r₀ and R are sufficiently small. Let M_ρ be a bound for F inD.

We now consider the following initial value problem inw(t, x):

(3.8)

⎧⎪

⎨

⎪⎩

∂w

∂t =

(j,α)∈∆0

t^kj,αb_j,α(t, x)w^j ∂w

∂x _α

, w

t=ε = u(ε, x).

Our goal is to show that the formal solution w(t, x) = _∞

k=0w_k(x)(t−ε)^k converges in some domain containing the origin. This then shows thatu(t, x) is analytically continued by w(t, x) up to some neighborhood of the origin.

First, sinceu(t, x) =O(|t|^σ) as t→0, there exists a constantAsuch that

|u(ε, x)| ≤Aε^σ uniformly inx. Hence, for some constantC₁, we have (3.9) u(ε, x) Aε^σC₁φ⁽¹⁾

x R

.

Without loss of generality, we may assume that σ < 1. (For if u(t, x) = O(|t|^σ) as t → 0 with σ ≥ 1, then it is also true that u(t, x) = O(|t|^σ) as t →0 withσ <1.)

To construct an inequality to be satisfied by the majorant function, we will first majorize the expression t^kj,αb_j,α(t, x) using the functionφ⁽⁰⁾(z). For brevity, let us set Z = (t−ε)/cr+x/R. Thent is majorized by

t = ε+ (t−ε) _ε (ε+ 4cr)

1 + t−ε 4cr

(3.10)

ε (ε+ 4cr) 4S φ⁽⁰⁾(Z).

Now, we may expand the function b_j,α(t, x) into

(3.11) b_j,α(t, x) =

∞ i=0

b_j,α⁽ⁱ⁾(x)tⁱ,

where each b_j,α⁽ⁱ⁾(x) is holomorphic in a neighborhood of {|x| ≤2R} and sat- isfies

(3.12) |b⁽ⁱ⁾_j,α(x)| ≤ M_ρ

ρ^j+α(2r₀)^i+kj,α . Using this estimate, we see that

(3.13) b⁽ⁱ⁾_j,α(x) M_ρC₁φ⁽⁰⁾(x/R) ρ^j+α(2r₀)^i+kj,α , where C₁ is the same constant as in (3.9).

Combining the above relations and settingε=cr/2 give t^kj,αb_j,α(t, x) _ε

∞ i=0

(ε+ 4cr) 4S φ⁽⁰⁾(Z)

_i+kj,α C₁M_ρφ⁽⁰⁾(Z) ρ^j+α(2r₀)^i+kj,α

(3.14)

ε C₁M_ρ

ρ^j+α φ⁽⁰⁾(Z) ∞

i=0

9crS r₀

_i+kj,α

By choosing and fixing r small enough so that 9rS < r0/2, we finally have for any 0< c≤1 and ε=cr/2 the relation

(3.15) t^kj,αb_j,α(t, x)_ε 2C₁M_ρ

ρ^j+α c^kj,αφ⁽⁰⁾(Z).

Thus, any function W(t, x) found to satisfy the majorant relations

(3.16)

⎧⎪

⎪⎨

⎪⎪

⎩

∂W

∂t _ε

(j,α)∈∆0

2C₁M_ρ

ρ^j+α c^kj,αφ⁽⁰⁾(Z)W^j ∂W

∂x _α

, W

t=ε ε Aε^σC₁φ⁽¹⁾ x

R

,

is one majorant function for the formal solution w(t, x). To finish the proof, we give the following proposition.

Proposition 3.4. By choosing suitable values for ρ > 0 and c > 0, and setting ε=cr/2, the function

(3.17) W(t, x) =Aε^σC₁φ⁽¹⁾ t−ε

cr + x R

satisfies the majorant relations given in (3.16).

From this proposition, we can see thatW(t, x) is holomorphic in a domain containing the origin; the same must be true for w(t, x).

The proof of the above proposition is quite similar to the given proof of the Cauchy-Kowalewski Theorem. Nevertheless, we will state below the gist of the computation. The left hand side is estimated as follows:

(3.18) ∂W

∂t = Aε^σC₁ 1 cr

dφ⁽¹⁾

dz (Z) _ε AC₁(cr)^σ−1

2^σ+3 φ⁽⁰⁾(Z).

On the other hand, the right hand side is majorized as follows:

R.H.S. ε

(j,α)∈∆0

2C₁M_ρ

ρ^j+α c^kj,αφ⁽⁰⁾(Z)

Aε^σC₁φ⁽⁰⁾(Z)_j (3.19)

×

Aε^σC₁

R φ⁽⁰⁾(Z) _α

ε 2C₁M_ρφ⁽⁰⁾(Z)

(j,α)∈∆0

c^{kj,α+σ(j+α)}

Ar^σC₁ 2^σρ

_jAr^σC₁ 2^σRρ

_α .

Having fixed r, we may take ρ large enough so that

(3.20) Ar^σC₁

2^σρ < 1

2 and Ar^σC₁ 2^σRρ < 1

2.

Now, if (j, α)∈∆ then k_j,α+ 1 +σ(j+α−1)≥δ(σ)>0 by assumption. If in case (j, α) ∈∆, that is, ifj+α≤1, then we havek_j,α+ 1 +σ(j+α−1)≥ 1−σ >0. Thus, the above simplifies into

(3.21) R.H.S. _ε 8C₁M_ρc^σ−1cmin{δ(σ),1−σ}φ⁽⁰⁾(Z).

Therefore, to satisfy the majorant relations in (3.16), we must force

(3.22) Ar^σ−1

2^σ+3 ≥8M_ρcmin{δ(σ),1−σ}.

Since the values of the other constants have been fixed, the above inequality may be satisfied by choosing a very small value for c >0.

4 Construction of Singular Solutions

In the previous section, we have shown that any solution u(t, x) to (e) which is holomorphic in Ω₊and which satisfiesu(t, x) =O(|t|^σ) may be analytically continued up to the origin, for as long as σ > σ_K. But what happens when σ = σ_K? In this section, we will try to construct a singular solution with growth order |t|^σK on the hypersurface {t= 0}. This in effect asserts that the index σ_K obtained by Kobayashi is optimal.

For simplicity, let us assume that ∆₀ is a finite set; thenσ_K is a negative rational number and so σ_K ∈ Z/L for some natural number L > 0. In the following, we will drop the subscript K and write σ for σ_K.

LetM={(j, α)∈∆; σ= (−k_j,α−1)/(j+α−1)} and set

(4.1) P(x, y, z) =

(j,α)∈M

b_j,α(0, x)y^jz^α.

We remark that k_j,α+ 1 +σ(j +α−1) = 0 whenever (j, α) is in M; it is strictly positive whenever (j, α) is in ∆₀\M.

We will try to construct a solution of (e) of the form

(4.2) u(t, x) =t^σ

ϕ(x) +w(t, x) ,

where ϕ(x) ≡ 0 and w(0, x)≡ 0. Since σ <0, this clearly has a singularity of order |t|^σ at t = 0. Substituting this to (e), we get an equation of the form

σϕ+t∂w

∂t +σw = P

x, ϕ+w, ∂ϕ

∂x +∂w

∂x (4.3)

+t^1/LR

t^1/L, x, ϕ+w, ∂ϕ

∂x +∂w

∂x

.

The above equation now has two unknownsϕandwwhich must satisfyϕ ≡0 and w(0, x)≡0. To find one solution to this equation, it is sufficient to find functions ϕ and wsatisfying the following two equations:

σϕ = P

x, φ,∂ϕ

∂x

and (e_I)

t∂

∂t+σ

w = A₁(x)w+A₂(x)∂w

∂x +G₂

x, ϕ, ϕ, w,∂w

∂x (e_II)

+ t^1/LR

t^1/L, x, ϕ+w, ∂ϕ

∂x +∂w

∂x

.

Here, A₁(x) = ∂P

∂y(x, ϕ, ϕ), A₂(x) = ∂P

∂z(x, ϕ, ϕ) and G₂ is the remainder term of the Taylor expansion of P. To summarize our goal, we have

Proposition 4.1. If equation(e_I)has a solutionϕ(x)which is not identically zero and equation (e_II) has a solution w(t, x) satisfying w(0, x)≡0, then we have succeeded in constructing a solution u(t, x) to (e) with singularity of order |t|^σ at {t= 0}.

Not much is known regarding the existence of solutions of the ordinary differential equation (e_I). If the equation is solvable in∂ϕ/∂x, then we could apply Cauchy’s Theorem and conclude that a holomorphic solution exists.

We now turn our attention to equation (e_II). By performing the variable change s^L=t, the said equation may be written in the form

(E_∗) t∂w

∂t = G

t, x, w,∂w

∂x

,

where G(t, x, w, v) is a function holomorphic in a neighborhood of the origin (0,0,0,0) in C⁴. We can check that G(0, x,0,0)≡0 holds near x= 0.

Thus, instead of analyzing the particular equation (e_II), it makes sense to study the more general equation (E_∗). This leads us to the next section.

5 Singular Partial Differential Equations

Once again, we state our assumptions in studying equation (E_∗). LetGbe a function holomorphic in some neighborhood of the origin in C⁴ and suppose G(0, x,0,0) is identically zero nearx= 0. We seek for a holomorphic solution w(t, x) of (E_∗) which satisfies w(0, x)≡0.

IfGis void ofxand∂w/∂x, i.e., (E_∗) is an ordinary differential equation, then we have the following result by Briot and Bouquet.

Theorem 5.1 (Briot-Bouquet, 1856). Let g(t, w) be holomorphic in a neighborhood of the origin (0,0) ∈ C² and suppose g(0,0) = 0. If ∂g/∂w does not take values in N^∗ at the origin, then the equation

(5.1) tdu

dt =g(t, u)

has a unique holomorphic solution u(t) satisfying u(0) = 0.

A generalization of this result into partial differential equations was proved by G´erard and Tahara. To state their result we first write G as

(5.2) G

t, x, w,∂w

∂x

= a(x)t+b(x)w+c(x)∂w

∂x +R₂

t, x, w,∂w

∂x

, where R₂ is the remainder of the Taylor expansion of G. Then we have Theorem 5.2 (G´erard-Tahara, 1990). Assume that the coefficient c(x) in (5.2) is identically zero and that b(0) ∈ N^∗. Then equation (E_∗) has a unique holomorphic solution satisfying w(0, x)≡0.

Proof of Theorem 5.2. Just like before, we assume a formal solution of the form w(t, x) =_∞

k=1w_k(x)t^k. The conditionb(0) ∈N^∗ assures the existence of a unique formal solution of this form.

We then expand G(t, x, w, ∂w/∂x) into (5.3) G

t, x, w,∂w

∂x

=a(x)t+b(x)w+

p+q+α≥2

a_p,q,α(x)t^pw^q ∂w

∂x _α

.

Let us suppose that this expansion is convergent in a neighborhood of the set {(t, x, w, ∂w/∂x); |t| ≤ r₀, |x| ≤ R₀, |w| ≤ ρ and |∂w/∂x| ≤ ρ}, and that G is bounded by M there.

Sinceb(0) ∈N^∗, we can find a constant A such that

(5.4) k

k−b(x)

≤A for all k ∈N^∗ and |x| ≤R₀.

Then, any function W(t, x) satisfying the following relations is a majorant of the formal solution:

(5.5)

⎧⎪

⎨

⎪⎩ t∂W

∂t AM

r₀

t

1−x/R₀ +

p+q+α≥2

AM r₀^pρ^j+α

t^pW^q 1−x/R₀

∂W

∂x _α

, W

t=0 ≡ 0.

Let 0< r < r₀ and 0< R < R₀. We shall show that one suchW(t, x) is the function

(5.6) W(t, x) =L t φ⁽¹⁾ t

cr + x R

,

where the constants Land cwill be suitably chosen later. For simplicity, set X =t/cr+x/R.

Substituting this to the left hand side of (5.5) gives t∂W

∂t = Lt φ⁽¹⁾(X) + Lt² cr

dφ⁽¹⁾ dz (X) (5.7)

Lt φ⁽¹⁾(X) + Lt²

8crφ⁽⁰⁾(X).

Meanwhile, the right hand side becomes

R.H.S. = AM

1−x/R0

t r0

+

p+q+α≥2

AM 1−x/R0

t r0

_pLt

ρ φ⁽¹⁾(X) _q (5.8) ×

×

Lt Rρ

dφ⁽¹⁾ dz (X)

_α

∞

p=1

AM 1−x/R₀

t r₀

_p

+

p+q+α≥2 q+α≥1

AM φ⁽⁰⁾(X) 1−x/R₀

t r₀

_p

×

×

Lt ρ

_qLt Rρ

_α . The two summations above are majorized separately as follows:

∞ p=1

AM 1−x/R₀

t r₀

_p

AM t

r₀

4S φ⁽¹⁾(X) 1−t/r₀−x/R₀ (5.9)

4S AM C₁

r₀ tφ⁽¹⁾(X).

and

p+q+α≥2 q+α≥1

AM φ⁽⁰⁾(X) 1−x/R₀

t r₀

_pLt ρ

_qLt Rρ

_α (5.10)

t r₀ +Lt

ρ + Lt Rρ

₂ AM φ⁽⁰⁾(X)

1−t/r₀ −Lt/ρ−Lt/Rρ−x/R₀ t

r₀ +Lt ρ + Lt

Rρ ₂

AM C₁φ⁽⁰⁾(X),

where the last simplification is possible if we assume that

(5.11) 1

r₀ + L ρ + L

Rρ ≤ 1 cr₀.

If we compare the majorant relations above to the one in equation (5.7), then it is easy to see that W(t, x) satisfies (5.5) if we could force

(5.12) L ≥ AM C₁4S

r₀ and L

8cr ≥ AM C₁ 1

r₀ + L ρ + L

Rρ ₂

. These and the added condition in (5.11) are satisfied by choosing L large enough, fixing it, and then choosing a sufficiently small value for c.

We thus have shown that the functionW(t, x) defined in (5.6) majorizes the formal solution w(t, x). This implies that the formal solution converges in some neighborhood of the origin.

Let us go back to equation (E_∗). If c(x) is not identically zero, then we may divide the situation into several possibilities. We write

(5.13) c(x) =x^pc(x),˜

where p is a nonnegative integer and ˜c(0) = 0.

In the possibility that p = 0, we have c(0) = 0, and so by the implicit function theorem, equation (E_∗) is solvable in ∂w/∂x. Thus, the original problem is equivalent to the following initial value problem:

(E_∗)

⎧⎪

⎨

⎪⎩

∂w

∂x = H

t, x, w, t∂w

∂t

, w

x=0 = ψ(t).

Here ψ(t) is some function satisfying ψ(0) = 0. (This condition on ψ is a compatibility condition since we are originally looking for a function w(t, x) which satisfies w(0, x)≡0.) By applying the Cauchy-Kowalewski Theorem, we have

Theorem 5.3. Supposep= 0 in(5.13). Then for any arbitrary holomorphic functionψ(t)withψ(0) = 0, equation(E_∗)has a unique holomorphic solution w(t, x) satisfying w(0, x)≡0 and w(t,0) =ψ(t).

Having stated this result, we are now left to consider cases when p ≥1.

There is yet no result covering all these. Chen and Tahara have considered the case when p = 1, while research is still being done on the cases when p > 1.

We now state the following result due to Chen and Tahara.

Theorem 5.4 (Chen-Tahara, 1999). Supposep= 1in(5.13). If a positive constant µ exists such that

(5.14) |k−b(0)−˜c(0)l| ≥ µ(k+l+ 1) for all (k, l)∈N^∗×N, then (E_∗) has one and only one holomorphic solution satisfying w(0, x)≡0.

Proof of Theorem 5.4. We first write the coefficients b(x) and ˜c(x) as (5.15) b(x) = b(0) +xβ(x) and c(x) = ˜˜ c(0) +xγ(x).

Then equation (E_∗) may now be written as t ∂w

∂t −b(0)w−c(0)˜ x∂w

∂x = xβ(x)w+xγ(x)

x∂w

∂x

+a(x)t (5.16)

+

p+q+α≥2

a_p,q,α(x)t^pw^q ∂w

∂x _α

. Just like before, let us assume that this expansion of G(t, x, w, ∂w/∂x) is valid in a neighborhood of {(t, x, w, ∂w/∂x); |t| ≤ r₀, |x| ≤ R₀, |w| ≤ ρ and |∂w/∂x| ≤ ρ}, and that G is bounded by M there. Furthermore, assume that a(x), β(x) and γ(x) are bounded on this domain by A, B and C, respectively.

The existence of a formal solutionw(t, x) = _∞

k=1w_k(x)t^k is guaranteed by the Poincar´e condition imposed in (5.14). Now, it can be shown that this formal solution is majorized by any function W(t, x) satisfying these relations:

(5.17)

⎧⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎩ µ

t ∂

∂t+x ∂

∂x + 1

W B xW +A t

1−x/R₀ + C x 1−x/R₀

x∂W

∂x

+

p+q+α≥2

M r₀^pρ^j+α

t^pW^q 1−x/R₀

∂W

∂x _α

, W

t=0 ≡ 0.

One suchW(t, x) may be found in the form (5.18) W(t, x) = L t φ⁽¹⁾

t cr + x

R

,

where 0 < r < r₀ and the positive constants L, c and R will have to satisfy some conditions which will later be specified. For brevity, we again set X = t/cr+x/R.

The computation involving t∂W/∂t, A t/(1−x/R₀) and the summation in equation (5.17) are basically the same as in the proof of Theorem 5.2, and hence may be omitted. In estimating the remaining terms, we will make use of the fact that

(5.19) z φ⁽⁰⁾(z) = 1 4S

∞ m=1

z^m (m+ 1)²

m+ 1 m

₂

4φ⁽⁰⁾(z).

We thus have, for some constant C₂ >0, B xW

1−x/R₀ + C x 1−x/R₀

x∂W

∂x (5.20)

BLC₂xt φ⁽⁰⁾(X) + CLC₂xt x

Rφ⁽⁰⁾(X) (BLC₂+ 4CLC₂)xt φ⁽⁰⁾(X).

On the other hand, the left hand side is estimated as follows:

(5.21) µx∂W

∂x = µLtx 1 R

dφ⁽¹⁾

dz (X) µLtx 1

8Rφ⁽⁰⁾(X).

Therefore, in order forW(t, x) to satisfy the majorant relations in (5.17), we must impose, aside from the conditions to be satisfied by L and c, the condition that

(5.22) BC₂+ 4CC₂ ≤ µ

8R.

Obviously, the above is satisfied if we take R < R₀ sufficiently small. This completes the proof.

6 Welcome to the World of “Singular PDEs”

In the previous section, we studied the holomorphic solutions to the singular partial differential equation (E_∗), a type of equation which naturally arose in the construction of singular solutions to (e). Recall that we had assumed in (4.2) a solution of the form

u(t, x) =t^σ

ϕ(x) +w(t, x) .

We originally requiredw(t, x) to be analytic near the origin, but this require- ment is not strictly necessary; it is actually enough to assume that w(t, x) satisfies the following two conditions:

i) w(t, x) is holomorphic in the domain{(t, x)∈ R(C\{0})×C; 0<|t|<

η(argt),|x|< R}, whereη(s) is a positive-valued, continuous function on Rand R >0, and

ii) for anyθ >0, we have sup

|x|<R|w(t, x)|=o(1) as t→0 in |argt|< θ.

Here R(C\{0}) denotes the universal covering space of C\{0}. Thus, we have come to

Problem. Determine all solutions w(t, x) to equation (E_∗) satisfying conditions i) and ii).

This problem is now the main subject of research in the theory of singular partial differential equations. The author hopes that young mathematicians would join him in doing research in this problem.

References

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[7] P. D. Lax, Nonlinear hyperbolic equations, Comm. Pure Appl. Math.

6 (1953), 231-258.

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