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arXiv:0805.0244v2 [math.AP] 22 Jan 2009

Derivative loss for Kirchhoff equations with

non-Lipschitz nonlinear term

Marina Ghisi

Universit` a degli Studi di Pisa

Dipartimento di Matematica “Leonida Tonelli”

PISA (Italy)

e-mail: ghisi@dm.unipi.it

Massimo Gobbino

Universit` a degli Studi di Pisa

Dipartimento di Matematica Applicata “Ulisse Dini”

PISA (Italy)

e-mail: m.gobbino@dma.unipi.it

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Abstract

In this paper we consider the Cauchy boundary value problem for the integro-differential

equation

??

with a continuous nonlinearity m : [0,+∞) → [0,+∞).

It is well known that a local solution exists provided that the initial data are regular

enough. The required regularity depends on the continuity modulus of m.

In this paper we present some counterexamples in order to show that the regularity

required in the existence results is sharp, at least if we want solutions with the same

space regularity of initial data. In these examples we construct indeed local solutions

which are regular at t = 0, but exhibit an instantaneous (often infinite) derivative loss

in the space variables.

utt− m

Ω

|∇u|2dx

?

∆u = 0in Ω × [0,T)

Mathematics Subject Classification 2000 (MSC2000): 35L70, 35L80, 35L90.

Key words: integro-differential hyperbolic equation, continuity modulus, Kirchhoff

equations, Gevrey spaces, derivative loss.

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1Introduction

In this paper we consider the hyperbolic partial differential equation

utt(t,x) − m

??

Ω

|∇u(t,x)|2dx

?

∆u(t,x) = 0∀(x,t) ∈ Ω × [0,T),(1.1)

where Ω ⊆ Rnis an open set, ∇u and ∆u denote the gradient and the Laplacian of

u with respect to the space variables, and m : [0,+∞) → [0,+∞). Equation (1.1) is

usually considered with initial conditions

u(x,0) = u0(x),ut(x,0) = u1(x)∀x ∈ Ω,(1.2)

and boundary conditions, for example of Dirichlet type (but the theory is more or less

the same also with Neumann or periodic boundary conditions)

u(x,t) = 0∀(x,t) ∈ ∂Ω × [0,T).(1.3)

From the mathematical point of view, (1.1) is probably the simplest example of quasi-

linear hyperbolic equation. From the mechanical point of view, this Cauchy boundary

value problem is a model for the small transversal vibrations of an elastic string (n = 1)

or membrane (n = 2). In the string context it was introduced by G. Kirchhoff in

[15].

A lot of papers have been devoted to existence of local or global solutions to (1.1),

(1.2), (1.3). The results can be divided into four main families.

(A) Local existence in Sobolev spaces. If m is a locally Lipschitz continuous function

such that m(σ) ≥ ν > 0 for every σ ≥ 0 (strict hyperbolicity), then (1.1), (1.2),

(1.3) admits a unique local solution for initial data in Sobolev spaces. This result

was first proved by S. Bernstein in the pioneering paper [3], and then extended

with increasing generality by many authors. The more general statement is prob-

ably contained in A. Arosio and S. Panizzi [1] (see also the references quoted

therein), where it is proved that the problem is locally well posed in the phase

space

Vβ:= Hβ+1(Ω) × Hβ(Ω) + boundary conditions

for every β ≥ 1/2.

(B) Global existence for analytic data. If m is a continuous function such that m(σ) ≥ 0

for every σ ≥ 0 (weak hyperbolicity), and u0(x) and u1(x) are analytic functions,

then (1.1), (1.2), (1.3) admits at least one global solution (T = +∞), which is

analytic in the space variables for any time.

This result was proved with increasing generality by S. Bernstein [3], S. I.

Pohozaev [19], A. Arosio and S. Spagnolo [2], P. D’Ancona and S. Spa-

gnolo [8, 9].

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(C) Local existence with assumptions between (A) and (B). More recently, H. Hiro-

sawa [13] considered the continuity modulus ω of the nonlinear term m. He proved

existence of local solutions in suitable classes of initial data, depending on ω, both

in the strictly hyperbolic and in the weakly hyperbolic case. From the point of

view of local solutions these results represent an interpolation between the results

of (A) and (B). The precise relations between ω and the regularity required on

initial data are stated in section 2. Roughly speaking, in the strictly hyperbolic

case the situation is summed up by the following scheme:

ω(σ) = o(1) → analytic data,

ω(σ) = σα(with α ∈ (0,1)) → Gevrey space Gs(Ω) with s = (1 − α)−1,

ω(σ) = σ|logσ| → C∞(Ω) (or Vβwith finite derivative loss),

ω(σ) = σ → V1/2(with no derivative loss).

More regularity is required in the weakly hyperbolic case, as shown in the following

scheme:

ω(σ) = o(1) → analytic data,

ω(σ) = σα(with α ∈ (0,1)) → Gevrey space Gs(Ω) with s = 1 + α/2,

ω(σ) = σ → Gevrey space G3/2(Ω).

(D) Global existence in special situations. Besides (B), there are four special cases in

which global solutions are known to exist. We refer the interested reader to the

quoted literature for the details. We just point out that in all these results the

nonlinearity m is assumed to be Lipschitz continuous and strictly positive.

• K. Nishihara [18] proved global existence for quasi-analytic initial data.

This class of functions strictly contains the space of analytic functions, but

it is strictly contained in every Gevrey class Gs(Ω) with s > 1.

• J. M. Greenberg and S. C. Hu [12], and then P. D’ancona and S.

Spagnolo [10] proved global existence for small initial data in Sobolev spaces

in the case Ω = Rn, where dispersion plays a crucial role. Later on this

dispersion-based approach was extended to external domains (see [22, 23]

and the references quoted therein).

• S. I. Pohozaev [20] proved global existence for initial data in the Sobolev

space V1in the case where m(σ) = (a + bσ)−2, with a > 0, b > 0. In this

particular case indeed equation (1.1) admits a second order invariant.

• In some recent papers R. Manfrin [16] (see also [17], [14], [11]) proved

global existence in a new class of nonregular initial data. Manfrin’s spaces

are small in the sense that they don’t contain any Gevrey class Gs(Ω) with

s > 1, but they are large in the sense that any initial condition (u0,u1) ∈ V1

is the sum of two initial conditions belonging to these spaces!

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Despite of the many positive results, as far as we know no paper has been devoted

to negative results. In particular in the literature we found no counterexample even

against the most optimistic conjecture, according to which a global solution to (1.1),

(1.2), (1.3) exists assuming only that m is a nonnegative continuous function, and the

initial data belong to the “energy space” V0.

In this paper we make a first step in the direction of counterexamples. We focus on

local solutions, and we prove that Hirosawa’s results [13] are sharp, both in the strictly

hyperbolic and in the weakly hyperbolic case.

In Theorem 3.1 and Theorem 3.4 we construct indeed local solutions u(x,t) of (1.1),

(1.2), (1.3) with very regular initial data, but such that for every t > 0 we have that

(u(x,t),ut(x,t)) belongs to the phase space Vβif and only if β ≤ 1/2. In a few words

these solutions exhibit an instantaneous derivative loss up to V1/2. In these examples

the maximal regularity admitted for the initial data depends on the continuity modulus

of m. Just to give some examples, in the strictly hyperbolic case we have the following

three situations (see Example 3.2).

• If m is continuous we can have derivative loss for quasi analytic initial data.

This proves in particular that in Nishihara’s result [18] the Lipschitz continuity

assumption on the nonlinear term cannot be relaxed to continuity, even when

looking for local solutions.

• If m is α-H¨ older continuous for some α ∈ (0,1), then we can have derivative loss

for initial data in any Gevrey space Gs(Ω) with s > (1 − α)−1.

• If m is α-H¨ older continuous for every α ∈ (0,1), then we can have derivative loss

for initial data in C∞(Ω). This proves in particular that in the results stated in (A)

(i.e., well posedness in Vβfor every β ≥ 1/2) the Lipschitz continuity assumption

on m cannot be relaxed to H¨ older continuity.

In the weakly hyperbolic case we have for example the following two situations (see

Example 3.5).

• If m is α-H¨ older continuous for some α ∈ (0,1), then we can have derivative loss

for initial data in any Gevrey space Gs(Ω) with s > 1 + α/2.

• If m is Lipschitz continuous we can have derivative loss for initial data in any

Gevrey space Gs(Ω) with s > 3/2. This proves in particular that in the results

stated in (A) the strict hyperbolicity cannot be relaxed to weak hyperbolicity.

Our examples don’t exclude that a local solution can always exist in V1/2. This

remains an open problem.

Our approach is based on the theory developed by F. Colombini, E. De Giorgi,

and S. Spagnolo [5], and then extended by F. Colombini and S. Spagnolo [6],

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and by F. Colombini, E. Jannelli, and S. Spagnolo [7]. They considered linear

equations with a time dependent coefficient such as

utt(t,x) − c(t)∆u(t,x) = 0.(1.4)

If c(t) is not Lipschitz continuous or vanishes for t = 0, they showed how to construct

“solutions” of (1.4) which are very regular at time t = 0, but very irregular (not even

distributions) for t > 0.

The construction of our counterexamples is divided into two steps. In the first step

we consider the linear equation (1.4), and we modify the parameters in the construction

described in [5, 6, 7] in order to obtain solutions of (1.4) which are very regular at

time t = 0, but with a prescribed minimal regularity (they belong to Vβ if and only

if β ≤ 1/2) for every t > 0. These counterexamples for the linear equation, stated in

Proposition 3.6 and Proposition 3.7, are maybe interesting in themselves because they

extend the theory developed in [5, 6, 7] to coefficients c(t) with arbitrary continuity

modulus. The assumptions on initial data in these counterexamples are sharp because

they are complementary to those required in the existence results (both for linear and

for Kirchhoff equations).

In the second step we show that these solutions, up to modifying one dominant

Fourier component, are solutions of (1.1) for a suitable choice of the nonlinearity m.

This paper is organized as follows. In section 2 we rephrase (1.1), (1.2), (1.3) as an

abstract evolution problem in a Hilbert space, and we restate Hirosawa’s local existence

results in the abstract setting. In section 3 we state our counterexamples, which we

prove in section 4. For the convenience of the reader in appendix A we sketch the main

point of the proof of the local existence results in the abstract setting.

2Preliminaries and local existence results

Continuity moduli

recalled throughout the paper.

Let ω : [0,+∞) → [0,+∞). The following assumptions are

(ω1) We have that ω ∈ C0([0,+∞)) is an increasing function such that ω(0) = 0, and

ω(a + b) ≤ ω(a) + ω(b) for every a ≥ 0 and b ≥ 0.

(ω2) The function σ → σ/ω(σ) is nondecreasing.

A function f : X → R (where X ⊆ R) is said to be ω-continuous if there exists a

constant L ∈ R such that

|f(a) − f(b)| ≤ Lω(|a − b|)

Two simple properties of continuity moduli are stated in Lemma A.2 of the appendix.

In particular from (A.4) it follows that the composition of a Lipschitz continuous function

and an ω-continuous function (in any order) is again an ω-continuous function.

∀a ∈ X, ∀b ∈ X.(2.1)

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It is not difficult to see that the set of ω-continuous functions only depends on the

values of ω in a right neighborhood of 0. For this reason, with a little abuse of notation,

we often consider continuity moduli which are defined, and satisfy (ω1) and/or (ω2), in

a right neighborhood of 0. For the same reason when needed we assume also that ω is

invertible.

Abstract setting for Kirchhoff equations

For every x and y in H, let |x| denote the norm of x, and let ?x,y? denote the scalar

product of x and y. Let A be an unbounded linear operator on H with dense domain

D(A). We always assume that A is self-adjoint and nonnegative, so that the power Aβ

is defined for every β ≥ 0 in a suitable domain D(Aβ).

We consider the second order evolution problem

Let H be a separable real Hilbert space.

u′′(t) + m(|A1/2u(t)|2)Au(t) = 0,∀ t ≥ 0,(2.2)

with initial data

u(0) = u0,u′(0) = u1.(2.3)

It is well known that (2.2), (2.3) is just an abstract setting of (1.1), (1.2), (1.3),

corresponding to the case where H = L2(Ω), and Au = −∆u, defined for every u in a

suitable domain D(A) depending on the boundary conditions (see [1]).

Functional spaces

able complete orthonormal system {ek}k≥1made by eigenvectors of A. We denote the

corresponding eigenvalues by λ2

Under this assumption (which in the concrete case corresponds to bounded domains)

we can work with Fourier series. However, any definition or statement of this section

can be easily extended to the general setting just by using the spectral theorem for

self-adjoint operators ([21, Theorem VIII.4, p. 260]).

By means of the orthonormal system every u ∈ H can be written in a unique way in

the form u =?∞

Aβu :=

?

so that we can consider the quantity

For the sake of simplicity, let us assume that H admits a count-

k, so that Aek= λ2

kekfor every k ≥ 1.

k=1ukek, where uk= ?u,ek? are the Fourier components of u. Moreover

for every β > 0 we have that

∞

k=1

λ2β

kukek,

?u?2

D(Aβ):=

∞

?

k=1

λ4β

ku2

k,

and characterize the spaces D(Aβ) and D(A∞) as follows

D(Aβ) :=

?

u ∈ H : ?u?2

D(Aβ)< +∞

?

,D(A∞) :=

?

β>0

D(Aβ).

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With these notations the phase spaces Vβare defined as Vβ:= D(A(β+1)/2)×D(Aβ/2).

Let now ϕ : [0,+∞) → [1,+∞) be any function. Then for every r > 0 and β > 0

we can set

∞

?

and then define the spaces

?u?2

ϕ,r,β:=

k=1

λ4β

ku2

kexp?rϕ(λk)?, (2.4)

Gϕ,r,β(A) :=?u ∈ H : ?u?2

If two continuous functions ϕ1(σ) and ϕ2(σ) coincide for every large enough σ, then

Gϕ1,r,β(A) = Gϕ2,r,β(A). For this reason, with a little abuse of notation, we consider these

spaces even if ϕ(σ) is defined, continuous, and greater than 1 only for large values of σ.

ϕ,r,β< +∞?,Gϕ,β(A) :=

?

r>0

Gϕ,r,β(A).

Hirosawa’s results

stated in the abstract setting. Since the abstract statements do not seem to follow

trivially from the ones in [13], we sketch the proofs in appendix A.

The first result concerns the strictly hyperbolic case (see Theorem 2.2 in [13]).

We are now ready to recall the main results proved in [13], re-

Theorem A (Strictly hyperbolic case) Let ω : [0,+∞) → [0,+∞) be a function

satisfying (ω1). Let m : [0,+∞) → (0,+∞) be an ω-continuous function satisfying the

strict hyperbolicity condition m(σ) ≥ ν > 0 for every σ ≥ 0.

Let ϕ : [0,+∞) → [1,+∞) be a function such that

σ

ϕ(σ)ω

limsup

σ→+∞

?1

σ

?

< +∞.(2.5)

Let us assume that

(u0,u1) ∈ Gϕ,r0,3/4(A) × Gϕ,r0,1/4(A)(2.6)

for some r0> 0.

Then there exist T > 0, and R > 0 with RT < r0 such that problem (2.2), (2.3)

admits at least one local solution

u ∈ C1?[0,T];Gϕ,r0−Rt,1/4(A)?∩ C0?[0,T];Gϕ,r0−Rt,3/4(A)?.

Remark 2.1 Condition (2.7), with the range space depending on time, simply means

that

u ∈ C1?[0,τ];Gϕ,r0−Rτ,1/4(A)?∩ C0?[0,τ];Gϕ,r0−Rτ,3/4(A)?

for every τ ∈ (0,T].

This amounts to say that scales of Hilbert spaces are the natural setting for this

problem.

(2.7)

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Example 2.2 Let us give some examples in order to clarify the interplay between as-

sumptions (2.5) and (2.6).

• If ω(σ) = o(1) as σ → 0+(which simply means that m is continuous), then (2.5)

holds true with ϕ(σ) = σ. In this case (2.6) means that u0and u1are analytic,

and one has re-obtained the classical local existence result for analytic data in the

strictly hyperbolic case. In (2.5) one can also take ϕ(σ) = σω(1/σ), thus obtaining

local existence for a larger class of initial data.

• If ω(σ) = σαfor some α ∈ (0,1) (which means that m is H¨ older continuous), then

(2.5) holds true with ϕ(σ) = σ1−α. In this case (2.6) means that one can take

initial data in the Gevrey space Gs(A) with s = (1 − α)−1. We recall that Gs(A)

is the space Gϕ,β(A) corresponding to ϕ(σ) = σ1/s(β is insignificant in this case).

• If ω(σ) = σ|logσ| (which means that m is log-Lipschitz continuous), then (2.5)

holds true with ϕ(σ) = logσ. In this case Gϕ,r,β(A) = D(Aβ+r/4). One can

therefore take initial data in D(A∞) and obtain a solution in the same space, or

even initial data in Vγ for some γ > 1/2 and obtain a solution with a possible

progressive derivative loss (due to the term r0− Rt in (2.7)).

• If ω(σ) = σ (which means that m is Lipschitz continuous), then (2.5) holds true

with ϕ(σ) ≡ 1. Therefore (2.6) means that one can take initial data in V1/2. This

is of course the classical existence result in Sobolev spaces.

Since we are interested in local solutions, Theorem A can be applied also in the

mildly degenerate case, namely when m may vanish but m(|A1/2u0|2) > 0.

The following result (see Theorem 2.1 of [13]) concerns the weakly hyperbolic case,

and it is essential to deal with the really degenerate case, i.e., when m(|A1/2u0|2) = 0.

Theorem B (Weakly hyperbolic case) Let ω : [0,+∞) → [0,+∞) be a function

satisfying (ω1). Let m : [0,+∞) → [0,+∞) be an ω-continuous function.

Let ϕ : [0,+∞) → [1,+∞) be a function such that

?

?ω(1/σ)

Let (u0,u1) and r0> 0 be such that (2.6) holds true.

Then there exist T > 0, and R > 0 with RT < r0 such that problem (2.2), (2.3)

admits at least one local solution u satisfying (2.7).

limsup

σ→+∞

σϕ

?

σ

??−1

< +∞.(2.8)

Example 2.3 Let us give some examples.

• If ω(σ) = o(1) as σ → 0+, then (2.8) holds true with ϕ(σ) = σ. Once again (2.6)

means that u0 and u1 are analytic, and one has re-obtained the classical local

existence result for analytic data in the weakly hyperbolic case.

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• If ω(σ) = σαfor some α ∈ (0,1], then (2.8) holds true with ϕ(σ) = σ2/(α+2).

Therefore (2.6) means that one can take initial data in the Gevrey space Gs(A)

with s = 1 + α/2. This is true in particular for α = 1 (i.e., when m is Lipschitz

continuous). In this case we have local existence for initial data in G3/2(A).

Remark 2.4 One cannot expect local solutions to be unique if m is not Lipschitz

continuous (see some examples in [2]). However the existence results rely on an a priori

estimate (see Proposition A.1) which implies in particular that under assumptions (2.5)

or (2.8) any local solution satisfying the minimal regularity requirement (A.1) also

satisfies the stronger condition (2.7). This is a sort of propagation of regularity: if the

space in (2.6) is strictly contained in D(A3/4) × D(A1/4), then also solutions lie in a

scale of spaces which is strictly contained in D(A3/4) × D(A1/4). We are going to see

that this is no more true when condition (2.5) or (2.8) are not satisfied.

3Statements of counterexamples

The first counterexample shows the optimality of Theorem A in the non-Lipschitz case.

Theorem 3.1 (Strictly hyperbolic case) Let A be a self adjoint linear operator on

a Hilbert space H. Let us assume that there exist a countable (not necessarily complete)

orthonormal system {ek}k≥1 in H, and an increasing unbounded sequence {λk}k≥1 of

positive real numbers such that Aek= λ2

Let ω : [0,+∞) → [0,+∞) be a function satisfying (ω1) and (ω2).

Let ϕ : [0,+∞) → [1,+∞) be a function such that

σ

ϕ(σ)ω

kekfor every k ≥ 1.

lim

σ→+∞

?1

σ

?

= +∞. (3.1)

Then there exist a function m : [0,+∞) → [1/2,3/2], a real number T0> 0, and a

function u : [0,T0] → H such that

(i) m is ω-continuous;

(ii) (u(0),u′(0)) ∈ Gϕ,r,3/4(A) × Gϕ,r,1/4(A) for every r > 0;

(iii) u ∈ C1([0,T0];D(A1/4)) ∩ C0([0,T0];D(A3/4)) is a solution of (2.2);

(iv) for every t ∈ (0,T0] we have that (u(t),u′(t)) ?∈ Vβfor every β > 1/2.

Example 3.2 Let us consider some examples.

• The assumptions of Theorem 3.1 are satisfied if we take

ω(σ) =

1

|logσ|1/2,ϕ(σ) =

σ

logσ.

In this case m is continuous, the initial data are quasi analytic, and the solution

u has an instantaneous infinite derivative loss.

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• Let α ∈ (0,1). The assumptions of Theorem 3.1 are satisfied if we take

ω(σ) = σα,ϕ(σ) =σ1−α

logσ.

In this case m is α-H¨ older continuous, the initial data are in the Gevrey space

Gs(A) for every s > (1 − α)−1, and the solution u has an instantaneous infinite

derivative loss.

• The assumptions of Theorem 3.1 are satisfied if we take

ω(σ) = σ|logσ|3,ϕ(σ) = log2σ.

In this case m is α-H¨ older continuous for every α ∈ (0,1) (but not log-Lipschitz

continuous), the initial data are in D(A∞), and once again the solution u has an

instantaneous infinite derivative loss.

Remark 3.3 Theorem 3.1, as it is stated, is void in the log-Lipschitz case. Indeed when

ω(σ) = σ|logσ| and ϕ satisfies (3.1), then all initial data satisfying (ii) belong to V1/2

but not to Vβfor β > 1/2. In a certain sense they have no derivatives to loose!

On the other hand, a careful inspection of the proof reveals that Theorem 3.1 can be

improved as follows. Given any function ψ : [0,+∞) → [1,+∞) such that ψ(σ) → +∞

as σ → +∞ we can find a solution satisfying (i), (ii), (iii), and

(iv’) for every t ∈ (0,T0] we have that (u(t),u′(t)) ?∈ Gψ,r,3/4(A) × Gψ,r,1/4(A) for every

r > 0.

Thus for example we can take

ω(σ) = σ|logσ|,ϕ(σ) = log|logσ|,ψ(σ) = log|log|logσ||,

and obtain a solution with initial data in Gϕ,r,3/4(A)×Gϕ,r,1/4(A) which for every positive

time doesn’t belong even to the weaker scale Gψ,r,3/4(A)×Gψ,r,1/4(A). So also in the log-

Lipschitz case we have a (infinitesimally small) derivative loss.

We spare the reader from this generalization.

The second counterexample shows the optimality of Theorem B.

Theorem 3.4 (Weakly hyperbolic case) Let A, {ek}, {λk} be as in Theorem 3.1.

Let ω : [0,+∞) → [0,+∞) be a function satisfying (ω1) and (ω2).

Let ϕ : [0,+∞) → [1,+∞) be a function such that

?

?ω(1/σ)

Then there exist a function m : [0,+∞) → [0,3/2], a real number T0 > 0, and a

function u : [0,T0] → H such that

lim

σ→+∞σϕ

?

σ

??−1

= +∞.(3.2)

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(i) m is ω-continuous;

(ii) (u(0),u′(0)) ∈ Gϕ,r,3/4(A) × Gϕ,r,1/4(A) for every r > 0;

(iii) u ∈ C1([0,T0];D(A1/4)) ∩ C0([0,T0];D(A3/4)) is a solution of (2.2);

(iv) for every t ∈ (0,T0] we have that (u(t),u′(t)) ?∈ Vβfor every β ≥ 1;

(v) there exists a sequence τk→ 0+such that |A(β+1)/2u(τk)| is unbounded for every

β > 1/2.

Example 3.5 Let α ∈ (0,1]. The assumptions of Theorem 3.1 are satisfied if we take

ω(σ) = σα,ϕ(σ) =σ2/(α+2)

logσ

.

In this case m is α-H¨ older continuous (Lipschitz continuous if α = 1), the initial

data are in the Gevrey space Gs(A) for every s > 1 + α/2, and the solution u has an

instantaneous infinite derivative loss.

In particular, in contrast to the strictly hyperbolic case (see Remark 3.3), The-

orem 3.4 provides examples of infinite derivative loss even in the Lipschitz (or log-

Lipschitz) case.

The counterexamples of Theorem 3.1 and Theorem 3.4 originate from two counterex-

amples for the linear equation

v′′+ c(t)Av = 0.(3.3)

This equation is well studied in mathematical literature. It is well known for example

that, if the coefficient c(t) is ω-continuous, then the Cauchy problem is well posed in

Gϕ,β(A) provided that ϕ and ω satisfy (2.5) in the strictly hyperbolic case, and (2.8)

in the weakly hyperbolic case. The main argument is that the approximated energy

estimates introduced in [5] can be extended word-by-word to arbitrary continuity moduli

as we do in the appendix below.

This result is sharp. Indeed if ϕ and ω do not satisfy (2.5) or (2.8), hence if they

satisfy (3.1) or (3.2) (see also Remark 3.9 below), then the Cauchy problem is not well

posed in Gϕ,β(A). In literature we found a lot of counterexamples for special choices of

ω and ϕ (see for example [4] and the references quoted therein), but we didn’t find the

general counterexamples under assumptions (3.1) or (3.2).

In the following two propositions we state them in the form which is needed in the

proof of Theorem 3.1 and Theorem 3.4.

Proposition 3.6 (Strictly hyperbolic case) Let A, {ek}, {λk}, ω, ϕ be as in The-

orem 3.1.

Then there exist c : [0,1] → [1/2,3/2], and v : [0,1] → H such that

(SH-1) c is ω-continuous;

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(SH-2) (v(0),v′(0)) ∈ Gϕ,r,3/4(A) × Gϕ,r,1/4(A) for every r > 0;

(SH-3) v ∈ C1([0,1];D(A1/4)) ∩ C0([0,1];D(A3/4)) is a solution of (3.3);

(SH-4) for every t ∈ (0,1] we have that (v(t),v′(t)) ?∈ Vβfor every β > 1/2.

Proposition 3.7 (Weakly hyperbolic case) Let A, {ek}, {λk}, ω, ϕ be as in The-

orem 3.4.

Then there exist c : [0,1] → [0,3/2], and v : [0,1] → H such that

(WH-1) c is ω-continuous;

(WH-2) (v(0),v′(0)) ∈ Gϕ,r,3/4(A) × Gϕ,r,1/4(A) for every r > 0;

(WH-3) v ∈ C1([0,1];D(A1/4)) ∩ C0([0,1];D(A3/4)) is a solution of (3.3);

(WH-4) for every t ∈ (0,1] we have that (v(t),v′(t)) ?∈ Vβfor every β ≥ 1;

(WH-5) there exists a sequence τk→ 0+such that |A(β+1)/2v(τk)| is unbounded for every

β > 1/2.

Remark 3.8 Proposition 3.6 and Proposition 3.7 are strongly based on the counterex-

amples shown in [5, 6, 7]. On the other hand, besides the fact that we deal with arbitrary

continuity moduli, and arbitrary sequences of eigenvalues, there are some differences we

would like to point out.

• In [5, 6, 7] the derivative loss is bigger because solutions instantaneously lie outside

the space of distributions. Here we need to be more careful since we want solutions

to lie in V1/2and nothing more.

• “Derivative loss” has a slightly different meaning in [5, 6, 7] and in this paper.

Losing the m-th derivative in [5, 6, 7] means that there exists a sequence τk→ 0+

such that the norm of (v(τk),v′(τk)) in Vm tends to +∞. This of course may

happen also if v(τk) and v′(τk) are in D(A∞) for every k. In other words, what is

actually lost is a uniform bound on the m-th derivative.

In statements (SH-4) and (WH-4), and in the corresponding statements of Theo-

rem 3.1 and Theorem 3.4, we lose the m-th derivative in a stronger sense, namely

(v(t),v′(t)) ?∈ Vmfor every t > 0. On the contrary in statement (WH-5), and in the

corresponding statement of Theorem 3.4, we are forced to lose the last derivatives

only in the weaker sense.

Remark 3.9 A careful inspection of the proofs shows that the same conclusions hold

true also if the limit in (3.1) and (3.2) is replaced by the corresponding limsup computed

along the sequence {λk}.

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4Proofs

The proof is organized as follows. In section 4.1 we construct functions c(t) and v(t) de-

pending on several parameters. In Proposition 4.2, Proposition 4.3 and Corollary 4.4 we

relate the regularity properties of c(t) and v(t) to suitable conditions on the parameters.

Then in section 4.2 we choose the parameters in order to prove Proposition 3.6. In sec-

tion 4.3 we do the same for Proposition 3.7. Finally in section 4.4 we prove Theorem 3.1

and Theorem 3.4.

4.1General Construction

IngredientsThe starting point of the construction are the functions

b(ε,t) := 1 − 4εsin(2t) − ε2(1 − cos(2t))2,

w(ε,t) := sint · exp?ε?t −1

introduced in section 7 of [5].

Then the construction is based on seven sequences {tk}, {sk}, {τk}, {ηk}, {δk}, {εk},

{ak} satisfying the following assumptions.

(Hp-tk) The sequence {tk} is a decreasing sequence of positive real numbers (which

we think as times) such that t0= 1, and tk→ 0+as k → +∞.

(Hp-sk) The sequence {sk} is a sequence of positive real numbers (which we think as

times) such that tk< sk< tk−1for every k ≥ 1.

(Hp-τk) The sequence {τk} is a sequence of positive real numbers (which we think as

times) such that tk< τk< skfor every k ≥ 1.

(Hp-ηk) The sequence {ηk} is an increasing subsequence of the sequence {λk} of the

eigenvalues of A1/2.

(4.1)

(4.2)

2sin(2t)??,

(Hp-δk) The sequence {δk} is a nonincreasing sequence of positive real numbers with

δ0= 1. Moreover we require that√δkηktk/(2π) and√δkηksk/(2π) are inte-

gers, and 2√δkηkτk/π is an odd integer for every k ≥ 1.

(Hp-εk) The sequence {εk} is a sequence of positive real numbers such that εk≤ 1/16

for every k ≥ 1. Moreover we require that εkδk→ 0 and that {√δkεkηk} is

a nondecreasing sequence.

No special assumption is required on the sequence {ak}. We denote the limit of {δk}

by δ∞.

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Definition and properties of c(t)

and for every k ≥ 1

Let c : [0,1] → R be the function defined by c(0) = δ∞,

c(t) :=

δkb?εk,√δkηkt?

δk−1− δk

tk−1− sk(t − sk) + δk

if t ∈ [tk,sk],

if t ∈ [sk,tk−1].

(4.3)

Note that in the interval [sk,tk−1] the function c(t) is the affine interpolation between

δkand δk−1.

From the assumptions on the parameters we have that c(tk) = c(sk) = δk. Moreover

δk− 8εkδk≤ c(t) ≤ δk+ 8εkδk

δk≤ c(t) ≤ δk−1

∀t ∈ [tk,sk], (4.4)

∀t ∈ [sk,tk−1]. (4.5)

Since εkδk→ 0, and δk→ δ∞, estimates (4.4) and (4.5) imply that c(t) is continuous

in [0,1]. Since εk≤ 1/16 we have also that

1

2δk≤ c(t) ≤3

2δk

∀t ∈ [tk,sk],(4.6)

and globally

1

2δ∞≤ c(t) ≤3

2

∀t ∈ [0,1].(4.7)

Concerning the derivative we have that

|c′(t)| ≤ 16εkηkδ3/2

k

∀t ∈ (tk,sk),(4.8)

and of course

c′(t) =δk−1− δk

tk−1− sk

∀t ∈ (sk,tk−1). (4.9)

The ω-continuity of c(t) in the whole interval [0,1] can be deduced from the ω-

continuity in the intervals [tk,tk−1] provided that the following uniform estimates hold

true (we omit the easy and classical proof).

Lemma 4.1 Let c : [0,1] → R be any function. Let {tk} be any sequence satisfying

(Hp-tk).

Then c is ω-continuous in [0,1] if and only if there exists a constant L such that

(i) |c(ti) − c(tj)| ≤ Lω(|ti− tj|) for every pair of positive integers i and j;

(ii) |c(a) − c(b)| ≤ Lω(|a − b|) for every k ≥ 1, and every a and b in [tk,tk−1].

2

Applying the lemma we find the following sufficient condition for the for the ω-

continuity of c(t).

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