Sign-changing solutions of the nonlinear heat equation with persistent singularities

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Sign-changing solutions of the nonlinear heat equation with persistent singularities

Thierry Cazenave, Flavio Dickstein, Ivan Naumkin, Fred B. Weissler

To cite this version:

Thierry Cazenave, Flavio Dickstein, Ivan Naumkin, Fred B. Weissler. Sign-changing solutions of the

nonlinear heat equation with persistent singularities. ESAIM: Control, Optimisation and Calculus of

Variations, EDP Sciences, 2020, 26, pp.126. �10.1051/cocv/2020082�. �hal-02893405�

https://doi.org/10.1051/cocv/2020082

www.esaim-cocv.org

SIGN-CHANGING SOLUTIONS OF THE NONLINEAR HEAT EQUATION WITH PERSISTENT SINGULARITIES

Thierry Cazenave

, Fl´ avio Dickstein

, Ivan Naumkin

and Fred B. Weissler

Abstract. We study the existence of sign-changing solutions to the nonlinear heat equation

=

∆u +

on

,

3, with

, where

=

(

), which are singular at

= 0 on an interval of time. In particular, for certain

0 that can be arbitrarily large, we prove that for any

L

(

in a neighborhood of 0, there exists a local (in time) solution

of the nonlinear heat equation with initial value

, which is sign-changing, bounded at infinity and has the singularity

at the origin in the sense that for

0,

as

0, where

=

(N

2

). These solutions in general are neither stationary nor self-similar.

Mathematics Subject Classification. 35K91, 35K58, 35C06, 35K67, 35A01, 35A21.

Received June 29, 2020. Accepted November 18, 2020.

Dedicated to Enrique Zuazua on the occasion of his 60th birthday.

1. Introduction

In this paper, we study the nonlinear heat equation

∂

u = ∆u + |u|

u (1.1)

∗Research supported by the “Brazilian-French Network in Mathematics”.

∗∗Flavio Dickstein was partially supported by CNPq (Brasil).

∗∗∗Ivan Naumkin is a Fellow of Sistema Nacional de Investigadores. He was partially supported by project PAPIIT IA101820.

Keywords and phrases: Nonlinear heat equation, sign-changing solutions, singular self-similar solutions, singular stationary solutions, persistent singularities.

1 Sorbonne Universit´e, CNRS, Universit´e de Paris, Laboratoire Jacques-Louis Lions, B.C. 187, 4 place Jussieu, 75252 Paris Cedex 05, France.

2 Instituto de Matem´atica, Universidade Federal do Rio de Janeiro, Caixa Postal 68530, 21944-970 Rio de Janeiro, RJ, Brazil.

3 Departamento de F´ısica Matem´atica, Instituto de Investigaciones en Matem´aticas Aplicadas y en Sistemas, Universidad Nacional Aut´onoma de M´exico, Apartado Postal 20-126, Ciudad de M´exico 01000, Mexico.

4 Universit´e Sorbonne Paris Nord, CNRS UMR 7539 LAGA, 99 Avenue J.-B. Cl´ement, 93430 Villetaneuse, France.

*∗∗∗Corresponding author:[email protected]

c

The authors. Published by EDP Sciences, SMAI 2020

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

on R

with

N ≥ 3 and 2

N − 2 < α < α

, (1.2)

where

α

= 4

N − 4 + 2 √

N − 1 . (1.3)

(Note that

< α

<

, see Lem. 2.1 (i).) We are interested in sign-changing solutions of (1.1) which have a singularity at x = 0 on an interval of time.

Positive solutions of (1.1) with a standing or moving singularity have been well studied. The simplest, for all α >

, is the homogeneous stationary solution β

|x|

where

β = 2 α

N − 2 − 2 α

> 0. (1.4)

Moreover, for all

< α <

, (1.1) has a one-parameter family (U

)

⊂ C

( R

\ {0}) of radially sym- metric, positive, singular stationary solutions satisfying |x|

U

(x) → β

as x → 0 and |x|

U

(x) → λ as

|x| → ∞. Furthermore, for this range of α, the family (U

)

and β

|x|

constitute all the positive, radially symmetric, singular stationary solutions of (1.1). See ([21], Prop. 3.1). Under the stronger assumption (1.2), these solutions can be used as prototypes to construct positive solutions of (1.1) with a moving singularity, i.e.

a singularity located at x = ξ(t) for every t in some interval, under appropriate conditions on the function ξ(·).

See [17, 20]. Positive self-similar solutions of (1.1), both forward and backward, with a standing or moving sin- gularity, have also been constructed, see [18, 19]. The finite-time blowup and the long-time asymptotic behavior of positive singular solutions of (1.1) are studied in [7, 16].

Sign changing stationary solutions of equation (1.1) have been less studied. We show here that for all

<

α <

, equation (1.1) has sign-changing, radially symmetric, stationary solutions that are singular at x = 0.

These solutions behave like β

|x|

at the origin and oscillate indefinitely as |x| → ∞. See Proposition 6.1 and Corollary 6.2.

For the same range

< α <

, equation (1.1) also has sign-changing, radially symmetric, self-similar solutions which are singular for all positive time. More precisely, it follows from ([5], Thm. 1.3) that there exist an integer m ≥ 0 and an increasing sequence (µ

)

⊂ (0, ∞), µ

→ ∞ as m → ∞, such that for each µ

there exists a radially symmetric self-similar solution

U (t, x) = t

f |x|

√ t

of (1.1) in the sense of distributions, where the profile f ∈ C

(0, ∞) has exactly m zeros and satisfies f

+ N − 1

r + r

2

f

+ 1

α f + |f |

f = 0. (1.5)

Moreover, r

f (r) → β

as r → 0 and r

f(r) → (−1)

µ

as r → ∞. It follows that |x|

U (t, x) → β

as

|x| → 0 for all t > 0, and U(t, ·) → (−1)

µ

| · |

in L

( R

) as t → 0.

The purpose of this article is to construct sign-changing solutions of (1.1) which are singular at x = 0 for

small positive time, and which are neither stationary nor self-similar. The construction of these solutions is

based on the following perturbation result.

Theorem 1.1. Assume (1.2). Let S > 0 and

U ∈ L

((0, S) × R

) ∩ C((0, S) × ( R

\ {0})) (1.6) satisfy for some constant C

| |x|

U (t, x) − β

| ≤ C h |x|

|x| + √ t

+ |x|

i

, (1.7)

for 0 < t < S and x 6= 0, where

ρ = 2

α − N − 2

2 −

r (N − 2)

4 − β(α + 1), (1.8)

and β is given by (1.4). (ρ > 0 by Lemma 2.1 (iii) below.) Assume further that there exists U

∈ L

( R

) such that

U (t, ·) −→

t→0

U

(·), (1.9)

in L

( R

). Given δ > 0 and u

∈ L

( R

) ∩ L

({|x| > δ}) such that u

(x) = U

(x) a.e. on {|x| < δ},

there exist T ∈ (0, S) and a solution u ∈ L

((0, T ) × R

) ∩ C((0, T ) × ( R

\ {0})) of

∂

u − ∆u = |u|

u − [∂

U − ∆U − |U |

U], (1.10) in the sense of distributions D

((0, T ) × R

), such that

u(t) −→

t→0

u

in L

( R

), (1.11)

and

|u(t, x) − U (t, x)| ≤ C(1 + |x|

), 0 < t < T, x 6= 0, (1.12) where

η = N − 2

2 −

r (N − 2)

4 − β(α + 1) > 0. (1.13)

In Theorem 1.1, the choice of u

is both flexible and rigid. On the one hand, for |x| > δ there is complete freedom to choose u

as long as it is bounded. On the other hand, for |x| < δ, u

must agree precisely with U

, the initial value of the given function U . The following remark gives information on the relationship between U and u, and how u

affects this relationship.

Remark 1.2. (i) In Theorem 1.1, U need not be a solution of (1.1), so that u need not be a solution of (1.1).

However, if U solves (1.1), then so does u.

(ii) In Theorem 1.1, U need not be radially symmetric. Even if U is radially symmetric, u is not radially symmetric if u

is not. (Recall that u

is specified only in the ball of radius δ.)

(iii) The function u cannot be a stationary solution of (1.1), unless u

is a stationary solution of (1.1), by (1.11).

(iv) The function u cannot be a self-similar solution of (1.1), unless u

is homogeneous. Indeed, recall that the initial value of a self-similar solution, if it exists, is always homogeneous.

(v) Since η <

<

by (1.2) and (2.2), it follows from (1.7) and (1.12) that u given by Theorem 1.1 is singular for all 0 < t < T at x = 0, and has the same singular behavior as U(t), i.e.

|x|

u(t, x) −→

x→0

β. (1.14)

In order to use Theorem 1.1 to construct sign-changing solutions of (1.1) with a singularity at x = 0, we consider separately the cases where U is a radially symmetric stationary solution of (1.1), and where U is a radially symmetric self-similar solution of (1.1) with singular profile. This gives the following two theorems.

Theorem 1.3. Assume (1.2). Let U be a radially symmetric, stationary solution of (1.1) that is singular at x = 0. Given δ > 0 and u

∈ L

( R

) ∩ L

({|x| > δ}) such that

u

(x) = U (x) a.e. on {|x| < δ},

there exist T > 0 and a solution u ∈ L

((0, T ) × R

) ∩ C((0, T ) × ( R

\ {0})) of (1.1) in the sense of distributions, such that (1.12) holds and u(0) = u

in the sense (1.11). Moreover, u(t) is singular at x = 0 for all t < T and satisfies (1.14).

We stress the fact that there exist sign-changing stationary solutions U to which Theorem 1.3 applies, by Corollary 6.2. In this case, if δ is sufficiently large, then the solution u is sign-changing for small time by (1.11).

See below for further discussion of this point.

Theorem 1.4. Assume (1.2). Let f ∈ C

(0, ∞) be a solution of the equation (1.5) having the singularity r

f (r) → β

as r → 0, let

µ = lim

r→∞

r

f (r) ∈ R , (1.15)

which exists by Proposition 7.2. Let

U (t, x) = t

f |x|

√ t

, t > 0, x 6= 0, (1.16)

so that U is a self-similar solution of (1.1) by Proposition 7.2. Given δ > 0 and u

∈ L

( R

) ∩ L

({|x| > δ}) such that

u

(x) = µ|x|

a.e. on {|x| < δ},

there exist T > 0 and a solution u ∈ L

((0, T ) × R

) ∩ C((0, T ) × ( R

\ {0})) of (1.1) in the sense of distributions, such that (1.12) holds and u(0) = u

in the sense (1.11). Moreover, u(t) is singular at x = 0 for all t < T and satisfies (1.14).

There exist sign-changing self-similar solutions U to which Theorem 1.4 applies, by Proposition 7.2. In this case, the solution u is necessarily sign-changing for small time, see the discussion below. Moreover, Proposition 7.2 states that (1.15) can be achieved by a sign-changing profile for a sequence µ = µ

→ ∞.

Note that, for a given solution U , Theorems 1.3 and 1.4 produce many different solutions of (1.1) with the same singularity. Indeed, we can choose u

arbitrarily for |x| > δ as long as u

∈ L

({|x| > δ}). That two different choices of u

produce two different solutions of (1.1) follows from (1.11).

We observe that using Remark 1.2 and Theorems 1.3 and 1.4 we do indeed obtain sign-changing solutions

of (1.1) with persistent singularities, which are neither stationary nor self-similar. We may assume that u

is

neither homogeneous, nor a stationary solution of (1.1). That u is sign-changing is of course true by (1.11) if u

is sign-changing, and this is always possible since u

is prescribed only for |x| < δ. Furthermore, in Theorem 1.3, if U is sign-changing and δ is chosen sufficiently large, then u

is necessarily sign-changing. Finally, even if u

> 0, Theorem 1.4 produces sign-changing solutions. Indeed, suppose the profile f is sign-changing (there exist such profiles, see [5], Thm. 1.3), and let τ, ε > 0 be such that f (τ ) = −ε. Given any x

∈ R

such that |x

| = 1, it follows from (1.12) with x = τ √

tx

that

|t

u(t, τ √

tx

) + ε| = t

|u(t, τ √

tx

) − U (t, τ √

tx

)| ≤ C(t

+ τ

t

) ≤ ε 2 for t > 0 small, so that u(t, τ √

tx

) ≤ −

t

< 0 for t > 0 small.

We formalize some of the previous observations with the following corollary to Theorem 1.4.

Corollary 1.5. There exists a sequence (µ

)

⊂ (0, ∞), µ

→ ∞, such that if u

∈ L

( R

) ∩ L

({|x| > 1}) satisfies

u

(x) = µ

|x|

a.e. on {|x| < δ},

for some δ > 0 and n ≥ 1, then there exist T > 0 and a sign-changing solution u ∈ L

((0, T ) × R

) ∩ C((0, T ) × ( R

\ {0})) of (1.1) in the sense of distributions, which is not self-similar (or stationary), such that u(0) = u

in the sense (1.11) and u satisfies (1.14) for all 0 < t < T . In particular, u(t) is singular at x = 0 for all 0 < t < T .

In Theorem 1.4, the solution u(t) has the spatial singularity µ|x|

when t = 0, and the singularity β

|x|

when t > 0. Since µ can be chosen arbitrarily large by Proposition 7.2, we see that the singularity of u at t = 0 can be greater than the singularity at t > 0.

Let α > 0 and let u

∈ L

( R

\ {0}) ∩ L

({|x| > 1}) equal µ|x|

near the origin with µ > 0. If µ is sufficiently large, then there is no positive (possibly singular) solution of (1.1) with the initial value u

. See ([5], Prop. A.1) or ([6], Cor. 2.7). (Note that this is not in contradiction with the results in [17–20], since all the positive, singular solutions of (1.1) constructed there have an initial value u

= u(0) which behaves like β

|x|

near the origin.) On the other hand, if α <

, then there exist sign-changing, local in time solutions (regular for positive time) of (1.1) with the initial value u

. See ([6], Thm. 5.1). It follows from Corollary 1.5 that, at least for certain arbitrarily large µ and under assumption (1.2), there also exist local in time, sign-changing solutions of (1.1) with the initial value u

, which are singular at the origin for positive time.

We observe that for positive data (U ≥ 0 and u

≥ 0), Theorem 1.3 is weaker than ([20], Thm. 1.1). Indeed, in ([20], Thm. 1.1), the singularity of u can move with time, and u

need not be equal to U (0) in a neighborhood of the origin, but sufficiently close to U (0). A technical reason for this difference is that Theorem 1.3 allows sign-changing solutions so that we cannot apply the powerful comparison arguments used in [20].

Also, it is natural to ask if u(t, x) → 0 as |x| → ∞ in Theorems 1.3 and 1.4, assuming u

(x) → 0 as |x| → ∞.

Our construction of the solution u does not answer this question. The analogous property for perturbations of self-similar solutions with regular profile is true, see ([6], Thm. 5.1).

We now describe our strategy to prove Theorem 1.1. We construct u as a perturbation of U in the form u = U + w.

The resulting equation for w is

∂

w − ∆w = |U + w|

(U + w) − |U |

U. (1.17)

The leading term on the right-hand side of (1.17) is (α + 1)|U |

w, which by (1.7) behaves like β(α + 1)|x|

w

near the origin. This makes it delicate to apply a standard perturbation argument to (1.17). It turns out to

be helpful to subtract the term β(α + 1)|x|

w from both sides of the equation, leading to the following heat equation with inverse square potential

∂

w − ∆w − β (α + 1)|x|

w = Mw, (1.18)

where

Mw = |U + w|

(U + w) − |U |

U − β(α + 1)|x|

w. (1.19) We observe that the operator −∆ − β(α + 1)|x|

in (1.18) has good properties only if

β(α + 1) < (N − 2)

4 , (1.20)

the constant in Hardy’s inequality. Inequality (1.20) is equivalent to α < α

, where α

is given by (1.3). (See Lem. 2.1 below.) Note that α

>

, since N > 2. Under the assumption (1.20), the operator H on L

( R

) defined by

( D(H) = {u ∈ H

( R

); ∆u + β (α + 1)|x|

u ∈ L

( R

)}

Hu = ∆u + β (α + 1)|x|

u, u ∈ D(H ) (1.21)

is a negative self-adjoint operator, hence the generator of a C

semigroup of contractions (e

)

, which is an analytic semigroup on L

( R

). Moreover, there exist two constants A > 0 and a > 0 such that the corresponding heat kernel K(t, x, y) satisfies the estimate

0 < K(t, x, y) ≤ At

e

2

at

h(t, x)h(t, y), (1.22)

where

h(t, x) = 1 +

√ t

|x|

, (1.23)

and η is given by (1.13). See ([10], Thm. 1.2), ([12], Thm. 3), ([13], Thm. 3.10).

Using the kernel K we write equation (1.18) in the integral form w(t) =

Z

R^N

K(t, x, y)w

(y) dy + Z

0

Z

R^N

K(t − s, x, y)Mw(s, y) dy ds, (1.24) where w

= u(0) − U (0). Since K is bounded from below by a term similar to the right-hand side of (1.22), it follows that K(t, x, y) has the singularity |x|

as |x| → 0 and similarly in y. Thus we see that the kernel of the operator e

associated with equation (1.18) is more singular than the heat kernel associated with equation (1.17). Of course, the right-hand side of (1.18) is less singular than the right-hand side of (1.17). In fact the worst term in Mw is of order (

|x|+√

t

)

|x|

w, where ω > 0 is given by (3.14) below. (See (4.14), (4.16), (4.17).) At positive times, this term is better than |x|

w. However, as t → 0, it behaves like |x|

w.

This, combined with the singularity of the kernel, excludes the possibility of carrying out a standard contraction

mapping argument based on (1.24). Our solution to this difficulty is taken from [6] and involves a contraction

mapping argument in a class of functions w that are sufficiently small as (t, x) → (0, 0) so as to balance the

singularity of Mw. The key point is to find such a class which is preserved by the iterative process. The fixed

point w thus obtained satisfies the integral equation (1.24) and in fact solves (1.18) in the sense of distributions.

Therefore u = U + w satisfies (1.10) in the sense of distributions.

We do not know if the condition α < α

in Theorem 1.1 is necessary. However, if α > α

(i.e. β(α + 1) >

(N−2)²

4

), then our proof breaks down from the beginning, since in this case the linear heat equation with potential β (α + 1)|x|

is ill-posed, see [1, 23].

The results in this paper are motivated by our article [6] where we prove an analogue of Theorem 1.4 where the self-similar solution U has a regular profile. Such self-similar solutions have a singularity at (t, x) = (0, 0), which introduces some limitations in our results. In particular, we are led to consider in [6] initial values w

that equal U (0, ·) in a neighborhood of the origin. The same limitation appears here, and we do not know if it is technical or not.

The rest of this paper is organized as follows. Section 2 is devoted to some properties of the parameters we use throughout the paper. In Section 3, we establish some specific estimates for the nonhomogeneous heat equation with inverse square potential. In Section 4, we introduce the setting for the fixed point argument that we use in Section 5 to prove Theorem 1.1. In Section 6, we give a description of the radially symmetric, stationary solutions of (1.1), showing the existence of sign-changing solutions. In Section 7, we deduce Theorems 1.3 and 1.4 from Theorem 1.1. We collect in Appendix A some general properties, concerning mostly the linear heat equation with inverse square potential, that we use in the paper and for which we did not find a reference.

2. Elementary inequalities

This section is devoted to the following elementary properties.

Lemma 2.1. Suppose N ≥ 3, and let α

> 0 be defined by (1.3) and Λ ∈ R by Λ = 1

α − N − 2 4

− N − 2

2 − 1

α

. (2.1)

The following properties hold.

(i) α

satisfies

2

N − 2 < α

< 4

N − 2 . (2.2)

(ii) For 0 < α <

, the three properties

α < α

, (2.3)

Λ > 0, (2.4)

and (1.20) (where β is given by (1.4)), are equivalent.

(iii) Let 0 < α < α

and let Λ > 0 be given by (2.1). If µ

, µ

are defined by µ

= 1

α − N − 2

4 − √

Λ, (2.5)

µ

= 1

α − N − 2

4 + √

Λ, (2.6)

then 0 < µ

< µ

. Moreover,

− β (α + 1) > 0 by Property (ii) above, and if ρ is given by (1.8), then

ρ = 2µ

(2.7)

so that ρ > 0.

Proof. Since

2 √

N − 1 = p

N

− (N − 2)

we see that 2 √

N − 1 < N . Therefore N − 4 + 2 √

N − 1 < 2N − 4, hence α

>

. Moreover, 2 √

N − 1 > 2, so that N − 4 + 2 √

N − 1 > N − 2, hence α

<

. This proves Property (2.2).

We now prove Property (ii). We have

(N − 2)

4 − β(α + 1) = 4Λ (2.8)

so that (2.4) and (1.20) are equivalent. We write 1 α

= N − 2

4 +

√ N − 1 − 1 2 and

1 α = 1

α

+ ε so that

Λ = ε

+ ε √ N − 1.

It follows that Λ > 0 if and only if either ε > 0, i.e. α < α

, or else ε < − √

N − 1. In this last case, 1

α < 1 α

− √

N − 1 = N − 2 4 − 1

2 −

√ N − 1

2 < N − 2 4 so that α >

. Thus we see that (2.3) and (2.4) are equivalent.

To prove Property (iii), we observe that µ

> µ

. Moreover, Λ < (

−

)

. Thus √

Λ <

−

so that µ

> 0. Finally, formula (2.8) yields ρ = 2µ

.

3. The nonhomogeneous heat equation with inverse square potential

In this section, we assume (1.2), and we use the operator H defined by (1.21) and the corresponding semigroup (e

)

with the kernel K(t, x, y). We let ρ, η > 0 be defined by (1.8) and (1.13), respectively. This section is devoted to estimates of

Z

e

f (s) ds,

for some specific right-hand sides f .

Lemma 3.1. Let c ≥ 1, 0 ≤ b ≤ 2 and κ ≥ 0 (κ > 0 if b = 2) satisfy

b + (c − 1)η − κ < 2. (3.1)

Since 2η < N − 2 by (1.13), we have b + (c + 1)η − κ < N , and we fix 0 ≤ ε < min n 1

4 , N − (b + (1 + c)η − κ) o

. (3.2)

Define Ψ(t) for t > 0 by

Ψ(t, x) = h(t, x)

|x|

|x|

|x| + √ t

, x ∈ R

(3.3)

where h is given by (1.23). It follows that for every m > κ − 2 there exists B

> 0 such that for all 0 < r ≤ ∞, Z

0

e

1

Ψ(s, ·)s

ds ≤ B

r

t

|x|

h(t, x), (3.4) for all t > 0 and x ∈ R

, with B

→ 0 as m → ∞.

Proof. We write using (1.22),

e

1

Ψ(s)(x) ≤ Ah(t − s, x)(t − s)

Z

R^N

e

2

a(t−s)

1

h(t − s, y)Ψ(s, y) dy

= A(aπ)

h(t − s, x)e

[1

h(t − s, ·)Ψ(s, ·)](x), so that

e

1

Ψ(s) ≤ Ch(t − s, x)e

h

1

h(t − s, ·)h(s, ·)

| · |

| · |

| · | + √ s

i . Since

h(t − s, y)h(s, y)

≤ C

1 + (t − s)

|y|

1 + s

|y|

≤ C

1 + (t − s)

|y|

+ s

|y|

+ (t − s)

s

|y|

,

we deduce that

e

1

Ψ(s) ≤ Ch(t − s, x)[I

+ (t − s)

I

+ s

I

+ (t − s)

s

I

] (3.5) where

I

= e

1

| · |

| · |

| · | + √ s

, I

= e

1

| · |

| · |

| · | + √ s

, I

= e

1

| · |

| · |

| · | + √ s

, I

= e

1

| · |

| · |

| · | + √ s

.

We note that

1

≤ r

|y|

, (3.6)

and we recall that if 0 ≤ p < N , then

e

| · |

≤ C(t + |x|

)

. (3.7) See e.g. ([4], Cor. 8.3).

Let κ

= min{κ, b}. We have I

≤ r

e

| · |

| · |

| · | + √ s

≤ r

s

e

(| · |

)

≤ Cr

s

(t − s + |x|

)

≤ Cr

|x|

s

(t − s)

,

(3.8)

where we used (3.6), (3.7) and the property 0 ≤ b + ε − κ

≤ b + ε ≤

< N.

Similarly, letting κ

= min{κ, b + η},

I

≤ r

s

e

(| · |

) ≤ Cr

s

(t − s + |x|

)

≤ Cr

|x|

s

(t − s)

, (3.9) where we used (3.6), (3.7) and the property

0 ≤ b + η + ε − κ

≤ b + η + ε ≤ 9

4 + η ≤ 9

4 + N − 2 2 < N.

Next, setting κ

= min{κ, b + cη},

I

≤ r

s

e

(| · |

) ≤ Cr

s

(t − s + |x|

)

≤ Cr

|x|

s

(t − s)

(t − s + |x|

)

≤ Cr

|x|

s

(t − s)

1

|x|

h(t − s, x) .

(3.10)

Here we used (3.6), (3.7) and the property 0 ≤ b +cη +ε −κ

< N . The last inequality is immediate if κ

= b + cη;

and if κ

= κ, it follows from b + cη + ε − κ ≤ b + (1 + c)η + ε − κ < N by (3.2).

Furthermore, setting κ

= min{κ, b + (1 + c)η},

I

≤ r

s

e

(| · |

) ≤ Cr

s

(t − s + |x|

)

≤ Cr

|x|

s

(t − s)

(t − s + |x|

)

≤ Cr

|x|

s

(t − s)

1

|x|

h(t − s, x) . (3.11) In (3.11) we used (3.6), (3.7) and the property 0 ≤ b + (1 + c)η + ε − κ

< N. The last inequality is immediate if κ

= b + (1 + c)η; and if κ

= κ, it follows from (3.2). We deduce from (3.5), (3.8), (3.9), (3.10) and (3.11) that

r

|x|

e

1

Ψ(s) ≤Ch(t − s, x) h

s

(t − s)

+ s

(t − s)

i + C|x|

h

s

(t − s)

+ s

(t − s)

i

.

Since h(t − s, x) ≤ h(t, x) and |x|

≤ t

h(t, x), we deduce that r

|x|

e

1

Ψ(s) ≤Ch(t, x) h

s

(t − s)

+ s

(t − s)

+t

s

(t − s)

+ s

(t − s)

i . Multiplying by s

and integrating in s ∈ (0, t), we obtain (3.4) with

B

= C

2

X

j=1

Z

σ

m−κj

2

(1 − σ)

b−κj

2

dσ + C

4

X

j=3

Z

σ

m+ηc−κj

2

(1 − σ)

b+(c−1)η−κj

2

dσ. (3.12)

The integrals in (3.12) are finite. Indeed, for all j ∈ {1, 2, 3, 4}, min n m − κ

2 , m + ηc − κ

2

o ≥ m − κ 2 > −1.

Moreover,

<

≤ 1 for j = 1, 2. Furthermore, if κ

= κ for j = 3 or j = 4, then b + (c − 1)η − κ

2 = b + (c − 1)η − κ 2 < 1

by (3.1). If κ

= b + cη, then

= −

< 0 < 1. Similarly if κ

= b + (1 + c)η, then

=

−η < 0 < 1. Finally, the property B

→ 0 follows by dominated convergence.

Lemma 3.2. Let

(b

, c

, κ

) = (2, 1, ρ), (b

, c

, κ

) = (0, 1 + α, 0), (b

, c

, κ

) = (

, 2, 0), (b

, c

, κ

) = (2, 1, αρ).

It follows that b

< 2 and that the triplets (b

, c

, κ

) satisfy (3.1) for j = 1, 2, 3, 4.

Proof. We first note that by (1.13)

η < N − 2 2 < 2

α , (3.13)

since α <

. Next,

b

+ (c

− 1)η − κ

= 2 − ρ < 2.

For j = 2,

b

+ (c

− 1)η − κ

= αη < 2 by (3.13). For j = 3,

b

+ (c

− 1)η − κ

= 2 − 2

α + η < 2

by (3.13). For j = 4,

b

+ (c

− 1)η − κ

= 2 − αρ < 2.

This completes the proof.

Lemma 3.3. Let

ω = ρ min{1, α} (3.14)

and let

g(t, x) = |x|

|x| + √ t

|x|

+ (M h(t, x))

+ M e g(t, x) (3.15)

for a.a. t > 0, x ∈ R

, where M ≥ 0, h is given by (1.23), and

e g(t, x) =

( 0 α ≤ 1

|x|

h(t, x) α > 1.

It follows that there exists a constant A such that

g(t, x) ≤ A(1 + |x|

) (3.16)

g(t, x)h(s, x) ≤ A(1 + |x|

) (3.17)

g(t, x)h(s, x)h(σ, x) ≤ A(1 + |x|

) (3.18)

for a.a. t, s, σ ∈ (0, 1) and x ∈ R

, where

ϑ = N − 2

2 − η > 0. (3.19)

Moreover, there exists ε

> 0 such that if 0 ≤ ε < ε

is fixed, then for every m > max{ρ, αρ} − 2, Z

0

e

1

g(s)h(s)s

ds ≤ R

r

t

|x|

h(t, x) (3.20) where

R

−→

m→∞

0. (3.21)

Proof. We first prove (3.16)–(3.18). It follows from (1.23) and (3.15) that for 0 ≤ t ≤ 1

g(t, x) ≤ C(1 + |x|

+ |x|

+ |x|

) (3.22)

g(t, x)h(s, x) ≤ C(1 + |x|

+ |x|

+ |x|

) (3.23)

g(t, x)h(s, x)h(σ, x) ≤ C(1 + |x|

+ |x|

+ |x|

) (3.24)

Since

α < 4

N − 2 and η = N − 2

2 − ϑ < N − 2

2 , (3.25)

we see that ηα < 2 and

− η > 0, so that (3.16) follows from (3.22). Using again (3.25), we obtain 2 + η <

, (α + 1)η <

and 2 −

+ 2η <

, hence (3.17) follows from (3.23). Moreover, (3.25) yields

2 + 2η = N − 2ϑ ≤ N − ϑ, (α + 2)η ≤ N − ϑ 2N

N − 2 ≤ N − ϑ, 2 − 2

α + 3η ≤ N − 3ϑ ≤ N − ϑ, so that (3.18) follows from (3.24).

Estimate (3.20) follows from Lemma 3.1. For the term (

|x|+√

t

)

|x|

we apply Lemma 3.1 with (b, c, κ) = (2, 1, ρ) if ω = ρ and (b, c, κ) = (2, 1, αρ) if ω = αρ. This is possible by Lemma 3.2. For the term h

we apply Lemma 3.1 with (b, c, κ) = (0, 1 + α, 0). This is again possible by Lemma 3.2. Finally for the term e g, we need only consider the case α > 1 and we apply Lemma 3.1 with (b, c, κ) = (

, 2, 0), which is possible by Lemma 3.2.

4. The setting for the fixed-point argument

In this section, we assume (1.2), and we use the operator H defined by (1.21) and the corresponding semigroup (e

)

with the kernel K(t, x, y). We introduce the framework for the fixed-point argument that we use for the proof of Theorem 1.1.

We begin with the definition of several auxiliary functions. Let δ > 0, and set

a

= 2

δ (4.1)

for j ≥ 0. Define the sequence (χ

)

⊂ L

( R

) by

χ

(x) =

( 0 |x| ≤ a

1 |x| > a

. (4.2)

Given T > 0 and an integer m ≥ 1, we set

Θ(t, x) = h(t, x)



t

+

m

X

j=1

t

χ



 , (4.3)

for 0 ≤ t ≤ T and x ∈ R

, where h is given by (1.23). Given M > 0, we define

E = {w ∈ L

((0, T ) × R

); |w| ≤ M Θ}, (4.4) and

d(w, z) =

w − z Θ

((0,T)×R^N)

, w, z ∈ E, (4.5)

so that (E, d) is a complete metric space.

Lemma 4.1. Let δ > 0. With the notation (4.1)–(4.2), it follows that there exists B

> 0 such that for all j ≥ 0 and all 0 ≤ t, s ≤ 1

e

χ

≤ B

h(t, x) e

a2 j+1

2at

+ χ

(4.6) where a is the constant in (1.22) and h is given by (1.23).

Proof. Applying (1.22), Z

R^N

K(t, x, y)χ

(y) dy ≤ Ct

h(t, x) Z

{|y|>aj}

e

2

at

h(t, y) dy. (4.7)

We also write

Z

R^N

K(t, x, y)χ

(y) dy ≤ Ct

h(t, x) Z

R^N

e

2

at

h(t, x − y) dy. (4.8)

We deduce from (4.8) that Z

R^N

K(t, x, y)χ

(y) dy ≤ Ct

h(t, x) Z

R^N

e

2 at

1 +

√ t

|x − y|

dy

= Ch(t, x) Z

R^N

e

2 a

1 + 1

|(x/ √ t) − y|

dy.

Since η < N by (1.13), we see that Z

R^N

e

2 a

1 + 1

|z − y|

≤ 2

Z

R^N

e

2 a

+ 2

Z

|z−y|<1

|z − y|

≤ C

independent of z ∈ R

, and it follows that Z

R^N

K(t, x, y)χ

(y) dy ≤ Ch(t, x) for all x ∈ R

. In particular, if |x| > a

, then

Z

R^N

K(t, x, y)χ

(y) dy ≤ Ch(t, x)χ

(x). (4.9) Next, if |x| ≤ a

and |y| ≥ a

, then

|x − y| ≥ |y| − a

= |y| − 1 2 a

≥ 1

2 |y| ≥ a

and so

e

2

at

= e

2 2at

e

2

2at

≤ e

2 8at

e

a2 j+1

2at

. (4.10)

If |x| ≤ a

, then we deduce from (4.7), (4.10), and h(t, ·) ≤ h(1, ·) that Z

R^N

K(t, x, y)χ

(y) dy ≤ Ch(t, x)e

a2 j+1 2at

t

Z

R^N

e

2

8at

h(1, y) dy ≤ Ch(t, x)e

a2 j+1

2at

. (4.11) Then (4.6) follows from (4.9) and (4.11).

Lemma 4.2. Let S > 0 and let U ∈ L

((0, S) × R

) + L

((0, S) × R

) satisfy (1.7) for a.a. 0 < t < S and x 6= 0, where ρ is given by (1.8). Let T ≤ S,

0 < T < 1

4 , (4.12)

δ > 0, M > 0, and let E be defined by (4.4). It follows that Mw ∈ L

((0, T ) × R

) for all w ∈ E, where Mw is defined by (1.19). Moreover,

|Mw(t, ·) − Mz(t, ·)| ≤ B

|w(t, ·) − z(t, ·)|g(t, ·), (4.13) for all w, z ∈ E , where g(t, x) is given by (3.15) and B

is independent of T , m, M , w and z.

Proof. Let w, z ∈ E. Set

f (s) = |s|

s and define

Z(x) = β

|x|

, V = U − Z. (4.14)

It follows that

Mw = f (Z + V + w) − f (Z + V ) − f

(Z)w, so that

Mw − Mz = f (Z + V + w) − f (Z + V + z) − f

(Z)(w − z)

= (w − z) Z

0

[f

(Z + V + sw + (1 − s)z) − f

(Z)] ds. (4.15) Since f

(s) = (α + 1)|s|

and

| |x|

− |y|

| ≤

( α(|x|

+ |y|

)|x − y| if α ≥ 1,

|x − y|

if 0 < α ≤ 1, for all x, y ∈ R , we deduce from (4.15) that

|Mw − Mz| ≤ C|w − z| ×

( |V |

+ |w|

+ |z|

+ Z

(|V | + |w| + |z|) if α ≥ 1,

|V |

+ |w|

+ |z|

if 0 < α ≤ 1. (4.16)

By (1.7)

|V (t, x)| ≤ C h |x|

|x| + √ t

|x|

+ 1 i

. (4.17)

On the other hand, (4.3) and (4.12) yield

|w| + |z| ≤ 2M Θ(t, x) ≤ 2M h(t, x)

m

X

j=0

t

< 4Mh(t, x). (4.18)

if t < T . From (4.17) and (4.18) we get (recall that h ≥ 1)

|V |

+ |w|

+ |z|

≤ C |x|

|x| + √ t

|x|

+ (Mh(t, x))

(4.19)

and

Z

(|V | + |w| + |z|) ≤ C |x|

|x| + √ t

|x|

+ r

M h(t, x)

. (4.20)

Estimate (4.13) follows from (4.16), (4.19), (4.20) and (3.15).

Lemma 4.3. Let δ > 0, T > 0, and m > max{ρ, αρ} − 2, m ≥ 1. Let U ∈ L

((0, T ) × R

) + L

((0, T ) × R

) satisfy (1.7) for a.a. 0 < t < T and x 6= 0. Let χ

defined by (4.2) and Θ by (4.3). Suppose w

∈ L

( R

) satisfies |w

| ≤ Cχ

and w ∈ L

((0, T ) × R

) satisfies |w| ≤ CΘ for some constant C. It follows that

W (t, x) =: e

w

+ Z

0

e

Mw(s) ds, (4.21)

where M is given by (1.19), is well defined and |W | ≤ Ch e with C > e 0 and h given by (1.23). In particular, W ∈ L

((0, T ) × R

) + L

((0, T ) × R

).

Moreover, |x|

W ∈ L

((0, T ) × R

) and

∂

W − ∆W − β(α + 1)|x|

W = Mw (4.22)

in D

((0, T ) × R

). In addition, W, Mw ∈ L

((0, T ) × {|x| > ε}) for every ε > 0.

Proof. The contribution of w

is estimated by (4.6). Next, using (4.13) with z = 0, and the inequality Θ(s, y) ≤ s

+ C

χ

(y), we have

|Mw(s, y)| ≤ B

g(s, y)|w(s, y)| ≤ CB

g(s, y)Θ(s, y)

≤ CB

s

g(s, y) + CB

C

χ

(y)g(s, y)

≤ C

s

g(s, y) + C

χ

(y),

(4.23)

for some constants C

, C

. (In the last inequality, we used the fact that g defined by (3.15) is bounded on the

support of χ

.) Since h ≥ 1, the contribution of s

g(s, y) is estimated by (3.20) with ε = 0 and r = ∞; and

the contribution of χ

is estimated by (4.6) and the inequality h(t − s, x) ≤ h(t, x). It follows that |W | ≤ Ch,

and the L

regularity of W follows from h(t, x) ≤ C + C|x|

and η <

. The same estimate also shows that

|x|

W ∈ L

((0, T ) × R

). That W, Mw ∈ L

((0, T ) × {|x| > ε}) for every ε > 0 follows from |W | ≤ Ch and from (4.13) with z = 0.

We now prove (4.22). The term corresponding to w

in (4.21) satisfies the homogeneous equation in D

((0, T ) × R

) by Lemma A.4, so we now assume w

= 0. We fix a function

ζ ∈ C

((0, T ) × R

).

Next, let ξ ∈ C

( R

) satisfy 0 ≤ ξ ≤ 1, ξ(x) = 1 for |x| ≤ 1 and ξ(x) = 0 for |x| ≥ 2, and set ψ

(x) = ξ(nx), θ

(x) = 1 − ξ x

n

, ρ

(x) = 1 − ψ

(x) − θ

(x), for x ∈ R

and n ≥ 1. It follows that

kψ

k

−→

n→∞

0 (4.24)

for all 1 ≤ r < ∞, that ρ

is supported in {

≤ |x| ≤ 2n}, and that 0 ≤ θ

≤ 1 and θ

(x) = 0 for |x| ≤ n. We write

W = V

+ W

+ Z

, where

V

= Z

0

e

(ψ

Mw)(s, ·) ds, W

=

Z

e

(ρ

Mw)(s, ·) ds, Z

=

Z

e

(θ

Mw)(s, ·) ds.

Since |Mw| ≤ CgΘ and Θ ≤ Ch, it follows from (3.17) that

|Mw| ≤ C(1 + |x|

). (4.25)

In particular, we see that ρ

Mw ∈ L

((0, T ), L

( R

)). Applying Lemma A.5, we deduce that Z

0

Z

R^N

W

(−∂

ζ − ∆ζ − β(α + 1)|x|

ζ) = Z

0

Z

R^N

ζρ

Mw.

Since N ≥ 3, the right-hand side of (4.25) is in L

( R

), so that ζMw ∈ L

((0, T ) × R

). Since 0 ≤ ρ

≤ 1 and ρ

→ 1 a.e., we deduce by dominated convergence that

Z

Z

R^N

W

(−∂

ζ − ∆ζ − β(α + 1)|x|

ζ) −→

n→∞

Z

Z

R^N

ζMw. (4.26)

Next, we let 0 < r < 2

δ, so that 1

χ

≡ 0 by (4.2); and so it follows from (4.23) that

1

|Mw(s, y)| ≤ C

s

1

g(s, y).

Since ψ

≤ 1

n}

and h ≥ 1, we deduce that for n ≥

ψ

|Mw(s, y)| ≤ C

s

1

n}

g(s, y) ≤ C

s

1

n}

g(s, y)h(s, y). (4.27) We fix 0 < ε ≤ ε

, where ε

is given by Lemma 3.3, sufficiently small so that

2 + η + ε < N. (4.28)

(This is possible, since η <

.) It follows from (4.27), (3.20) and (1.23) that V

(t, x) ≤ Cn

|x|

h(t, x) ≤ Cn

|x|

, on the support of ζ, and we conclude using (4.28) that

Z

Z

R^N

V

(−∂

ζ − ∆ζ − β(α + 1)|x|

ζ) −→

n→∞

0. (4.29)

Moreover, 0 ≤ θ

≤ 1

, so that by Lemma A.3

sup

0≤t≤T

1

h(T ) Z

(t)

≤ C Z

0

1 +

√ s n

e

2

s

ds −→

n→∞

0.

Since

≥ ε > 0 on the support of ζ, and since the support of ζ is included in {|x| ≤

)} for n large, we deduce that

Z

Z

R^N

Z

(−∂

ζ − ∆ζ − β (α + 1)|x|

ζ) −→

n→∞

0. (4.30)

Applying (4.26), (4.29) and (4.30), we see that Z

0

Z

R^N

W (−∂

ζ − ∆ζ − β(α + 1)|x|

ζ) = Z

0

Z

R^N

ζMw,

Since ζ ∈ C

((0, T ) × R

) is arbitrary, this proves that W is a solution of (4.22) in D

((0, T ) × R

).

5. Proof of Theorem 1.1

We prove Theorem 1.1 by using a fixed point argument. We set w

= u

− U (0), so that

w

∈ L

( R

) and w

= 0 on {|x| < δ}. (5.1) We let

M = 2B

kw

k

, (5.2)