2 The Lifespan Problem

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A Constructive Approach to Obtaining the Lifespan of Solutions to Cauchy Problems with Small Data

John Paolo O. Soto University of the Philippines Diliman

Quezon City, Philippines [email protected]

Jose Ernie C. Lope University of the Philippines Diliman

Mark Philip F. Ona University of the Philippines Diliman

Abstract

This paper revisits the proof of the lifespan of solutions to second order nonlinear Cauchy problems with small analytic data. The Cauchy problem is solved using the method of successive approximations, thereby constructively obtaining the approximate solutions as well as estimates of their lifespan.

Keywords: lifespan of solutions, Cauchy problem, successive approximations 2020 MSC:35B30, 35G25, 35A10

1 Introduction

The lifespan problem for Cauchy problems has been studied extensively by many authors.

In some works, semilinear wave equations are studied in detail and functional analytic methods are used to obtain estimates of the lifespan of their solutions [1, 2, 3]. D’Ancona and Spagnolo stated that the lifespan of solutions of ordinary differential equations becomes larger as the data become smaller, which also holds true in the case of partial differential equations under some hyperbolicity assumption [4].

Gourdin and Mechab [5, 6] also presented several results regarding the lifespan of solutions of certain Cauchy problems. In [5], they considered the generalized Kirchoff equations in the real-analytic category and studied the lifespan of their solutions under certain smallness assumptions. They proved that under some conditions on the equation, or on the Cauchy data, there is an existence-uniqueness result for which the lifespan can also be estimated. In [6], they considered equations of general order without the hyperbolicity condition. They obtained solutions that are holomorphic intand of Gevrey class with respect tox. Furthermore, in the case where the right-hand side is independent oft, they were able

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to compute the lifespan of the solution. Lastly, in [6], they considered the case when the Cauchy data are in some Gevrey class and they were also able to show that the lifespan is of some order with respect to 1/ε, provided that the right-hand side of the equation is independent oft.

Yamane [7] considered the second-order nonlinear equation ((∂_t²u−P(∂))u=F ∇u,∇²u

u(0, x) =φ(x), ∂tu(0, x) =ψ(x), (CP) where (t, x)∈R×Rⁿ, ∇u = (∂x_iu)_1≤i≤n, ∇²u = (∂x_i∂x_ju)_1≤i,j≤n, and P(∂) is some second-order linear partial differential operator. He formulated an existence-uniqueness result for (CP), together with an estimate for the lifespan of the solution. His result stated that if the Cauchy data satisfy a smallness condition in the form of a Cauchy type estimate, then the lifespan of the solution is of order1with respect to1/ε. This result was improved in subsequent papers [8, 9], as the right hand-side of the equation in these works now also depend on the time derivative of the unknown functionu(t, x).

In the abovementioned papers, Gourdin and Mechab, and Yamane used the fixed point theorem in proving their results, that is, they constructed a Banach algebra derived from some majorant series and defined a contraction map on the Banach space. A more recent work by Tolentino, Bacani and Tahara [10] that considered a more general Cauchy problem also employed a fixed point approach. The equations considered in these works are non- singular, but uniformly analytic solutions have also been constructed for singular equations [11].

The main goal of this paper is to provide an alternative approach in establishing the unique solvability of second order nonlinear Cauchy problems and constructively obtain estimates of the lifespan of their solutions. In particular, we will show that as the data become smaller in some sense, the lifespan of the solution becomes longer, and so the solution becomes global int. Furthermore, we will show that the lifespan is indeed of order 1 with respect to 1/ε. The Banach Fixed Point Theorem was used in [12] and [7], which required them to choose a predetermined value forT dependent on1/εand introduce spaces of functions defined on the interval(−T, T). While we will adopt the same spaces used in [7], we can more concretely acquire the order of the lifespan with respect to1/ε since our method of proof is constructive. We see merit in this approach because we are able to obtain concretely the sequence of approximate solutions as well as the estimates of their norms, which in turn leads to the estimate of their lifespan.

2 The Lifespan Problem

Let(t, x)∈R×R,Ωan open subset ofR, and N={0,1,2, . . . ,}.

Definition 2.1. AC^∞ functionφ(x)is said to be uniformly analytic onΩ if there exists a positive constantCsuch that for allα∈N,

sup

x∈Ω

|∂_x^αφ(x)| ≤C^α+1α!.

We denote byA(Ω)the collection of all uniformly analytic functions onΩ.

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Definition 2.2. Letk∈N. A continuous function functionu(t, x)onΩT = (−T, T)×Ωis said to belong to the classC^k(T;A(Ω))if

a. for allj ∈ {0,1, . . . , k} andα∈N, ∂^j_t∂^α_xu∈C(Ω_T),

b. for allT^′ ∈(0, T), there existsC=C_T^′ >0such that for allj∈ {0,1, . . . , k} and for allα∈N,

sup

|t|≤T^′, x∈Ω

|∂^j_t∂^α_xu(t, x)| ≤C^α+1α!.

Letγ∈C. We consider the second order nonlinear Cauchy problem ((∂_t²−γ∂_x²)u=F(∂_xu, ∂_x²u)

u(0, x) =φ(x), ∂tu(0, x) =ψ(x), (M) under the following assumptions:

(A1) The functionF(X) is analytic in a neighborhood around the origin, and vanishes of second order atX = 0;

(A2) The Cauchy dataφ(x)andψ(x)are uniformly analytic in an open subsetΩofR. Yamane was able to prove the following result.

Theorem 2.3([7]). Suppose that (A1) and (A2) hold. Then there existsµ >0such that the following holds for all0< ε <1: if the Cauchy data satisfysup_x∈Ω|∂_x^αφ(x)| ≤ε^α+1α! and sup_x∈Ω|∂_x^αψ(x)| ≤ ε^α+1α! for all α∈ N, then (M) has a unique solution in C²(T;A(Ω)), withT =µ/ε.

Remark 2.4.We note that the Cauchy-Kowalevsky Theorem (see, e.g., [13]) guarantees the local existence and uniqueness of the analytic solution (intandx) to (M). The current setup only considers the one-dimensional ‘space’ variable version of Yamane’s problem because our main focus is to illustrate the constructive approach in obtaining the lifespan of solutions.

Yamane remarked in [9] that the best possible order of the lifespan of the solution is of order 1 with respect to 1/ε. This claim is illustrated in the following example: Let Ω = (¹₄,¹₂), and consider the Cauchy problem given by







∂_t²u=−(∂xu)²

u(0, x) = log(2−εx), ∂_tu(0, x) = −ε 2−εx.

The above Cauchy problem has a solution given by u(t, x) = log|2−εt−εx|, which is undefined att= ¹_ε(1−^ε₂x). This gives the lifespan of the solution to beTε= ¹_ε(1−^ε₄), since u(t, x)will be only be defined on ΩT = (−T, T)×(¹₄,¹₂)when T ≤Tε. Note thatTε→ ∞ asε→0, andTε is of order 1 with respect to1/ε.

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3 Preliminaries

Given two formal power series f(X) =P

akX^k, ak ∈C, andg(X) =P

bkX^k,bk ∈R, we say thatf ≪g if and only if|ak| ≤bk for all k. Also, set the operatorsD⁻¹ andD to be the anti-differentiation and differentiation operators, respectively.

In the following, we recall some of the results of Yamane [7]. We will also provide an improvement to one of his propositions.

Define the series

ϕ(X) = 1 K

X

k∈N

X^k

(k+ 1)², whereK= 4π²/3. (1) This series is due to Lax [14], except for the constantK, which was later added to simplify calculations. The functionϕ is convergent on the unit disk and satisfies ϕ² ≪ϕ for this choice ofK.

Definition 3.1([7]). Letζ >0andT >0. A functionu(t, x)is said to belong to the space GT ,ζ(Ω) if it is continuous onΩ_T, infinitely differentiable in x, and there exists a constant C >0 such that for allα∈Nandt∈(−T, T),

sup

x∈Ω

|∂_x^αu(t, x)| ≤Cζ^αD^αϕ(|t|/T),

whereϕis the one defined in (1).

For f ∈ GT ,ζ(Ω) and g ∈ H(Ω), we say that f(t, x) ≪ g(t, x) if for all α ∈ N and t∈(−T, T)

sup

x∈Ω

|∂_x^αu(t, x)| ≤∂_x^αg(t,0).

SetϕT ,ζ(t, x) =ϕ(|t|/T+ζx).Clearly, for anyα∈N,∂_x^αϕT ,ζ(t,0) =ζ^αD^αϕ(|t|/T). There- fore,u(t, x)∈ GT ,ζ(Ω)if there existsC >0such thatu(t, x)≪CϕT ,ζ(t, x). Define the norm

∥u∥as the infimum of suchC’s. Then from [7], the spaceGT ,ζ(Ω)becomes a Banach algebra with respect to this norm.

Moreover, by equipping the direct sum LG_{T ,ζ}(Ω) with norm given by ∥⃗τ(t, x)∥_N = maxj∥τj(t, x)∥, where⃗τ(t, x) = (τ1(t, x), . . . , τN(t, x)), the following estimates can be shown.

Proposition 3.2([7]). LetF(X) =F(X1, . . . , XN) =P

|α|≥2aαX^αbe a convergent power series which vanishes of second order atX = 0. If⃗τ(t, x),⃗σ(t, x)∈LG_{T ,ζ}(Ω) have sufficiently small norms, thenF(⃗τ(t, x))andF(⃗σ(t, x))are well-defined as elements ofG_{T ,ζ}(Ω).

Moreover, there exists a constantA=A(F)>0, depending only on F and independent of

⃗τ , ⃗σ, T, ζ andΩ, such that

∥F(⃗τ(x))∥ ≤A∥⃗τ∥²_N,

∥F(⃗τ(x))−F(⃗σ(x))∥ ≤A∥⃗τ−⃗σ∥_N(∥⃗τ∥_N +∥⃗σ∥_N).

The next proposition will help us deal with derivatives of functions inGT ,ζ(Ω).

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Proposition 3.3([7]). Let k, α∈Nwith−k+α≤0. The operator∂_t^−k∂^α_x is an endomor- phism of the Banach spaceGT ,ζ(Ω). Moreover, there existsB >0 such that

∂^−k_t ∂_x^αu

≤BT^kζ^α∥u∥

for anyk, α satisfyingα≤k≤2.

Yamane showed in [7] that for allT >0 andζ >0,G_{T ,ζ}(Ω)⊆C(T;A(Ω)). He further noted that for certain values ofζ, the converse inclusion also holds. Since φ(x) ∈ A(Ω), there exist positive constantsp(φ)andq(φ), not necessarily unique, such that for allα∈N,

sup

x∈Ω

|∂^α_xφ(x)| ≤p(φ)q(φ)^αα!. (2) The almost converse inclusion is as follows:

Proposition 3.4 ([7]). If ψ(x) ∈ A(Ω), then for all T > 0 and ζ ≥ e²q(ψ), we have ψ(x) ∈ GT ,ζ(Ω) and ∥ψ∥ ≤ Kp(ψ), where K is the one in (1). Here, e is the Euler’s number.

We now state a slightly improved version of a proposition in [7], in the case when the estimate in (2) takes a particular form.

Proposition 3.5. Assumeφ(x)∈ A(Ω)satisfies (2)with p(φ) =q(φ) =ε, and letm be a positive integer. Then forj= 1,2, . . . , m:

p(∂_x^jφ) =m!ε^j+1 and q(∂_x^jφ) = (m+ 1)!ε.

Proof. Fixj ∈ {1,2, . . . , m}. Sinceφ(x)∈ A(Ω), so are its higher-order derivatives. More- over, by (2),

sup

x∈Ω

|∂_x^α(∂_x^jφ(x))| ≤ε^α+j+1(α+j)!

=ε^j+1 (m+ 1)!ε^α(α+j)(α+j−1)· · ·(α+ 1) (m+ 1)!^α α!

≤m!ε^j+1 (m+ 1)!εα

α!

from which we obtain the desired result. Here, we used the fact that iff_j(α) := (α+j)(α+ j−1)· · ·(α+1)[(m+1)!]^−α, thenf_j(α)≤f_m(α), for anyj= 1, . . . , mandα∈N. Moreover, the sequence{fm(α)}α∈N is decreasing on N, and attains the maximum value of m!when α= 0.

Remark 3.6. Note that the coefficient ofε^j+1 in our estimate ofp(∂_x^jφ)is always m! for anyj ≤m, as opposed to the one in [7] where the coefficient depends on the order of the derivative. Although this is a slight improvement, it will result to simpler computations if we are to consider equations of general order.

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4 Proof of Theorem 2.3

We now prove Theorem 2.3 through the method of successive approximations. For the preliminary step, we first reduce the problem into an initial-value problem as in [7] and afterwards define our approximate solutions. Throughout the proof, we let the constantsA andB, defined in Propositions 3.2 and 3.3 be sufficiently large. For brevity of notations, setD^1,2u= (∂_xu, ∂_x²u).

Letv(t, x) =u(t, x)−φ(x)−tψ(x). This implies that v(0, x) = 0 = ∂_tv(0, x). Hence, (M) becomes

∂²_tv=γ∂_x²(v+φ+tψ) +F D^1,2(v+φ+tψ) .

Furthermore, setw(t, x) =∂²_tv(t, x)and sov(t, x) =∂_t⁻²w(t, x). The original equation (M) is then reduced to an equation of the formw=L(w), where the operator Lis defined as

L(w) :=γ∂_x²(∂_t⁻²w+φ+tψ) +F D^1,2 ∂_t⁻²w+φ+tψ . Define the approximate solutions{wk}_k≥0 such that

w₀(t, x) =L(0) and w_n(t, x) =L(w_n−1), forn≥1.

Furthermore, define the sequence{dk}k≥0 as follows:

d0(t, x) =w0 and dn(t, x) =wn−w_n−1, forn≥1.

It then follows that forn≥1,

d_n=γ∂²_x(∂_t⁻²d_n−1) +H_n, whereHn=F D^1,2 ∂_t⁻²w_n−1+φ+tψ

−F D^1,2 ∂_t⁻²w_n−2+φ+tψ

andw₋₁≡0.

Let 0 < ε < 1. Set ζ = 6e²ε and T = µ/ε^σ, for some σ >0. We will show that the approximate solutions converge to the true solution, sayw(t, x), by showing that the partial sums of {dn} converge in GT ,ζ(Ω). In the following, we will show that for each n ∈ N,

∥dn∥ ≤Crⁿ, for some 0< r <1 andC >0, independent ofε.

Lett∈(−T, T). We start with the casek= 0. Under the choiceζ = 6e²ε, we can use Propositions 3.4 and 3.5 to obtain forα= 1,2,

∥∂_x^α(φ+tψ)∥ ≤2K(1 +T)ε³. Moreover, from Proposition 3.2,

F(D^1,2(φ+tψ)) ≤A

D^1,2(φ+tψ)

2 2

≤A 2K(1 +T) max

ε², ε³ ²

≤4AK²(1 +T)²ε⁴. Thus, by the triangle inequality, we have

∥d0∥=∥w0∥=

γ∂_x²(φ+tψ) +F(D^1,2(φ+tψ))

≤2|γ|K(1 +T)ε³+ 4AK²(1 +T)²ε⁴

≤CK²ε³(1 +T) (1 +ε(1 +T)),

whereC= max{2|γ|,4A}. Finally, sinceT =µ/ε^σ, if we assume thatµ <1, we have

∥d0∥ ≤CK²(ε³+µε^3−σ+ε⁴+ 2µε^4−σ+µ²ε^4−2σ)

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≤6CK²ε^4−2σ.

Since we want the bound to remain finite for anyε∈(0,1), we must haveσ≤2. For the casen≥1, we have the following proposition.

Proposition 4.1. Letµ be sufficiently small such that

72e²BC²K²µ² 3e²+ 18e²BCK²µ²+ 4K

<1

2, (3)

whereA andB are the constants defined in Propositions 3.2 and 3.3, C= max{2|γ|,4A}, andK is the one in (1). Then the following estimate holds for all n≥1:

∥dn∥ ≤1 2

ⁿ

ε^(6·2ⁿ^{−2)−(6·2}ⁿ^−4)σ.

Proof. Recall that d1satisfies the estimate

d1=γ∂_x²(∂_t⁻²d0) +H1.

Since|γ| ≤C, by Proposition 3.3, it is seen that γ∂_x²∂_t⁻²d0

≤ |γ|BT²ζ²∥d0∥= 36|γ|e⁴Bµ²ε^2−2σ∥d0∥

≤216e⁴BC²K²µ²ε^6−4σ.

From our previous computation, we have

D^1,2(φ+tψ)

₂ ≤ 2K(ε²+ε^2−σ) ≤ 4Kε^2−σ. Also, since terms with T²ζ² are smaller than those with T²ζ, we have

D^1,2∂_t⁻²d0

₂ ≤ ∂x∂_t⁻²d0

≤36e²BCK²µ²ε^5−4σ.Lastly, sinceA≤C, by Proposition 3.2 we have

∥H₁∥=

F(D^1,2(∂_t⁻²d₀+φ+tψ))−F(D^1,2(φ+tψ))

≤A

D^1,2∂_t⁻²d0

₂

D^1,2(∂_t⁻²d0+φ+tψ) ₂+

D^1,2(φ+tψ) ₂

≤36e²BC²K²µ²ε^5−4σ· 36e²BCK²µ²ε^5−4σ+ 8Kε^2−σ

= 1296e⁴B²C³K⁴µ⁴ε^10−8σ+ 288e²BC²K³µ²ε^7−5σ.

Clearly, for the bound of∥d1∥ to be finite for anyε∈ (0,1), we must have that σ≤5/4.

This implies thatmax{ε^6−4σ, ε^7−5σ} ≤ε^10−8σ and thus we have

∥d₁∥ ≤

γ∂_x²∂⁻²_t d₀

+∥H₁∥

≤216e⁴BC²K²µ²ε^10−8σ+ 1296e⁴B²C³K⁴µ⁴ε^10−8σ+ 288e²BC²K³µ²ε^10−8σ

≤72e²BC²K²µ²ε^10−8σ 3e²+ 18e²BCK²µ²+ 4K

≤ 1

2

ε^10−8σ by our choice ofµin (3).

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Suppose now that the claim holds forn=k. We will show that the claim also holds for k=n+ 1. The functiondn+1satisfies the equation

d_n+1=γ∂_x²(∂_t⁻²d_n) +F D^1,2(∂_t⁻²w_n+φ+tψ)

−F D^1,2(∂_t⁻²w_n−1+φ+tψ) .

By the inductive hypothesis, γ∂²_x(∂_t⁻²d_n)

≤ |γ|BT²ζ²· 1

2 ⁿ

ε^(6·2ⁿ^{−2)−(6·2}ⁿ^−4)σ

= 1

2 ⁿ

·36e⁴BCµ²ε^(6·2ⁿ^)−(6·2ⁿ^−2)σ.

Since the functionf(x) = 1 + 2(6x−4)⁻¹is decreasing onZ⁺, we see that for the bound of∥dn∥ to be finite for anyε∈(0,1), σshould satisfy σ≤(6·2ⁿ−2)/(6·2ⁿ−4). Using this, we can obtain the following estimate for∥w_i∥ for anyi≤n:

∥wi∥ ≤

n

X

k=0

∥dk∥ ≤

6CK²+

n

X

k=1

1 2

k

·ε^(6·2ⁿ^{−2)−(6·2}ⁿ^−4)σ

≤12CK²ε^(6·2ⁿ^{−2)−(6·2}ⁿ^−4)σ.

Hence, by Proposition 3.3, we have for anyi≤n, D^1,2∂_t⁻²wi

₂≤BT²ζ·12CK²ε^(6·2ⁿ^{−2)−(6·2}ⁿ^−4)σ

≤72e²BCK²µ²ε^(6·2ⁿ^{−1)−(6·2}ⁿ^−2)σ.

Similarly, using Proposition 3.3, we have D^1,2∂_t⁻²dn

₂≤

1 2

ⁿ

·6e²Bµ²ε^(6·2ⁿ^{−1)−(6·2}ⁿ^−2)σ.

Therefore, by Proposition 3.2 and using again the fact that

D^1,2(φ+tψ)

₂≤4Kε^2−σ, we get

∥Hn+1∥=

F(D^1,2(∂_t⁻²wn+φ+tψ)−F(D^1,2(D_t⁻²(wn−1+φ+tψ))

≤A

D^1,2∂_t⁻²d_n ₂

D^1,2∂_t⁻²w_n ₂+

D^1,2∂_t⁻²w_n−1 ₂+ 2

D^1,2(φ+tψ) ₂

≤ 1

2 ⁿ

·48e²BCKµ²

18e²BCKµ²ε^(6·2ⁿ⁺¹^{−2)−(6·2}ⁿ⁺¹^−4)σ+ε^(6·2ⁿ^+1)−(6·2ⁿ^−1)σ .

We again see that for the bound of∥dn+1∥ to be finite for anyε∈(0,1),σ should satisfy σ≤(6·2ⁿ⁺¹−2)/(6·2ⁿ⁺¹−4). Finally, by the choice ofµin (3), and the fact thatC, K≥1, we have

∥dn+1∥ ≤

γ∂_x²(∂_t⁻²dn)

+∥Hn+1∥

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≤ 1

2 n

·12e²BCµ²ε^(6·2ⁿ⁺¹^{−2)−(6·2}ⁿ⁺¹^−4)σ 3e²+ 72e²BCK²µ²+ 4K

≤ 1

2 ⁿ⁺¹

·ε^(6·2ⁿ⁺¹^{−2)−(6·2}ⁿ⁺¹^−4)σ.

This proves the existence of a solution in (M); as earlier remarked, uniqueness follows from the Cauchy-Kowalevsky Theorem [13]. Since the sequence1+2(6·2ⁿ−2)⁻¹is decreasing and tends to1 asngoes to infinity, we conclude thatσ= 1. This implies furthermore that T=µ/ε, we haveT → ∞as ε→0.

Remark 4.2. In this section, we only considered the real-analytic case. It was stated in [7]

that the holomorphic case follows similarly by defining auniformly holomorphic function and using the same machinery discussed in the real-analytic case. Furthermore, the proof for the complex-analytic case is done as in the real case.

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