Remarks on the Cauchy problem for the one-dimensional quadratic (fractional) heat equation

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Remarks on the Cauchy problem for the

one-dimensional quadratic (fractional) heat equation

Luc Molinet, Slim Tayachi

To cite this version:

Luc Molinet, Slim Tayachi. Remarks on the Cauchy problem for the one-dimensional quadratic

(fractional) heat equation. Journal of Functional Analysis, Elsevier, 2015, 269, pp.2305-2327. �hal-

00807047�

ONE-DIMENSIONAL QUADRATIC (FRACTIONAL) HEAT EQUATION

LUC MOLINET AND SLIM TAYACHI

Abstract.

We prove that the Cauchy problem associated with the one dimensional quadratic (fractional) heat equation:

ut

=

Dx^2αu∓u², t ∈

(0, T ), x

∈

R ^or T ^{, with} 0

< α≤

1 is well-posed in

H^s

for

s ≥

max(−α, 1/2

−

2α) except in the case

α

= 1/2 where it is shown to be well-posed for

s > −1/2 and ill-posed for s

=

−1/2. As a

by-product we improve the known well-posedness results for the heat equation (α = 1) by reaching the end-point Sobolev index

s

=

−1. Finally, in the case 1/2< α≤

1, we also prove optimal results in the Besov spaces

B₂^s,q.

Keywords: Nonlinear heat equation, Fractional heat equation, Ill-posedness, Well- posedness, Sobolev spaces, Besov spaces.

2000 AMS Classification: 35K15, 35K55, 35K65, 35B40

1. Introduction and main results

The Cauchy problem for the quadratic fractional heat equation reads

(1.1) u

_t

− D

_x^2α

u = ∓u

²

,

(1.2) u(0, ·) = u

₀

,

where u = u(t, x) ∈ _R , α ∈]0, 1], t ∈ (0, T ), T > 0, x ∈ _R or T and D

^2α_x

is the Fourier multiplier by |ξ|

^2α

. In this paper, we consider actually the corresponding integral equation which is given by

(1.3) u(t) = S

_α

(t)u

₀

∓

Z

_t

0

S

_α

(t − σ) u

²

(σ) dσ,

where S

α

(t) is the linear fractional heat semi-group and are interested in local well- posedness and ill-posedness results in the Besov spaces B

₂^s,q

(K) with s ∈ _R , q ∈ [1, ∞[

and K = R or T .

1

Let us recall that the Cauchy problem associated with the nonlinear heat equation in R

ⁿ

(1.4) u

_t

− ∆u = ∓u

^k

has been studied in many papers (see for instance [3, 4, 5, 6, 7, 9, 11, 12, 13, 14, 18, 19, 20, 21] and references therein). It is well-known that this equation is invariant by the space- time dilation symmetry u(t, x) 7→ u

_λ

(t, x) = λ

^k−12

u(λ

²

t, λx) and that the homogeneous Sobolev space ˙ H

^n2−k−12

is invariant by the associated space dilation symmetry ϕ(x) 7→

λ

^k−12

ϕ(λx). The Cauchy problem (1.4) is known to be well-posed in H

^s

for s ≥ s

_c

=

n

2

−

_k−1²

except in the case (n, k) = (1, 2). Indeed, in this case the well-posedness is only known in H

^s

for s > −1 and in [9] it is proven that the flow-map cannot be of class C

²

below H

⁻¹

. Hence, this result is close to be optimal if one requires the smoothness of the flow-map. Recently, it was proven in [8] that the associated solution-map : u

0

7→ u cannot be even continuous in H

^s

for s < −1. The first aim of this work is to push down the well-posedness result to the end point H

⁻¹

. The second step is to extend these type of results for the one-dimensional quadratic fractional heat equation (1.1). Indeed we will derive optimal results for the Cauchy problem (1.1) in the scale of the Besov spaces B

₂^s,q

in the case

¹₂

< α ≤ 1. In particular we will prove that the lowest reachable Sobolev index is −α that is strictly bigger then the critical Sobolev index for dilation symmetry that is 1/2 − 2α.

To reach the end-point index H

^−α

we do not follow the classical method for parabolic equations (cf. [4, 12, 21]) that does not seem to be applicable here. We rather rely on an approach that was first introduced by Tataru [16] in the context of wave maps. Note that we mainly follow [10] where this method has been adapted for dispersive-dissipative equations. The fact that our equation is purely parabolic enables us to simplify the proof.

The optimality of our results follows from an approach first introduced by Bejenaru-Tao [1] for a one-dimensional quadratic Schr¨ odinger equation. This approach is based on a high-to low frequency cascade argument.

Finally we consider the case 0 < α ≤ 1/2. By classical parabolic methods we obtain the well-posedness in the Sobolev space

¹

H

^s

( _R ), s ≥ 1/2 − 2α, unless α = 1/2. On the other hand, following a very nice result by Iwabuchi-Ogawa [8], we prove that (1.1) is ill-posed in H

^−1/2

( R ) for α = 1/2. It is worth noticing that (1/2, −1/2) is the intersection of the straight borderlines for well-posedness that are s = −α and s = 1/2 − 2α.

1

Recall that 1/2

−

2α is the critical Sobolev index for dilation symmetry.

Before stating our main result, let us give the precise definition of well-posedness we will use in this paper.

Definition 1.1. We will say that the Cauchy problem (1.1)-(1.2) is (locally) well-posed in some normed function space B if, for any initial data u

₀

∈ B , there exist a radius R > 0, a time T > 0 and a unique solution u to (1.3), belonging to some space-time function space continuously embedded in C([0, T ]; B), such that for any t ∈ [0, T ] the map u

₀

7→ u(t) is continuous from the ball of B centered at u

₀

with radius R into B. A Cauchy problem will be said to be ill-posed if it is not well-posed.

Theorem 1. Let K = R or T and α ∈]1/2, 1]. The Cauchy problem (1.1) is locally well- posed in the Besov space B

₂^s,q

(K) if and only if (s, q) ∈ _R × [1, +∞[ satisfies s > −α or s = −α and q ∈ [1, 2].

Remark 1.2. Our negative results can be stated more precisely in the following way : For any couple (s, q) ∈ _R × [1, +∞[ satisfying, s < −α or s = −α and q > 2, there exists T > 0 such that the flow-map u

0

7→ u(t) is not continuous at the origin from B

₂^s,q

(K) into D

⁰

(K) for any t ∈]0, T [.

This paper is organized as follows. In the next section we define our resolution spaces in the case K = R . In Section 3 we derive the needed linear estimates on the free term and the retarded Duhamel operator and in Section 4 we prove our well-posedness result.

Section 5 is devoted to the non-continuity results for the same range of α. In Section 6 we complete the well-posedness results by considering the case 0 < α ≤ 1/2. First, by classical parabolic methods, we prove that we can reach the critical Sobolev index for dilation symmetry that is 1/2 − 2α unless α = 1/2. Then, following [8], we prove that (1.1) is ill-posed in H

^−1/2

( _R ) for α = 1/2. Finally we explain the needed adaptations in the periodic case K = _T .

Throughout the paper, we will write f . g, whenever a constant C ≥ 1, only depending on parameters and not on t or x, exists such that f ≤ Cg. We write f ∼ g if f . g, and g . f. If C depends on parameters a, we write f .

a

g, instead.

2. Resolution Space We use the following definition for the Fourier transform

F (f )(ξ) = Z

R

f (x)e

^−ixξ

dx, and the inverse Fourier transform is

F

⁻¹

(f )(x) = 1 2π

Z

R

F(f )(ξ)e

^ixξ

dξ,

for f in S( _R ), the Schwartz space of rapidly decreasing smooth functions, and by duality if f in S

⁰

( R ), the space of tempered distributions. We denote sometimes F(f) by ˆ f . The fractional power of the Laplacien can be defined by the Fourier transform: For α ∈ _R ,

F (−∂

_x²

)

^α

f

(ξ) = |ξ|

^2α

F(f )(ξ).

Let s be a real number. The Sobolev space H

^s

( _R ) is defined by H

^s

( R ) = {u ∈ S

⁰

( R ) |

Z

R

(1 + |ξ|

²

)

^s

|F (u)(ξ)|

²

dξ < ∞}

where F (u) is the Fourier transform of u. The norm on H

^s

( R ) is defined by

||u||

_Hs(R)

= Z

R

(1 + |ξ|

²

)

^s

|F (u)(ξ)|

²

dξ

1/2

.

We will need a Littlewood-Paley analysis. Let η ∈ C

₀^∞

( R ) be a non negative even function such that supp η ⊂ [−2, 2] and η ≡ 1 on [−1, 1]. We define ϕ(ξ) = η(ξ/2) − η(ξ) and the Fourier multipliers

F (∆

_j

u)(ξ) = ϕ(2

^−j

ξ)F u(ξ), j ≥ 0, and F(∆

₋₁

u)(ξ) = η(ξ)Fu(ξ) .

For any s ∈ _R and q ≥ 1, the Besov space B

₂^s,q

( R ) is defined as the completion of S( _R ) for the norm

kuk

_B^s,q

2 (R)

= X

j≥−1

2

^jsq

||∆

_j

u||

^q_L2(R)

1/q

.

For s ∈ _R , s

₁

< s

₂

, 1 ≤ q

₁

≤ q

₂

and q ≥ 1 we have the following embeddings B

₂^s,q1

, → B

₂^s,q2

and B

₂^s2,q

, → B

₂^s1,1

.

Moreover, It is well-known that the H

^s

( R )-norm is equivalent to the B

₂^s,2

-norm so that H

^s

( R ) = B

₂^s,2

( R ).

Finally, for 1 ≤ p ≤ ∞ we consider the space-time space ˜ L

^p

( _R ; B

^s,q₂

) equipped with the norm

kuk

_˜

L^p_tB₂^s,q

= h X

j≥−1

2

^sjq

k∆

_j

u(t)k

^q

L^p_tL²_x

i

1/q

.

We are now able to define our resolution space. For T > 0 fixed, we consider the space X

_α,T^s,q

= ˜ L

^∞_T

B

₂^s,q

∩ L ˜

²_T

B

₂^s+α,q

equipped with the norm:

kuk

_X^s,q

α,T

= hX

j

sup

t∈]0,T[

2

^sjq

k∆

_j

u(t)k

^q

L²_x

i

1/q

+ hX

j

2

^j(s+α)q

k∆

_j

uk

^q

L²_TL²_x

i

1/q

.

Let us also consider the space Y

_α^s,q

:=

n

u ∈ L ˜

¹

R

^∗₊

; B

^s+2α,q₂

( R )

and ∂

t

u ∈ L ˜

¹

R

^∗₊

; B

^s,q₂

( R ) o

equipped with the norm:

kuk

_Y^s,q

α

= hX

j

2

^(s+2α)jq

k∆

_j

uk

^q

L¹_tL²_x

i

1/q

+ hX

j

2

^sjq

k∆

_j

u

_t

k

^q

L¹_tL²_x

i

1/q

For T > 0, the restriction space Y

_α,T^s,q

of Y

α^s,q

is endowed with the usual norm kuk

_Y^s,q

α,T

= inf

v∈Y

{kvk

_Y^s,q

α

, v ≡ u on ]0, T [ } .

For T > 0 our resolution space will be E

_α,T^s,q

= X

_α,T^s,q

+ Y

_α,T^s,q

endowed with the usual norm for a sum space :

kuk

_E^s,q

α,T

:= inf

u=v+w

(kvk

_X^s,q

α,T

+ kwk

_Y^s,q

α,T

).

3. Linear estimates We first establish the following lemma.

Lemma 3.1. Let 0 < T ≤ 1 and ϕ ∈ B

₂^s,q

. Then we have

(3.1) kS

_α

(t)ϕk

_X^s,q

α,T

. kϕk

_B^s,q

2

.

Proof. The standard smoothing effect of the (fractional) heat semi-group is not sufficient here since we have

kS

_α

(t)ϕk

_Bs+α,q

2

. t

⁻¹²

kϕk

_B^s,q

2

and the right hand side of this inequality is not square integrable near t = 0. Integrating by parts the linear fractional heat equation

(3.2) ∂

_t

u − D

^2α_x

u = 0

on ]0, t[× _R , t > 0, against u and using that u(0) = ϕ, we obtain Z

R

u

²

(t, x) dx + Z

t

0

Z

R

|D

^α_x

u(s, x)|

²

dx ds = Z

R

ϕ

²

(x) dx .

Using that for each j ∈ _N , ∆

j

S

α

(t)D

^s_x

ϕ satisfies the linear fractional heat equation (3.2) with D

^s_x

ϕ as initial datum, powering in q/2 and then summing in j ≥ 0, we get for any T > 0,

X

j≥0

2

^sjq

k∆

_j

S

_α

(t)ϕk

^q_L∞ TL²_x

1/q

+ X

j≥0

2

^(s+α)jq

k∆

_j

S

_α

(t)ϕk

^q

L²_TL²_x

1/q

. X

j≥0

2

^sjq

k∆

_j

ϕk

^q_L2

x

1/q

.

On the other hand, for j = −1 we write k∆

₋₁

S

_α

(t)ϕk

_L^∞

TL²_x

+ k∆

₋₁

S

_α

(t)ϕk

_L2

TL²_x

≤ 2 T

^1/2

k∆

₋₁

S

_α

(t)ϕk

_L^∞

TL²_x

≤ 2 T

^1/2

k∆

₋₁

ϕk

_L2 x

and the result follows.

As a direct consequence we get the following estimate on the semi-group : Let 0 < T ≤ 1 and ϕ ∈ B

₂^s,q

then it holds

(3.3) kS

_α

(t)ϕk

_E^s,q

α,T

. kS

_α

(t)ϕk

_X^s,q

α,T

. kϕk

_B^s,q

2

.

Let us now define the operator L

_α

by

(3.4) L

_α

(f )(t, x) =

Z

t 0

S

α

(t − t

⁰

)f (t

⁰

)dt

⁰

. Then we have

Lemma 3.2. Let 0 < T ≤ 1 and f ∈ E

_α,T^s,q

. Then we have

(3.5) kL

_α

(f)k

_Y^s,q

α,T

. (1 + T )kf k

_L˜1 TB₂^s,q

.

Proof. It suffices to prove the result for a time extension of L

_α

(f ). More precisely, it suffices to prove that

kηL

_α

(f )k

_Y^s,q

α,T

. kf k

_L˜1 t>0B₂^s,q

,

for any f ∈ L ˜

¹_t>0

B

₂^s,q

supported in time in [0, 1] and where η ∈ C

₀^∞

( R ) is defined in Section 2 . Let u be the solution of the Cauchy problem

∂

t

u − D

^2α_x

u = f, u(0) = 0.

It is easy to check that u = L

_α

(f ). Multiplying this equation by u and integrating by parts, we get

1 2

d dt

Z

R

u

²

+ Z

R

(D

^α_x

u)

²

= Z

R

f u .

Applying this equality to localizing in frequencies equation and using Bernstein inequality and the Cauchy-Schwarz one, we get for any j ∈ _N ,

1 2

d dt

Z

R

u

²_j

+ 2

^2αj

Z

R

u

²_j

≤ Z

R

f

_j²

1/2

Z

R

u

²_j

1/2

. Here u

_j

= ∆

_j

u, f

_j

= ∆

_j

f . If R

R

u

²_j

1/2

6= 0 we divide this last inequality by R

R

u

²_j

1/2

to obtain

d dt

Z

R

u

²_j

1/2

+ 2

^2αj

Z

R

u

²_j

1/2

≤ Z

R

f

_j²

1/2

. On the other hand, the smoothness and non negativity of t 7→ ku

_j

(t)k

²_L2

x

forces

_dt^d

ku

_j

(t)k

²_L2

x

= 0 as soon as ku

_j

(t)k

_L2

x

= 0. This ensures that the above differential inequality is actually valid for all t > 0. Integrating this differential inequality in time we get for any j ∈ _N ,

(3.6) 2

^2αj

ku

_j

k

_L1

tL²_x

. kf

_j

k

_L1 tL²_x

.

Now, in the case j = −1, we get in the same way k∆

−1

uk

_L2

. k∆

−1

f k

_L1

tL²_x

. Integrating on [0, 2T ] this leads to kη∆

₋₁

uk

_L1

tL²_x

. T kf

_j

k

_L1 tL²_x

.

Finally, in view of the linear fractional heat equation, the triangle inequality leads to (3.7) k∂

_t

(ηu

_j

)k

_L1

tL²_x

. kη∂

_t

u

_j

k

_L1

tL²_x

+ ku

_j

k

_L1

tL²_x

. kf

_j

k

_L1 tL²_x

.

Since u = L

_α

(f ), summing in j ∈ _N using Bernstein inequalities and recalling the expres- sion of the norm in Y

^s,α

, we conclude that

(3.8) kηL

_α

(f)k

_Y^s,q

α,T

. kf k

_L˜1 tB^s,qx

.

Lemma 3.3. Let 0 < T ≤ 1 and u ∈ Y

_α,T^s,q

. Then it holds

(3.9) kuk

_X^s,q

α,T

= kuk

_L˜∞

TB₂^s,q

+ kuk

_L˜2

TB₂^s+α,q

. kuk

_Y^s,q

α,T

. In particular, E

_α,T^s,q

, → X

_α,T^s,q

.

Proof. Again it suffices to prove this estimate for the non restriction spaces. Actually, by localizing in space frequencies it suffices to prove that for any function u ∈ L

¹

( R

+

; L

²

( R )) with u

t

∈ L

¹

( R

+

; L

²

( R )) it holds

(3.10) kuk

_L^∞

t L²_x

. ku

_t

k

_L1

tL²_x

and kuk

²_L2

t>0L²_x

. kuk

_L1

t>0L²_x

ku

_t

k

_L1 t>0L²_x

.

Indeed, applying (3.10) to the space frequency localization u

j

of u, Bernstein’s inequalities lead to

2

^js

ku

_j

k

_L^∞

t L²_x

. 2

^js

k∂

_t

u

_j

k

_L1

tL²_x

and 2

^jq(s+α)

ku

_j

k

^q

L²_t>0L²_x

. 2

^jq(s+2α)/2

ku

_j

k

^q/2

L¹_t>0L²_x

2

^jqs/2

k∂

_t

u

_j

k

^q/2

L¹_t>0L²_x

, which yields the result by summing in j and applying Cauchy-Schwarz in j on the right-

hand member of the second inequalities.

Let us now prove (3.10). The first part is a direct consequence of the equality u(t) =

− R

∞

t

u

_t

(s)ds and Minkowsky integral inequality. To prove the second part we notice that u

²

(t) = −u(t) R

∞

t

u

t

(s)ds so that we can write Z

∞

0

Z

R

u

²

(t, x) dx dt = Z

∞

0

Z

R

u(t, x) Z

t

0

u

_t

(s, x) ds dx dt .

Z

R

Z

∞ 0

|u(t, x)| dt Z

∞

0

|u

_t

(t, x)| dt dx . k

Z

∞ 0

|u(t, ·)| dtk

_L2 x

k

Z

∞ 0

|u

_t

(t, ·)| dtk

_L2 x

. kuk

_L1

t>0L²_x

ku

_t

k

_L1 t>0L²_x

,

where we used Minkowsky integral inequality in the last step.

4. Well-posedness for 1/2 < α ≤ 1 According to Lemma 3.2 we easily get for 0 < T ≤ 1,

kL

_α

(u

²

)k

_E^s,q

α,T

. kL

_α

(u

²

)k

_Y^s,q

α,T

. X

j

2

^jsq

k∆

_j

(u

²

)k

^q

L¹_TL²_x

1/q

. X

j

2

^jq(s+1/2)

k∆

_j

(u

²

)k

^q

L¹_TL¹_x

1/q

. (4.1)

Now, by para-product decomposition we have k∆

_j

(u

²

)k

_L1

TL¹_x

. kuk

_L2

T ,x

k∆

_j

uk

_L2

T ,x

+ X

|k−k⁰|≤3, k&j

k∆

_k

uk

_L2

T ,x

k∆

_k⁰

uk

_L2 T ,x

.

The contribution to (4.1) of the first term of the above right-hand side member can be estimated by

kuk

_L2 T ,x

X

j

2

^jq(s+1/2)

k∆

_j

uk

^q

L²_TL²_x

1/q

= kuk

_L2 T ,x

kuk

_˜

L²_TB^s+1/2,q₂

which is acceptable as soon as α ≥ 1/2 and (s > −α or s = −α and 1 ≤ q ≤ 2 ). Indeed, this last condition ensures that kuk

_L2

T ,x

. kuk

_L˜2

TB₂^s+α,q

. For the second term, we notice that for α > 1/2, we can estimate its contribution by

kuk

_L˜2

TB^0,∞₂

kuk

_L˜2 TB₂^s+α,q

X

j

2

^jq(1/2−α)

1/q

≤ C(α) kuk

_L˜2

TB^0,∞₂

kuk

_L˜2

TB₂^s+α,q

, where C(α) > 0 only depends on α > 1/2.

In view of Lemma 3.3, this proves that for α > 1/2, (4.2) kL

_α

(u

²

)k

_E^s,q

α,T

. kuk

_L˜2 T x

kuk

_L˜2

TB^s+α,q₂

. kuk

_E−α,2 α,T

kuk

_E^s,q

α,T

,

where the implicit constants only depends on α. In the same way, for any α ∈]1/2, 1], there exists C

α

> 0 such that

kL

_α

(uv)k

_E^s,α

T

. kuk

_L˜2 T x

kvk

_L˜2

TB₂^s+α,q

+ kvk

_L˜2 T x

kuk

_L˜2 TB₂^s+α,q

≤ C

α

kuk

_E^−α,2

α,T

kvk

_E^s,q

α,T

+ kvk

_E^−α,2

α,T

kuk

_E^s,q

α,T

. (4.3)

Let us now fixed α ∈]1/2, 1]. (4.3) together with (3.3) lead to the existence of β > 0 such that for all u

₀

∈ B

₂^−α,2

( R ) with

(4.4) ku

₀

k

_B^−α,2

2 (R)

≤ β , the mapping

u 7→ S

α

(·)u

₀

+ L

_α

(u

²

)

is a strict contraction in the ball of E

_α,1^−α,2

centered at the origin of radius (2C

_α

)

⁻¹

. Noticing that E

_α,1^s,q

, → E

_α,1^−α,2

as soon as

(4.5) (s ≥ −α and 1 ≤ q ≤ 2) or s > −α,

this ensures that the above mapping is also strictly contractive is a small ball of E

_α,T^s,q

as soon as (4.5)-(4.4) are satisfied. Since S

α

is a continuous semi-group in B

₂^s,q

( R ) and accord- ing to Lemma 3.3, E

_α,1^s,q

, → L ˜

^∞₁

B

₂^s,q

, this leads to the well-posedness result in B

₂^s,q

( _R ) under conditions (4.5) for initial data satisfying (4.4). The result for general initial data follows by a simple dilation argument. Indeed, the equation (1.1) is invariant under the dilation u(t, x) 7→ u

_λ

(t, x) = λ

^2α

u(λ

^2α

t, λx) whereas kλ

^2α

u

₀

(λ·)k

_B−α,2

2 (R)

≤ λ

^α−1/2

ku

₀

k

_B−α,2 2 (R)

→ 0 as λ & 0. Classical arguments then lead to the well-posedness result in B

₂^s,q

( _R ) for arbitrary large initial data with a minimal time of existence T ∼ 1 + ku

₀

k

_B^−α,2

2 (_R)

−_2α−1^4α

. Note that, the well-posedness being obtained by a fixed point argument, as a by-product we get that the solution-map : u

₀

7→ u is real analytic from B

₂^s,q

( R ) into C [0, T ]; B

^s,q₂

( R )

.

5. Ill-posedness results for 1/2 < α ≤ 1 .

In this section we prove discontinuity results on the flow map u

₀

7→ u(t) for any fixed t > 0 less than some T > 0. To clarified the presentation we separate the case s < −α and the case s = −α and q > 2.

5.1. The case s < −α. We take the counter example of [10] used for the KdV-Burgers equation.

We define the sequence of initial data {φ

_N

}

_N≥1

⊂ C

^∞

( _R ) via its Fourier transform by

(5.6) φ ˆ

N

(ξ) = N

^α

χ

I_N

(ξ) + χ

I_N

(−ξ) ,

where I

_N

= [N, N + 2] and χ

_IN

is the characteristic function of the interval I

_N

, χ

_IN

(ξ) =







1 if ξ ∈ I

N

, 0 if ξ 6∈ I

N

. That is

φ

N

(x) =







N^α π

sin(x) x

cos

(N + 1)x

if x 6= 0,

N^α

π

if x = 0.

Clearly φ

N

∈ C

0

( R ) :=

n

f ∈ C( R )| lim

_|x|→∞

f(x) = 0 o

. For any (s, q) ∈ _R × [1, +∞] we have kφ

_N

k

_B^s,q

2 (R)

∼ N

^α+s

and thus kφ

_N

k

_B−α,q

2 (R)

∼ 1

whereas φ

N

→ 0 in B

₂^s,q

( R ), for s < −α.

Let us consider the following bilinear operator, closely related to second iteration of the Picard scheme,

A

₂

(t, h, h) = 2 Z

t

0

S

_α

(t − t

⁰

)[S

_α

(t

⁰

)h]

²

dt

⁰

,

where S

α

is the semi-group of the linear heat equation. Let us denote by F

_x

the partial Fourier transform with respect to x. Recall that

F

_x

S

α

(t)ϕ

(ξ) = e

^−t[ξ|2α

F

_x

(ϕ)(ξ), ∀ ϕ ∈ S

⁰

( R ), and F

_x

(f g) = F

_x

(f ) ? F

_x

(g), where ? is the convolution product.

It follows that

F

_x

A

2

(t, φ

N

, φ

N

)

(ξ) = 2 Z

t

0

e

^{−(t−t0)|ξ|2α}

Z

R

e

^{−t0|ξ1|2α}

φ ˆ

N

(ξ

1

)e

^{−t0|ξ−ξ1|2α}

φ ˆ

N

(ξ − ξ

1

)dξ

1

dt

⁰

= 2

Z

R

φ ˆ

_N

(ξ

1

) ˆ φ

_N

(ξ − ξ

1

) Z

t

0

e

^{−(t−t0)|ξ|2α}

e

^{−[|ξ1|2α+|ξ−ξ1|2α]t0}

dt

⁰

dξ

1

= 2

Z

R

φ ˆ

_N

(ξ

₁

) ˆ φ

_N

(ξ − ξ

₁

) e

^{−[|ξ1|2α+|ξ−ξ1|2α]t}

− e

^−|ξ|2αt

Θ

α

(ξ, ξ

1

)

dξ

₁

, (5.7)

where

Θ

α

(ξ, ξ

1

) = |ξ|

^2α

− |ξ

₁

|

^2α

− |ξ − ξ

1

|

^2α

. Note that the integrand is nonnegative. In particular, F

_x

A

₂

(t, φ

_N

, φ

_N

)

(ξ) = |F

_x

A

₂

(t, φ

_N

, φ

_N

) (ξ)|.

Let

K

₁^N

(ξ) = n

ξ

1

| (ξ − ξ

1

, ξ

1

) ∈ I

N

× I

N

or (ξ − ξ

1

, ξ

1

) ∈ I

−N

× I

−N

o

and

K

₂^N

(ξ) = n

ξ

₁

| (ξ − ξ

₁

, ξ

₁

) ∈ I

_N

× I

−N

or (ξ − ξ

₁

, ξ

₁

) ∈ I

−N

× I

_N

o For any |ξ| ≤

¹₂

, K

₁^N

(ξ) = ∅ and thus

F

_x

A

₂

(t, φ

_N

, φ

_N

)

(ξ) = 2 Z

K₂^N(ξ)

φ ˆ

_N

(ξ

₁

) ˆ φ

_N

(ξ − ξ

₁

) e

^{−[|ξ1|2α+|ξ−ξ1|2α]t}

− e

^−|ξ|2αt

Θ

_α

(ξ, ξ

₁

)

dξ

₁

.

On the other hand, for any (a, b) ∈ _R

₊

× _R

₋

one has obviously,

|a|

^2α

+ |b|

^2α

− |a + b|

^2α

≥ |a| ∧ |b|

2α

.

Moreover, it is easy to check that |K

₂^N

(ξ)| ≥ 1 and that in K

₂^N

(ξ) it holds N

^2α

≤

|Θ

_α

(ξ, ξ

₁

)| ≤ 2(N + 2)

^2α

Hence, fixing t ∈]0, 1[, it holds F

_x

A

₂

(t, φ

_N

, φ

_N

)

(ξ) ≥ e

^−t/2

N

^2α

1 − e

^−N2αt

2(N + 2)

^2α

≥ 1

4 e

^−t/2

, ∀ξ ∈ [−1/2, 1/2],

for any N > 0 large enough. This ensures that for any fixed (s, q) ∈ _R × [1, +∞] and any fixed t ∈]0, 1[,

(5.8) kA

₂

(t, φ

_N

, φ

_N

)k

_B^s,q

2

≥ 1

4 e

^−t/2

for N > 0 large enough. Taking s < −α this proves the discontinuity of the map u

₀

7→ u(t) in B

₂^s,q

. To prove the discontinuity with value in D

⁰

( R ), we proceed as follows. Let g ∈ S ( R ) be such that ˆ g is positive equal to 1 on [−1/4, 1/4] and supported in [−1/2, 1/2]. We obtain for N > 0 large enough,

| Z

R

A

2

(t, φ

N

, φ

N

)(x)g(x)dx| ≥ 1 8 e

^−t/4

.

On the other hand the analytical well-posedness ensures that A

₂

(t, φ

_N

, φ

_N

) is bounded in B

₂^−α,1

uniformly in N . Then, since D( _R ) is dense in S( _R ), there exists ϕ ∈ D( _R ) such that (5.9)

Z

R

A

₂

(t, φ

_N

, φ

_N

)(x)ϕ(x)dx ≥ 1

2

⁴

e

^−t/4

.

This shows that A

₂

(t, φ

_N

; φ

_N

) does not converge to 0 in D

⁰

( _R ) and proves the discon- tinuity from B

₂^s,q

( R ), s < −α into D

⁰

( R ).

We now turn to prove the discontinuity of the flow-map u(t, ·) : B

₂^s,q

( R ) −→ B

₂^s,q

( R )

h 7−→ u(t, h) = S

_α

(t)h + R

t

0

S

_α

(t − σ) u

²

(σ) dσ.

By the theorem of well posedness, there exist T > 0 and

0

> 0 such that for any 0 < ≤

0

, khk

_B−α,1

2

≤ 1 and 0 ≤ t ≤ T,

u(t, h) = S

α

(t)h +

∞

X

k=2

^k

A

k

(t, h

^k

),

where h

^k

= (h, · · · , h), h

^k

7→ A

k

(t, h

^k

) are k−linear continuous maps from (B

₂^−α,1

( R ))

^k

into C([0, T ]; B

^−α,1₂

( R )) and the series converges absolutely in C([0, T ]; B

₂^−α,1

( R )).

Hence

u(t, φ

N

) −

²

A

2

(t, φ

N

, φ

N

) = S

α

(t)φ

N

+

∞

X

k=3

^k

A

k

(t, φ

^k_N

).

Using the inequalities

kS

_α

(t)φ

N

k

_Bs,1

2 (R)

≤ kφ

_N

k

_Bs,1

2 (R)

≤ 2N

^s+α

and

∞

X

k=3

^k

A

_k

(t, φ

^k_N

)

B₂^−α,1(R)

≤

₀

3

∞

X

k=3

^k₀

A

_k

(t, φ

^k_N

)

B₂^−α,1(R)

≤ C

³

,

where C is a positive constant, we deduce that for s ≤ −α,

(5.10) sup

t∈[0,T]

ku(t, φ

_N

) −

²

A

₂

(t, φ

_N

, φ

_N

)k

_Bs,1

2 (R)

≤ C

³

+ 2

₀

N

^s+α

. According to (5.8) this leads, for ≤

^C−1₂^e₅^−t/4

, to

ku(t, φ

_N

)k

_B^s,q

2 (R)

≥ C

0

²

/2 − 2

0

N

^s+α

.

By letting N → ∞ we obtain the discontinuity result since u(t, 0) = 0 and φ

N

→ 0 in B

₂^s,q

( _R ) for s < −α. The discontinuity of the flow-map from B

₂^s,q

( _R ) into D

⁰

( _R ) follows in the same way by combining (5.9) and (5.10).

5.2. The case s = −α and q > 2. This case is similar to the precedent except that we have to change a little the sequence of initial data. Here we take the same sequence as in the work of Iwabuchi and Ogawa [8]. For any N ≥ 10 we define

ψ

N

= N

⁻¹²

X

N≤j≤2N

φ

₂^j

.

where φ

₂^j

is defined in (5.6).

Noticing that ∆

_k

φ

₂j

= δ

_k,j

φ

₂j

, we can easily check that kψ

_N

k

_B−α,q

2

∼ N

^−12+1q

. In particular, kψ

_N

k

_B^α,q

2

→ 0 for any q > 2 whereas kψ

_N

k

_B−α,2

2

= kψ

_N

k

_H−α

∼ 1. Since the equation is analytically well-posed in H

^−α

( _R ), in view of the preceding case, it suffices to prove that A

2

(ψ

N

, ψ

N

, t) does not tend to 0 in D

⁰

. By the localization, it holds φ

₂^j

?φ

₂j0

≡ 0 on ] − 1/2, 1/2[ as soon as j 6= j

⁰

≥ 10 and the same reasons as above lead to

F

_x

A

₂

(t, ψ

_N

, ψ

_N

)

(ξ) = N

⁻¹²

X

N≤j≤2N

Z

K₂^2j(ξ)

φ ˆ

₂^j

(ξ

₁

) ˆ φ

₂^j

(ξ − ξ

₁

) e

^{−[|ξ1|2α+|ξ−ξ1|2α]t}

− e

^−|ξ|2αt

Θ

_α

(ξ, ξ

₁

)

dξ

₁

≥ N

⁻¹²

e

^−t/2

N

¹²

N

^2α

1 − e

^−N2αt

2(N + 2)

^2α

≥ 1

4 e

^−t/2

, ∀ξ ∈ [−1/2, 1/2], for any N > 0 large enough. This completes the proof of the ill-posedness results for 1/2 < α ≤ 1.

6. Further remarks

6.1. Wellposedness results in the case 0 < α ≤ 1/2. In this case we only consider

the well-posedness results in the Sobolev spaces H

^s

( R ). We prove by standard parabolic

methods that one can reach the dilation critical Sobolev exponant s

c

= 1/2 − 2α except

in the case α = 1/2 where 1/2 − 2α = −α. See for instance [12], [4] or [21] for the same

kind of results in the case α = 1.

Theorem 2. Let (α, s) ∈ _R

²

be such that α ∈ (0, 1/2] and s ≥ 1/2 − 2α with s > −α.

Then the Cauchy problem (1.1) is locally well-posed in H

^s

( R ).

Proof. The proof is done using a fixed point argument on a suitable metric space. The case s > 1/2 is trivial since H

^s

( _R ) is an algebra and the semi-group S

_α

is contractive on H

^s

( _R ). One can thus simply perform a fixed point argument in C([0, T ]; H

^s

( _R )) on the Duhamel formula for a suitable T > 0 related to ku

₀

k

_Hs(R)

. The case s = 1/2 is also rather easy and is postponed at the end of the proof. So let us assume that

(6.1) 1/2 − 2α ≤ s < 1/2 if 0 < α < 1/2 and − 1/2 < s < 1/2 if α = 1/2 , that is α, s belong to the set

n

(α, s) ∈ _R

²

| 0 < α ≤ 1/2, s ≥ 1/2 − 2α and s > −α o .

For s fixed as above we take 0 < s

0

< 1/2 such that 0 < s

0

− s < α and 2s

0

− 1

2 < s .

This is obviously possible for α = 1/2 since s > −1/2, and for 0 < α < 1/2 since s + 1/2 > s + α ≥ (1/2 − 2α) + α = 1/2 − α > 0. We first establish the existence and uniqueness of a solution of (1.3) in

X

M,T

:=

n

u ∈ C (0, T ], H

^s0

( R )

| kuk

_XT

:= sup

t∈(0,T]

t

^s0

−s

2α

ku(t)k

_H^s0(R)

≤ M o

by proving that the mapping

Λ

_u0

(u)(t) = S

_α

(t)u

₀

∓ Z

t

0

S

_α

(t − σ) u

²

(σ) dσ, is a strict contraction in X

_M,T

for suitable M > 0, T > 0.

From classical regularizing effects for the fractional heat equation it holds (6.2) kS

_α

(t)f k

_H^s2(R)

≤ Ct

^{−s22α−s1}

kf k

_H^s1(R)

, ∀ s

₁

≤ s

₂

, ∀ f ∈ H

^s1

( _R ).

Applying (6.2) with (s

1

, s

2

) = (s, s

0

), yields

(6.3) t

^s0

−s

2α

kS

_α

(t)u

₀

k

_Hs0(_R)

. ku

₀

k

_Hs(_R)

. Now, according to [11], since 0 < s

₀

< 1/2, it holds :

(6.4) kuvk

H^2s0−12(R)

≤ Ckuk

_Hs0(R)

kvk

_Hs0(R)

,

where C is a positive constant. We thus obtain for any t > 0, t

^s02α−s

Z

t 0

S

_α

(t − t

⁰

)u

²

(t

⁰

) dt

⁰

H^s0(R)

. t

^s02α−s

Z

t

0

S

_α

(t − t

⁰

)u

²

(t

⁰

)

H^s0(R)

dt

⁰

. t

^s0

−s 2α

Z

t 0

(t − t

⁰

)

^s0

−1/2

2α

ku

²

(t

⁰

)k

_H2s0−1/2(R)

dt

⁰

. t

^s0

−s 2α

Z

_t

0

(t − t

⁰

)

^s0

−1/2

2α

ku(t

⁰

)k

²_Hs0(R)

dt

⁰

. sup

τ∈]0,t[

τ

^s0

−s

2α

ku(τ )k

_H^s0(R)

2

t

s−(1/2−2α) 2α

Z

1 0

(1 − θ)

^2s0

−1 4α

θ

s−s0

α

dθ

. t

^{s−(1/2−2α)2α}

kuk

²_X

(6.5)

t

where in the last step we used that 0 < s

0

− s < α and that s

0

> 1/2 − 2α since s

₀

> s ≥ 1/2 − 2α. In view of (6.5) we easily get for 0 < T < 1 and v

_i

∈ X

_T

, i = 1, 2, (6.6) kΛ

_u0

(v

i

)k

_XT

. kS

_α

(·)u

₀

k

_XT

+ T

^{s−(1/2−2α)2α}

kv

_i

k

²_X

T

and

(6.7) kΛ

_u0

(v

₁

− v

₂

)k

_XT

. T

^{s−(1/2−2α)2α}

(kv

₁

k

_XT

+ kv

₂

k

_XT

)kv

₁

− v

₂

k

_XT

.

Combining these estimates with (6.3) we infer that for s > 1/2 − 2α, Λ

u0

is a strict contraction on X

M,T

with M ∼ ku

₀

k

_Hs(R)

and T ∼ ku

₀

k

−2α s−(1/2−2α)

H^s(R)

if s > 1/2 − 2α. This leads to the existence and uniqueness in X

_T

for any u

0

∈ H

^s

( R ). For s = 1/2 − 2α, Λ

u0

is also a strict contraction on X

_M,T

with M ∼ ku

₀

k

_Hs(R)

and T ∼ 1 but only under a smallness assumption on ku

₀

k

_Hs(R)

. Hence, we get the existence in X

T

for any u

0

∈ H

^s

( R ) with small initial data. Now to prove that the solution u belongs to C([0, T ]; H

^s

( R )) we first notice that S

_α

(u

₀

) ∈ C( _R

₊

; H

^s

( _R )). Moreover , according to (6.2), we have

sup

t∈]0,T[

Z

t 0

S

α

(t − t

⁰

)(u

²

− v

²

)(t

⁰

) dt

⁰

H^s(_R)

. sup

t∈]0,T[

Z

t 0

S

α

(t − t

⁰

)(u

²

− v

²

)(t

⁰

)

H^s(_R)

dt

⁰

. sup

t∈]0,T[

Z

t 0

(t − t

⁰

)

^min(0,(2s0

−1/2)−s

2α )

ku

²

− v

²

k

_H2s0−1/2(R)

dt

⁰

. sup

t∈]0,T[

Z

t 0

(t − t

⁰

)

^min(0,(2s0

−1/2)−s

2α )

ku − vk

_H^s0(R)

ku + vk

_H^s0(R)

dt

⁰

. ku + vk

_XT

ku − vk

_XT

T

^min(1+

s−s0

α ,^{s−(1/2−2α)}_2α )

Z

1 0

(1 − θ)

^min(0,

(2s0−1/2)−s

2α )

θ

s−s0

α

dθ

. T

^{min(1+s−sα0,s−(1/2−2α)2α )}

ku + vk

_XT

ku − vk

_XT

(6.8)

where in the last step we used that 0 < s

0

− s < α and that

^{2s0−1/2−s}_2α

> −1 since 2s

0

− s > s ≥ 1/2 − 2α. This ensures that starting with a continuous function v ∈ C [0, T ]; H

^s

( _R )

∩ X

_M,T

, the sequence of function constructed by the Picard sheme that converges to the solution in u ∈ X

_T

is a Cauchy sequence in C [0, T ]; H

^s

( _R )

and thus u ∈ C [0, T ]; H

^s

( R )

. The continuous dependence with respect to initial data in H

^s

( R ) follows also easily from (6.8).

It remains to handle the case of arbitrary large initial data in H

^sc

( _R ) when s = s

_c

= 1/2 −2α. We first notice that, according to (6.6)-(6.7), Λ

_u0

is a strict contraction in X

M,T

as soon as M = 2kS

_α

(·)u

₀

k

_XT

is small enough. Then, fixing u

0

∈ H

^sc

( R ), by the density of H

^s0

( _R ) in H

^sc

( _R ) we infer that for any ε > 0 there exists u

_0,ε

∈ H

^s0

( _R ) such that ku

₀

− u

_0,ε

k

_Hsc(_R)

< ε. Since u

_0,ε

∈ H

^s0

( R ) it holds kS

_α

(·)u

_0,ε

k

_XT

≤ T

^s02α−s

ku

_0,ε

k

_Hs0(_R)

. This leads to

kS

_α

(·)u

₀

k

_XT

. T

^s0

−s

2α

ku

_0,ε

k

_Hs0(_R)

+ ε .

Noticing that the right-hand side member of the above inequality can be made arbi- trary small by choosing suitable ε > 0 and T > 0, this proves the local existence in C [0, T ]; H

^sc

( R )

∩ X

T

for arbitrary large initial data in H

^sc

( R ). Note that here T > 0 does not depend only on ku

₀

k

_Hsc(_R)

but on the Fourier profile of u

₀

. The uniqueness holds in {f ∈ X

_T

/ kf k

_Xt

→ 0 as t & 0} . This completes the proof for (α, s) satisfying (6.1).

Finally for s = 1/2 we apply the fixed point argument in X ˜

_M,T

:=

n

u ∈ C (0, T ], H

^1+α2

( R )

| kuk

_X˜

T

:= sup

t∈(0,T]

t

¹⁴

ku(t)k

H^1+α2 (R)

≤ M o

.

Using that H

^1+α2

( _R ) is an algebra we easily get

t

¹⁴

Z

t 0

S

α

(t − t

⁰

)u

²

(t

⁰

) dt

⁰

H^1+α2 (R)

. t

¹⁴

Z

t

0

ku

²

k

H^1+α2 (R)

dt

⁰

. t

¹⁴

Z

_t

0

kuk

²

H^1+α2 (R)

dt

⁰

. t

³⁴

kuk

²_˜

Xt

Z

1 0

θ

^−1/2

dθ . (6.9)

This gives the local existence and uniqueness in ˜ X

_M,T

for M ∼ ku

₀

k

_H1/2(R)

and T ∼ ku

₀

k

−4 3

H^1/2(R)

. The fact that the solution u belongs to C [0, T ]; H

¹²

( _R )

and the continuous

dependence with respect to initial data in H

¹²

( R ) follows by noticing that sup

t∈]0,T[

Z

t 0

S

α

(t − t

⁰

)(u

²

− v

²

)(t

⁰

) dt

⁰

H¹²(R)

. sup

t∈]0,T[

Z

t 0

kS

_α

(t − t

⁰

)(u

²

− v

²

)(t

⁰

)k

H¹²(R)

dt

⁰

. sup

t∈]0,T[

Z

t 0

k(u

²

− v

²

)(t

⁰

)k

H¹²(R)

dt

⁰

. sup

t∈]0,T[

Z

t 0

ku(t

⁰

)

²

− v(t

⁰

)

²

k

H^1+α2 (R)

dt

⁰

. sup

t∈]0,T[

Z

t 0

ku(t

⁰

) + v(t

⁰

)k

H^1+α2 (R)

ku(t

⁰

) − v(t

⁰

)k

H^1+α2 (R)

dt

⁰

. ku + vk

_X˜

T

ku − vk

_X˜

T

T

¹²

Z

1

0

θ

^−1/2

dθ . (6.10)

6.2. Illposedness result for α = 1/2 and s = −1/2. Let us now prove an ill-posedness result at the crossing point (α, s) = (1/2, −1/2) of the two lines s = −α and s = 1/2 − 2α.

Recall that there exists T

0

> 0 and R

0

> 0 such that the solution-map u

0

7→ u associated with (1.1) for α = 1/2 is well-defined and continuous from the ball B (0, R

0

)

_L²

of L

²

( R ) with values in C([0, T ]; L

²

( _R )). The following norm inflation result clearly disproves the continuity of this solution map from B(0, R

0

)

_L²

endowed with the H

^−1/2

-topology with values in C([0, T ]; H

^−1/2

), for any T ≤ T

0

.

Theorem 3. There exists a sequence T

N

& 0 and a sequence of initial data {φ

_N

} ⊂ L

²

( R ) such that the sequence of emanating solutions {u

_N

} of (1.1)is included in C([0, T

_N

]; L

²

( _R )) and satisfy

(6.11) kφ

_N

k

_H−1/2

→ 0 and ku

_N

(T

_N

)k

_H−1/2

→ +∞ as N → ∞ .

We follow exactly the very nice proof of Iwabuchi-Ogawa [8] that proved the ill-posedness in H

⁻¹

of the 2-D quadratic heat equation. Note that (1, −1) is the intersection of the two lines s = −α and s = 1 − 2α, this last line corresponding to the scaling critical Sobolev exponent in dimension 2. We need to introduce the rescaled modulation spaces (M

_2,1

)

_N

that are defined for any integer N ≥ 1 by

(M

_2,1

)

_N

:= n

u ∈ S

⁰

( R ) | kuk

_(M2,1)N

< ∞ o where

kuk

_(M2,1)N

:= X

k∈2^N_Z

kˆ uk

_L2(k,k+2^N)

.