Fractional heat equations with subcritical absorption having a measure as initial data

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Fractional heat equations with subcritical absorption having a measure as initial data

Huyuan Chen, Laurent Veron, Ying Wang

To cite this version:

Huyuan Chen, Laurent Veron, Ying Wang. Fractional heat equations with subcritical absorption having a measure as initial data. 2015. �hal-00937420v3�

Fractional heat equations with subcritical absorption having a measure as initial data

Huyuan Chen¹

Department of Mathematics, Jiangxi Normal University, Nanchang 330022, China

Laurent V´eron²

Laboratoire de Math´ematiques et Physique Th´eorique Universit´e Fran¸cois Rabelais, Tours, France

Ying Wang³

Departamento de Ingenier´ıa Matem´atica Universidad de Chile, Chile

Abstract

We study existence and uniqueness of weak solutions to (F)∂tu+ (−∆)^αu+

h(t, u) = 0 in (0,∞)×R^N, with initial condition u(0,·) = ν in R^N, where N ≥ 2, the operator (−∆)^α is the fractional Laplacian with α ∈ (0,1), ν is a bounded Radon measure and h : (0,∞)×R→ R is a continuous function satisfying a subcritical integrability condition.

In particular, ifh(t, u) =t^βu^p withβ >−1 and 0< p < p^∗_β := 1 +^2α(1+β)_N , we prove that there exists a unique weak solutionukto (F) withν =kδ0, where δ0 is the Dirac mass at the origin. We obtain thatuk → ∞in (0,∞)×R^N as k → ∞ for p∈(0,1] and the limit of uk exists as k→ ∞ when 1< p < p^∗_β, we denote it by u∞. When 1 +^2α(1+β)_N+2α := p^∗∗_β < p < p^∗_β, u∞ is the minimal self-similar solution of (F)^∞ ∂tu+ (−∆)^αu+t^βu^p= 0 in (0,∞)×R^N with the initial conditionu(0,·) = 0 inR^N \ {0}and it satisfies u^∞(0, x) = 0 forx6= 0.

While if 1< p < p^∗∗_β , thenu^∞≡Up, whereUp is the maximal solution of the differential equationy^′+t^βy^p= 0 onR₊.

Contents

1 Introduction 2

2 Linear estimates 8

2.1 The Marcinkiewicz spaces . . . 8 2.2 The non-homogeneous problem . . . 12

1chenhuyuan@yeah.net

2Laurent.Veron@lmpt.univ-tours.fr

3yingwang00@126.com

3 Proof of Theorem ?? 19

4 Dirac mass as initial data 27

5 Self-similar and very singular solutions 31 5.1 The case 1 +^2α(1+β)_N+2α < p <1 +^2α(1+β)_N . . . 33 5.2 The case 1< p <1 +^2α(1+β)_N+2α . . . 39 5.3 The self-similar equation . . . 41 Key words: Fractional heat equation, Radon measure, Dirac mass, Self-similar solution, Very singular solution

MSC2010: 35R06, 35K05, 35R11

1 Introduction

Let h : (0,∞)×R → R be a continuous function and Q_∞ = (0,∞)×R^N with N ≥ 2. The first object of this paper is to consider existence and uniqueness of weak solutions to fractional heat equations

∂_tu+ (−∆)^αu+h(t, u) = 0 in Q_∞,

u(0,·) =ν in R^N, (1.1)

whereνbelongs to the spaceM^b(R^N) of bounded Radon measures inR^N and (−∆)^α (0< α <1) is the fractional Laplacian defined by

(−∆)^αu(t, x) = lim

ǫ→0⁺(−∆)^α_ǫu(t, x), where, for ǫ >0,

(−∆)^α_ǫu(t, x) = Z

RN

u(t, x)−u(t, z)

|z−x|^N+2α χ_ǫ(|x−z|)dz and

χ_ǫ(r) =

(0 if r∈[0, ǫ], 1 if r > ǫ.

In a pioneering work, Brezis and Friedman [6] have studied the semilinear heat equation with measure as initial data

∂_tu−∆u+u^p = 0 in Q_∞,

u(0,·) =kδ0 in R^N, (1.2)

where k > 0 and δ0 is the Dirac mass at the origin. They proved that if 1< p <

(N + 2)/N, then for every k > 0 there exists a unique solutionu_k to (1.2). When p≥(N + 2)/N, problem (1.2) has no solution and even more, they proved that no nontrivial solution of the above equation vanishing onR^N\{0}att= 0 exists. When 1< p <1 + _N², Brezis, Peletier and Terman used a dynamical system technique in [7] to prove the existence of avery singular solution u_s to

∂_tu−∆u+u^p = 0 in Q_∞, (1.3)

vanishing at t= 0 onR^N\ {0}. This functionu_s is self-similar, i.e. expressed under the form

u_s(t, x) =t^−p−11 f √|x|

t

, (1.4)

and f is uniquely determined by the following conditions f^′′+

N−1

η +¹₂η

f^′+_p−1¹ f −f^p = 0 on R₊ f >0 and f is smooth on R₊

f^′(0) = 0 and lim_η→∞η^p−12 f(η) = 0.

(1.5)

Furthermore, it satisfies

f(η) =c₁e^−η2η^{p−12 −N}{1−O(|x|⁻²)} asη→ ∞

for some c₁ >0. Later on, Kamin and Peletier in [21] proved that the sequence of weak solutions u_k converges to the very singular solution us ask→ ∞. After that, Marcus and V´eron in [23] studied the equation in the framework of the initial trace theory. They pointed out the role of the very singular solution of (1.3) in the study of the singular set of the initial trace, showing in particular that it is the unique positive solution of (1.3) satisfying

t→0lim Z

Bǫ

u(t, x)dx=∞, ∀ǫ >0, B_ǫ =B_ǫ(0), (1.6) and

t→0lim Z

K

u(t, x)dx= 0 ,∀K⊂R^N \ {0}, K compact. (1.7) If one replaces u^p by t^βu^p with p ∈(1,1 + ^2(1+β)_N ), these results were extended by Marcus and V´eron (β ≥0) in [24] and then Al Sayed and V´eron (β > −1) in [1].

The initial data problem with measure and general absorption term

∂_tu−∆u+h(t, x, u) = 0 in (0,T)×Ω, u= 0 in (0, T)×∂Ω, u(0,·) =ν in Ω,

(1.8)

in a bounded domain Ω of R^N, has been studied by Marcus and V´eron in [24] in the framework of the initial trace theory. They proved that the following general integrability condition on h

0≤|h(t, x, r)|≤˜h(t)f(|r|) ,∀(x, t, r)∈Ω×R₊×R Z _T

0

˜h(t)f(σt^N2)t^−N2dt <∞ ,∀σ >0 either ˜h(t) =t^α with α≥0 or f is convex,

(1.9)

in order that the problem has a unique solution for any bounded measure. In the particular case with h(t, x, r) = t^β|u|^p−1u, it is fulfilled if 1 < p < 1 + ^2(1+β)_N and β >−1, and the very singular solution exists in this range of values.

Motivated by a growing number of applications in physics and by important links on the theory of L´evy process, semilinear fractional equations has been attracted much interest in last few years, (see e.g. [8, 9, 10, 12, 14, 17, 18, 19]). Recently, in [15] we obtained the existence and uniqueness of a weak solution to semilinear fractional elliptic equation

(−∆)^αu+f(u) =ν in Ω,

u= 0 in Ω^c, (1.10)

when ν is a Radon measure and f satisfies a subcritical integrability condition. In [14] we studied the the different types of isolated singularities whenf(u) =u^p where 1< p < _N−2α^N . In particular, assuming that 0∈Ω, we proved that the sequence of solutions {u_k} (k ∈N) of (1.10), with ν =kδ0 converges to infinity when k → ∞, if p ∈ (0,1 + ^2α_N) and it converges to a solution with a strong singularity at 0 if p∈(1 + ^2α_N,_N^N_−2α).

One purpose of this paper is to study the existence and uniqueness of weak solutions to semilinear fractional heat equation (1.1) in a measure framework. We first make precise the notion of weak solution of (1.1) that we will use in this note.

Definition 1.1 We say that u is a weak solution of (1.1), if for any T > 0, u ∈ L¹(Q_T), h(t, u)∈L¹(Q_T) and

Z

QT

(u(t, x)[−∂_tξ(t, x) + (−∆)^αξ(t, x)] +h(t, u)ξ(t, x))dxdt

= Z

R^N

ξ(0, x)dν− Z

R^N

ξ(T, x)u(T, x)dx ∀ξ∈Y_α,T,

(1.11)

where Q_T = (0, T)×R^N and Y_α,T is a space of functions ξ : [0, T]×R^N → R satisfying

(i) kξkL¹(QT)+kξkL^∞(QT)+k∂tξkL^∞(QT)+k(−∆)^αξkL^∞(QT)<+∞; (ii) for t∈(0, T), there exist M >0 andǫ₀ >0 such that for all ǫ∈(0, ǫ₀],

k(−∆)^α_ǫξ(t,·)kL^∞(RN)≤M.

Before stating our main theorems, we introduce the subcritical integrability con- dition for the nonlinearity h, that is,

(H) (i) The function h : (0,∞)×R → R is continuous and for any t∈ (0,∞), h(t,0) = 0 andh(t, r₁)≥h(t, r₂) if r₁ ≥r₂.

(ii) There existβ >−1 and a continuous, nondecreasing function g :R₊ → R₊ such that

|h(t, r)| ≤t^βg(|r|) ∀(t, r)∈(0,∞)×R

and Z _+∞

1

g(s)s^{−1−p∗β}ds <+∞, (1.12) where

p^∗_β = 1 + 2α(1 +β)

N . (1.13)

We denote by H_α : (0,∞) ×R^N ×R^N → R₊ the heat kernel for (−∆)^α in (0,∞)×R^N, byH_α[ν] the associated heat potential of ν ∈M^b(R^N), defined by

H_α[ν](t, x) = Z

R^N

H_α(t, x, y)dν(y)

and byHα[µ] the Duhamel operator defined for (t, x)∈Q_T and anyµ∈L¹(Q_T) by H^α[µ](t, x) =

Z t 0

H_α[µ(s, .)](t−s, x)ds= Z t

0

Z

RN

Hα(t−s, x, y)µ(s, y)dyds.

Now we state our first theorem as follows.

Theorem 1.1 Assume that ν ∈ M^b(R^N) and the function h satisfies (H). Then problem (1.1) admits a unique weak solution u_ν such that

H_α[ν]− H^α[h(.,H_α[ν+])]≤uν ≤H_α[ν]− H^α[h(.,−H_α[ν−])] inQ∞, (1.14) where ν₊ and ν₋ are respectively the positive and negative part in the Jordan de- composition of ν. Furthermore,

(i) if ν is nonnegative, so is u_ν;

(ii) the mapping: ν 7→ uν is increasing and stable in the sense that if {νn} is a sequence of positive bounded Radon measures converging toν in the weak sense of measures, then {u_νn} converges to u_ν locally uniformly inQ_∞.

According to Theorem 1.1, there exists a unique positive weak solution u_k to

∂_tu+ (−∆)^αu+t^βu^p = 0 in Q_∞,

u(0,·) =kδ0 in R^N, (1.15) where β >−1,k >0 and p∈(0, p^∗_β). We observe that u_k → ∞ in (0,∞)×R^N as k→ ∞forp∈(0,1], see Proposition 4.2 for details. Our next interest in this paper is to study the limit of u_k ask→ ∞ for p∈(1, p^∗_β), which exists since {u_k}k is an increasing sequence of functions, bounded by

1+β p−1

_p−1¹

t^−1+βp−1, and we set u∞= lim

k→∞u_k in Q∞. (1.16)

Actually, u_∞ and {u_k}k are classical solutions to equation

∂_tu+ (−∆)^αu+t^βu^p = 0 in Q_∞, (1.17) see Proposition 4.3 for details.

Definition 1.2 (i) A solution u of (1.17) is called a self-similar solution if u(t, x) =t^−1+βp−1u(1, t^−2α1 x) (t, x)∈Q_∞.

(ii)A solutionuof (1.17) is called a very singular solution if it vanishes onR^N\{0} at t= 0 and

t→0lim⁺

u(t,0)

Γ_α(t,0) = +∞, where Γ_α:=H_α[δ₀]is the fundamental solution of

∂_tu+ (−∆)^αu= 0 in Q_∞,

u(0,·) =δ0 in R^N. (1.18)

We remark that for p∈ (1, p^∗_β), a self-similar solution u of (1.17) is also a very singular solution, since

t→0lim⁺Γ_α(t,0)t^2αN =c₂, (1.19) for some c₂ >0. For any self-similar solution u of (1.17), v(η) := u(1, t^−2α1 x) with η =t^−2α1 xis a solution of the self-similar equation

(−∆)^αv− 1

2α∇v·η−1 +β

p−1v+v^p = 0 in R^N. (1.20)

Since

1+β p−1

_p−1¹

is a constant nonzero solution of (1.20), the function

U_p(t) :=

1 +β p−1

_p−1¹

t^−1+βp−1 t >0 (1.21) is a flat self-similar solution of (1.17). It is actually the maximal solution of the ODE y^′+t^βy^p = 0 defined on R₊. Our next goal in this paper is to study non-flat self-similar solutions of (1.17).

Theorem 1.2 Assume that β >−1, u_∞ is defined by (1.16) and p^∗∗_β < p < p^∗_β,

where p^∗∗_β = 1 + ^2α(1+β)_N+2α . Thenu_∞ is a very singular self-similar solution of (1.17) in Q_∞. Moreover, there exists c₃ >1 such that

c⁻¹₃

1 +|x|^N+2α ≤u_∞(1, x) ≤ c₃ln(2 +|x|)

1 +|x|^N+2α x∈R^N. (1.22) When p^∗∗_β < p < p^∗_β with β > −1, we observe that u_∞ and U_p are self-similar solutions of (1.17) and u∞is non-flat. Now we are ready to consider the uniqueness of non-flat self-similar solution of (1.17) with decay at infinity, precisely, we study the uniqueness of self-similar solution to

∂_tu+ (−∆)^αu+t^βu^p = 0 in Q_∞,

lim_|x|→∞u(1, x) = 0. (1.23) We remark that if u is self-similar, then the assumption lim_|x|→∞u(1, x) = 0 is equivalent to lim_|x|→∞u(t, x) = 0 for any t >0. Finally, we state the properties of u_∞when 1< p≤p^∗∗_β as follows.

Theorem 1.3 (i) Assume1< p < p^∗∗_β andu_∞is defined by (1.16). Thenu_∞=U_p, where U_p is given by (1.21).

(ii) Assume p=p^∗∗_β andu_∞ is defined by (1.16). Thenu_∞ is a self-similar solution of (1.17) such that

u_∞(t, x)≥ c₄t^−N+2α2α

1 +|t^−2α1 x|^N+2α (t, x)∈(0,1)×R^N, (1.24) for some c₄ >0.

We note that Theorem 1.3 indicates that there exists no self-similar solution of (1.17) with an initial data u(0,·) vanishing in R^N \ {0} if p ∈ (1, p^∗∗_β ), since u_∞ is the least self-similar solution. In Theorem 1.3 part (ii), we do not know if the self-similar solution is flat or not. From the above theorems, we have the following result.

Theorem 1.4 (i) Assume p^∗∗_β < p < p^∗_β. Then problem (1.20) admits a minimal positive solution v_∞ satisfying

lim

|η|→∞|η|^{2α(1+β)p−1} v_∞(η) = 0. (1.25) Furthermore,

c⁻¹₃

1 +|η|^N+2α ≤v_∞(η)≤ c₃ln(2 +|η|)

1 +|η|^N+2α ∀η∈R^N (1.26) (ii) Assume1< p < p^∗∗_β . Then problem (1.20) admits no positive solution satisfying (1.25).

The question of uniqueness of the very singular solution in the casep^∗∗_β < p < p^∗_β remains an open problem.

It is worth comparing the above theorems with the results obtained by Nguyen and V´eron [25] concerning the limit, when k→ ∞of the solutions u=u_k of

∂_tu−∆u+u(ln(u+ 1)))^α= 0 in Q_∞,

u(0, .) =kδ₀ in R^N, (1.27) whereα >0. Note thatu_k>0 and the sequence{u_k}is increasing. In this problem, they proved that the diffusion is dominating if 0 < α ≤ 1 and the limit of theu_k is infinite. If 1 < α ≤ 2 the absorption dominates, but the limit of the u_k is the maximal solution of the associated ODE, y^′+y(ln(y+ 1)))^α= 0 on R₊. Finally, if α > 2 the limit of the u_k is a solution with a strong isolated singularity at (0,0), which could be called a very singular solution, although it is not self-similar.

This paper is organized as follows. In Section 2 we introduce some properties of Marcinkiewicz spaces and Kato’s type inequality for non-homogeneous problems. In Section 3 we prove Theorem 1.1. Section 4 is devoted to investigate the properties of solutions to (1.15). In Section 5 we give the proof of Theorem 1.2 and Theorem 1.3. Finally, we prove Theorem 1.4.

2 Linear estimates

2.1 The Marcinkiewicz spaces

We recall the definition and basic properties of the Marcinkiewicz spaces.

Definition 2.1 LetΘ⊂R^N+1be an open domain andµbe a positive Borel measure in Θ. For κ >1, κ^′ =κ/(κ−1) and u∈L¹_loc(Θ, dµ), we set

kukM^κ(Θ,dµ) = inf (

c∈[0,∞] : Z

E|u|dµ≤c Z

E

dµ ¹

κ′

, ∀E ⊂Θ,E Borel set )

(2.1) and

M^κ(Θ, dµ) ={u∈L¹_loc(Θ, dµ) :kukM^κ(Θ,dµ)<∞}. (2.2) M^κ(Θ, dµ) is called the Marcinkiewicz space of exponent κ or weak L^κ space and k.kM^κ(Θ,dµ) is a quasi-norm. The following property holds.

Proposition 2.1 [3, 15] Assume that 1 ≤q < κ < ∞ and u ∈L¹_loc(Θ, dµ). Then there exists c₅ >0 dependent of q, κsuch that

Z

E|u|^qdµ≤c₅kukM^κ(Θ,dµ)

Z

E

dµ 1−q/κ

,

for any Borel set E of Θ.

Remark 2.1 If Ω is a smooth domain ofR^N, we denote byH_α^Ω : (0,∞)×Ω×Ω→ R₊ the heat kernel for (−∆)^α and, if ν ∈M^b(Ω), by H^Ω_α[ν] the corresponding heat potential of ν defined by

H^Ω_α[ν](t, x) = Z

Ω

H_α^Ω(t, x, y)dν(y).

When Ω =R^N, by Fourier transform, it is clear that H_α(t, x, y) = 1

(2π)^N/2 Z

R^N

ei(x−y)·ζ−t|ζ|^2αdζ =H_α(t, x−y,0).

Furthermore, kH_α(t, .,0)kL¹ is independent oft. This implies

kH^Ω_α[ν](t, .)kL^p ≤ kνkL^p, ∀1≤p≤ ∞, ∀ν∈L^p(R^N). (2.3) Since H^Ω

α[ν](t+s, .) =H^Ω

α[H^Ω

α[ν](s, .)](t, .) for all t, s >0 (semigroup property) and ν ≥0 =⇒H^Ω_α[ν](t, .)≥0 the semigroup {H^Ω_α[.](t, .)}t≥0 is sub-Markovian. Further- more, since the operator (−∆)^α is symmetric in L²(R^N), the above semigroup is

analytic in L^p(R^N) for all 1≤p < ∞: if 1< p <∞ it follows from a general result of Stein [27]) and for p= 1 it is a consequence of regularity result from fractional powers of operators theory (see e.g. [22]). For 1≤p < ∞ the generator A_p of the semigroup in L^p(R^N) is the operator −(−∆)^α with domain

D(A_p) :={ν ∈L^p(R^N) : (−∆)^αν∈L^p(R^N)}. (2.4) and D(Ap) is dense since it contains C₀^∞(R^N). If p= ∞, the natural space is the space C₀(R^N) of continuous functions in R^N tending to 0 at infinity. The domain of the corresponding operator A_c0 is

D(A_c0) :={ν ∈C₀(R^N) : (−∆)^αν ∈C₀(R^N)}. (2.5) This operator is densely defined in C₀(R^N). In order to avoid confusion, C_c(R^N) (resp. C_c^∞(R^N)) denotes the space of continuous (resp. C^∞) functions inR^N with compact support. It is a dense subset of C₀(R^N).

The following regularizing effect L^p(R^N) 7→ L^q(R^N) (1 ≤ p ≤ q ≤ ∞) is valid for any submarkovian semigroup of contractions in all L^p(R^N)-spaces which has a self-adjoint generator inL²(R^N) (see e.g. [26]).

Proposition 2.2 Assume 1 ≤ p ≤ q ≤ ∞, p 6= ∞. Then for any ν ∈ L^p(R^N), H_α[ν](t, .)∈L^q(R^N)∩D(A_q)for allt >0and there holds, for some positive constant c=c(α, N, p, q),

kH_α[ν](t, .)kL^q(R^N) ≤ c

t^{2αN(1p−1q)}kνkL^q(R^N). (2.6) Note also that the function (t, x) 7→ H_α[ν](t, x) is C^∞ in Q_∞ as a result of the analyticity on the semigroup {H_α[.](t)}t>0.

Proposition 2.3 For any β > −1 and T > 0, there exists c₆ > 0 dependent of N, α, β such that for ν ∈M^b(Ω),

kH^Ω_α[|ν|]k_M^p∗β(Q^Ω_T,t^βdxdt)≤c₆kνk^Mb(Ω), (2.7) where p^∗_β is defined by (1.13) and Q^Ω_T = (0, T)×Ω.

In order to prove this proposition, we introduce some notations. Forλ >0 and y∈Ω, let us denote

A^Ω_λ(y) ={(t, x) ∈Q^Ω_T :H_α^Ω(t, x, y)> λ} and m^Ω_λ(y) = Z

A^Ω_λ(y)

t^βdxdt.

We also set A^R_λ^N =A_λ andm^R_λ^N =m_λ.

Lemma 2.1 There exists c7 >0 such that for any λ >1, A_λ(y)⊂(0, c₇λ^−2αN]×B

c7λ^−N1 (y), (2.8) where B_r(y) is the ball with radius r and center y in R^N.

Proof. We observe that H_α(t, x, y) = t^−2αNΓ_α(1,(x−y)t^−2α1 ), where Γ_α is the fundamental solution of (1.18). From [4] (see also[13] for an analytic proof), there existsc₈ >0 such that

Γ_α(1, z)≤ c₈ 1 +|z|^N+2α. This implies in particular

H_α(t, x, y)≤ c8t^−2αN 1 +

t^−2α1 |x−y|N+2α. (2.9) On the one hand, for (t, x)∈A_λ(y), we have that

t^−2αNΓ_α(1,0)≥t^−2αNΓ_α(1,(x−y)t^−2α1 )> λ, which implies

t <Γ

2α

αN(1,0)λ^−2αN. (2.10)

On the other hand, letting r=|x−y|, c8t

t^1+2αN +r^N+2α ≥t^−2αNΓα(1,(x−y)t^−2α1 )> λ, then

r≤(c₈tλ⁻¹)^N+2α1 , (2.11)

which, together with (2.10), implies

r≤c₉λ^−N1,

for somec₉ >0.

Proof of Proposition 2.3. By Lemma 2.1, there existsc₁₀>0 such that m_λ(y)≤c₁₀λ^{−1−2α(1+β)N} .

Clearly

H_α^Ω(t, x, y)≤H_α(t, x, y), (2.12)

then for any Borel setE ⊂Q^Ω_T andy ∈Ω, we have that Z

E

H_α^Ω(t, x, y)t^βdxdt≤λ Z

E

t^βdxdt+ Z

Aλ(y)

Hα(t, x, y)t^βdxdt and

Z

Aλ(y)

H_α(t, x, y)t^βdxdt=− Z +∞

λ

sdm_s(y) =λm_λ(y) + Z +∞

λ

m_s(y)ds

≤c₁₀λ^{−2α(1+β)N} +c₁₀ Z +∞

λ

s^{−1−2α(1+β)N} ds

≤c₁₁λ^{−2α(1+β)N} , where c₁₁=c₁₀

1 + _2α(1+β)^N

. As a consequence, it follows Z

E

H_α^Ω(t, x, y)t^βdxdt≤λ Z

E

t^βdxdt+c₁₁λ^{−2α(1+β)N} . Taking λ= (R

Et^βdxdt)^{−N+2α(1+β)N} , we obtain that Z

E

H_α^Ω(t, x, y)t^βdxdt≤(c₁₁+ 1)(

Z

E

t^βdxdt)^{N+2α(1+β)2α(1+β)} . (2.13) Since, by Fubini’s theorem,

Z

E

H^Ω_α[|ν|](t, x)t^βdxdt = Z

E

Z

Ω

H_α^Ω(t, x, y)d|ν(y)|t^βdxdt

= Z

Ω

Z

E

H_α^Ω(t, x, y)t^βdxdtd|ν(y)|, together with (2.13), it yields

Z

E

H^Ω_α[|ν|](t, x)t^βdxdt≤(c₁₁+ 1)kνk^Mb(Ω)

Z

E

t^βdxdt

_N+2α(1+β)^2α(1+β) .

Thus,

kH^Ω

α[|ν|]k

M^1+2α(1+β)N (Q^Ω_T,t^βdxdt) ≤(c₁₁+ 1)kνk^Mb(Ω),

which ends the proof.

2.2 The non-homogeneous problem

In this section we consider the linear non-homogeneous problem

∂_tu+ (−∆)^αu=µ in Q_T,

u(0,·) =ν in R^N. (2.14)

Ifµ∈L¹(Q_T) andν ∈L¹(R^N) a functionudefined in Q_T is anintegral solution of (2.14) in Q_T if it is expressed by Duhamel’s formula, that is

u(t, x) =H_α[ν](t, x) +Hα[µ](t, x) a.e. in Q_T. (2.15) where, we denote by Hα the operator of L¹(Q_T) defined for all (x, t)∈Q_T by

Hα[µ](x, t) = Z t

0

H_α[µ(., s)](x, t−s)ds= Z t

0

Z

R^N

H_α(t−s, x, y)µ(s, y)dyds. (2.16) Notice that, by Duhamel’s formula, there holds

ku(t,·)kL¹(R^N)≤ kµkL¹(QT)+kνkL¹(R^N), ∀t∈(0, T), (2.17) and

kukL¹(QT) ≤T(kµkL¹(QT)+kνkL¹(RN)). (2.18) The advantage of this notion of solution is that Duhamel’s formula has a meaning as soon asµ and ν are integrable in their respective domains of definition. As for any continuous semigroup of bounded linear operators, a strong solution is an integral solution.

The following proposition is the Kato’s type estimate which is essential tool to prove the uniqueness of solutions to (1.1). For T >0, we denoteQ_T = (0, T)×R^N. Proposition 2.4 Assumeµ∈L¹(Q_T)andν∈L¹(R^N). Then there exists a unique weak solution u∈L¹(QT) to the problem (2.14) and there exists c12>0 such that

Z

QT

|u|dxdt≤c₁₂ Z

QT

|µ|dxdt+c₁₂ Z

R^N|ν|dx. (2.19) Moreover, for any ξ∈Y_α,T, ξ≥0, we have that

Z

QT

|u|(−∂tξ+ (−∆)^αξ)dxdt+ Z

RN|u(T, x)|ξ(T, x)dx

≤ Z

QT

ξsign(u)µdxdt+ Z

R^N

ξ(0, x)|ν|dx

(2.20)

and Z

QT

u+(−∂tξ+ (−∆)^αξ)dxdt+ Z

R^N

u+(T, x)ξ(T, x)dx

≤ Z

QT

ξsign₊(u)µdxdt+ Z

R^N

ξ(0, x)ν₊dx.

(2.21)

In order to prove Proposition 2.4, we introduce the following notations. We say that u:Q_T →Ris in C_t,x^σ,σ′(Q_T) for σ, σ^′ ∈(0,1) if

kuk_C^σ,σ′

t,x (QT):=kukL^∞(QT)+ sup

QT

|u(t, x)−u(s, y)|

|t−s|^σ +|x−y|^σ′ <+∞ and u∈C_t,x^{1+σ,2α+σ′}(Q_T) if

kuk_C^{1+σ,2α+σ′}

t,x (QT):=kukL^∞(QT)+k∂_tuk_C^σ,σ′

t,x (QT)+k(−∆)^αuk_C^σ,σ′

t,x (QT)<+∞. Lemma 2.2 Let µ∈C¹(Q_T)∩L^∞(Q_T),ν ∈L^∞(R^N) andu be an integral solution of problem (2.14), then there exists σ∈(0,1) such thatu∈C_t,x^1+σ,2α+σ in(ǫ, T)×R^N for any ǫ∈ (0, T). In particular, if kD²νkL^∞(RN)+k(−∆)^ανk_Cx^1−α₍R^N) <∞, then u∈C_t,x^1+σ,2α+σ(QT).

Proof. Step 1. When kD²νkL^∞(R^N)+k(−∆)^ανk_Cx^1−α₍^RN₎ <∞, it follows directly by [9, (A.1)] thatu∈C_t,x^1+σ,2α+σ(Q_T).

Step 2. Whenν ∈L^∞(R^N), we use [10, Theorem 6.1] to obtain thatu∈C

σ 2α,σ t,x (Q_T) for someσ > 0. For anyǫ∈(0, T), let η: [0, T]→[0,1] be a C² function such that η = 0 in [0,^ǫ₄] andη = 1 in [ǫ, T] and v=ηuinQ_T. Since η does not depend on x, we obtain that v satifies,

∂_tv+ (−∆)^αv =ηµ+η^′(t)u, ∀(t, x)∈Q_T, whereηµ+η^′(t)u∈C

σ 2α,σ

t,x (Q_T) andv(0,·) = 0 inR^N, Then we apply the argument in Step 1 to obtain thatv∈C_t,x^1+σ,2α+σ(Q_T). Therefore, uisC_t,x^1+σ,2α+σ in (ǫ, T)×R^N.

The proof is complete.

Lemma 2.3 (i) Let µ ∈ C¹(QT)∩ L^∞(QT) and ν ∈ C¹(R^N) ∩L^∞(R^N), then problem (2.14) admits a unique classical solution u.

(ii)Let µ∈C¹(QT)∩L^∞(QT)∩L¹(QT), ν∈C²(R^N)∩L^∞(R^N)∩L¹(R^N) andube the classical solution of (2.14), thenu isC_t,x^1+σ,2α+σ in (ǫ, T)×R^N for anyǫ∈(0, T)

and for any ξ∈Y_α,T, Z

QT

u(t, x)[−∂tξ(t, x) + (−∆)^αξ(t, x)]dxdt

= Z

QT

µ(t, x)ξ(t, x)dxdt+ Z

R^N

ξ(0, x)νdx− Z

R^N

ξ(T, x)u(T, x)dx.

(2.22)

Thus u is a weak solution and it belongs to Y_α,T.

(iii) Let µ˜∈C¹(Q_T)∩L^∞(Q_T) andν ∈C¹(R^N)∩L^∞(R^N), then problem

−∂tw+ (−∆)^αw= ˜µ in QT,

w(T,·) =ν in R^N (2.23)

admits a unique classical solution w ∈ C_t,x^1+σ,2α+σ(Q_T) for some σ ∈ (0,1). More- over, ifµ∈C¹(Q_T)∩L^∞(Q_T)∩L¹(Q_T)andν ∈C²(R^N)∩L^∞(R^N)∩L¹(R^N), then ξ is a weak solution and it belongs to Y_α,T.

Proof. (i) By [10, Theorem 3.3, Theorem 6.1], if µ and ν are continuous and bounded, there exists a unique viscosity solution u∈C(Q_T). The higher regularity is provided by [10, Theorem 6.1] which asserts that there existσ >0 and a positive constant c depending on N, τ ∈ (0, T) and α such that for all (t, x) and (s, y) belonging toQ^B_T¹_−τ, there holds

|u(t, x)−u(s, y)|

(|x−y|+|t−s|^2α1 )^σ ≤c kuk_L∞(Q^B_T²)+ sup

0≤t≤Tku(t, .)kL¹(RN +kµkL^∞(QT)

!

(2.24) whereQ^Ω_T = (0, T)×Ω. Thusu∈C

σ 2α,σ

t,x (Q_T). By Lemma 2.2 the integral solutionu belongs toC_t,x^{1+σ′,2α+σ′} in (ǫ, T)×R^N for anyǫ∈(0, T) and someσ^′∈(0,min{_2α^σ , σ}).

Then u is a classical solution of (2.14) and thus a viscosity solution.

(ii) By the definition of (−∆)^αu, u(t,·) ∈ L¹(R^N) for all t ∈ (0, T). As in [9, Appendix A.2] we have Duhamel formula, thus u ∈ L¹(Q_T) and it is an integral solution.

We claim thatk(−∆)^α_ǫu(t,·)kL^∞(R^N)is uniformly bounded with respect toǫ∈(0, ǫ₀).

Since u(t,·) ∈ C_x^2α+σ(R^N) for some σ ∈ (0,min{2−2α,1}), then for x ∈ R^N and

y∈B1(0), |u(x+y) +u(x−y)−2u(x)| ≤ ku(t,·)kCx^2α+σ(RN)|y|^2α+σ. Thus, k|(−∆)^α_ǫu(t,·)|kL^∞(RN) ≤ sup

x∈R^N

"

Z

RN\B1(0)

|u(x+y)−u(x)|

|y|^N+2α dy

+1 2

Z

B1(0)\Bǫ(0)

|u(x+y) +u(x−y)−2u(x)|

|y|^N+2α dy

#

≤ 2kukL¹(RN)+ Z

B1(0)|y|^σ−Ndyku(t,·)k_Cx^2α+σ₍R^N). Next we claim that

Z

QT

ξ(−∆)^α_ǫudxdt= Z

QT

u(−∆)^α_ǫξdxdt ∀ξ ∈Y_α,T. (2.25) Indeed, using the fact that for any t >0 there holds

Z

RN

Z

RN

[u(t, z)−u(t, x)]ξ(t, x)

|z−x|^N+2α χ_ǫ(|x−z|)dzdx

= Z

R^N

Z

R^N

[u(t, x)−u(t, z)]ξ(t, z)

|z−x|^N+2α χ_ǫ(|x−z|)dzdx, then we have

Z

R^N

ξ(t, x)(−∆)^α_ǫu(t, x)dx

=−1 2 Z

R^N

Z

R^N

(u(t, z)−u(t, x))ξ(t, x)

|z−x|^N+2α +(u(t, x)−u(t, z))ξ(t, z)

|z−x|^N+2α

χ_ǫ(|x−z|)dzdx

= 1 2 Z

RN

Z

RN

[u(t, z)−u(t, x)][ξ(t, z)−ξ(t, x)]

|z−x|^N+2α χ_ǫ(|x−z|)dzdx.

Similarly, Z

R^N

u(t, x)(−∆)^α_ǫξ(t, x)dx= 1 2 Z

R^N

Z

R^N

[u(t, z)−u(t, x)][ξ(t, z)−ξ(t, x)]

|z−x|^N+2α χ_ǫ(|x−z|)dzdx.

Then (2.25) holds. Since u is C_t,x^1+σ,2α+σ in (ǫ, T) ×R^N for any ǫ ∈ (0, T) and ξ belongs to Y_α,T, (−∆)^α_ǫξ(t,·) → (−∆)^αξ(t,·) and (−∆)^α_ǫu(t,·) → (−∆)^αu(t,·) as ǫ→0 in R^N and (−∆)^α_ǫξ(t,·), (−∆)^α_ǫu(t,·)∈L^∞(R^N) and ξ(t,·), u(t,·) ∈L¹(R^N), then it follows by the Dominated Convergence Theorem that

lim

ǫ→0⁺

Z

RN

ξ(t, x)(−∆)^α_ǫu(t, x)dx= Z

RN

ξ(t, x)(−∆)^αu(t, x)dx

and

ǫ→0lim⁺ Z

RN

(−∆)^α_ǫξ(t, x)u(t, x)dx= Z

RN

(−∆)^αξ(t, x)u(t, x)dx.

Combining this with (2.25), and letting ǫ→0⁺, we have that Z

R^N

ξ(t, x)(−∆)^αu(t, x)dx= Z

R^N

(−∆)^αξ(t, x)u(t, x)dx, integrating over [0, T] and by (2.14), we conclude that (2.22) holds.

(iii) End of the proof. Letu be the weak solution of problem (2.14) obtained from (ii) with ˜µ(T −t, .) =µ(t, .) and

w(t, x) =u(T −t, x) (t, x)∈[0, T]×R^N.

Then w is a solution of (2.23) and for someσ∈(0,1), wis C_t,x^1+σ,2α+σ(QT). On the contrary, ifwis a solution of (2.23), thenu(t, x) =w(T−t, x) for (t, x)∈[0, T]×R^N is a solution of (2.14), then the uniqueness holds since the solution of (2.14) is unique.

Sinceu∈C_t,x^1+σ,2α+σ(QT), then (−∆)^αu(t,·)∈C_x^σ and then (−∆)^α_ǫu(t,·) is bounded,

which implies u∈Y_α,T.

Proof of Proposition 2.4. Uniqueness. Letv∈L¹(Q_T) be a weak solution of

∂_tv+ (−∆)^αv= 0 in Q_T,

v(0,·) = 0 in R^N. (2.26)

We claim that v= 0 a.e. inQ_T.

In fact, let ω be a Borel subset ofQ_T and η_ω,n be the solution of

−∂_tu+ (−∆)^αu=ζ_n in Q_T,

u(T,·) = 0 in R^N, (2.27)

where ζ_n: ¯Q_T →[0,1] is a functionC_c¹(Q_T) such that ζ_n→χ_ω inL^∞( ¯Q_T) asn→ ∞. Then η_ω,n∈Y_α,T by Lemma 2.3, and

Z

QT

vζ_ndxdt= 0.

Passing to the limit whenn→ ∞, we derive Z

ω

vdxdt= 0.

This implies v= 0 a.e. in QT.

Existence and estimate (2.20). For δ >0, we define an even convex function φ_δ by φ_δ(t) =

(|t| −^δ₂ if |t| ≥δ,

t²

2δ if |t|< δ/2. (2.28)

Then for any t, s ∈ R, |φ^′_δ(t)| ≤ 1, φ_δ(t) → |t| and φ^′_δ(t) → sign(t) when δ → 0⁺. Moreover,

φ_δ(s)−φ_δ(t)≥φ^′_δ(t)(s−t). (2.29) Let{µ_n},{ν_n} be two sequences of functions in C₀²(Q_T), C₀²(R^N), respectively, such that

n→∞lim Z

QT

|µ_n−µ|dxdt= 0, lim

n→∞

Z

R^N|ν_n−ν|dx= 0.

We denote by u_n the corresponding solution to (2.14) where µ, ν are replaced by µ_n, ν_n, respectively. By Lemma 2.2 and Lemma 2.3(ii), u_n ∈ C_t,x^1+σ,2α+σ(Q_T) ∩ L¹(Q_T) and then we use Lemma 2.3 in [15] and Lemma 2.3 (ii) to obtain that for any δ >0 andξ ∈Y_α,T, ξ≥0,

Z

QT

φ_δ(u_n)[−∂_tξ+ (−∆)^αξ]dxdt+ Z

R^N

ξ(T, x)φ_δ(u_n(T, x))dx

= Z

QT

ξ[∂_tφ_δ(u_n) + (−∆)^αφ_δ(u_n)]dxdt+R

R^Nξ(0, x)φ_δ(ν_n)dx

≤ Z

QT

ξφ^′_δ(un)[∂tun+ (−∆)^αun]dxdt+ Z

RN

ξ(0, x)φ_δ(νn)dx

= Z

QT

ξφ^′_δ(u_n)µ_ndxdt+ Z

R^N

ξ(0, x)φ_δ(ν_n)dx.

Letting δ→0⁺, we obtain Z

QT

|un|[−∂tξ+ (−∆)^αξ]dxdt+ Z

R^N

ξ(T, x)|un(T, x)|dx

≤ Z

QT

ξsign(u_n)µ_ndxdt+ Z

R^N

ξ(0, x)|ν_n|dx.

(2.30)

Let η_k be the solution of

−∂_tu+ (−∆)^αu=ς_k in Q_T,

u(T,·) = 0 in R^N, (2.31)

whereς_k:QT →[0,1] is aC₀² function such that ς_k= 1 in (0, T)×B_k(0). From the proof of Lemma 2.3, ˜η_k(t, x) :=η_k(T −t, x) satisfies with ˜ς_k(t, x) =ς_k(T−t, x)

∂tu+ (−∆)^αu= ˜ς_k in QT, u(0,·) = 0 in R^N. By Lemma 2.2, ˜η_k ∈C_t,x^1+σ,2α+σ(Q_T) with some σ∈(0,1) and

0≤η˜_k(t, x) ≤ c₈ Z _T

t

Z

R^N

(s−t)^−2αN

1 +|(s−t)^−2α1 (y−x)|^N+2αdyds

≤ c₈ Z _T

t

Z

R^N

dz

1 +|z|^N+2αds

= c13(T −t).

Taking ξ=η_k in (2.30), we derive that Z

QT

|u_n|χ_(0,T)×Bk(0)dxdt≤c₁₃T Z

QT

|µ_n|dxdt+c₁₃T Z

RN|ν_n|dx.

Then, lettingk→ ∞, we have Z

QT

|u_n|dxdt≤c₁₃T Z

QT

|µ_n|dxdt+c₁₃T Z

R^N|ν_n|dx. (2.32) Similarly,

Z

QT

|un−um|dx≤c13T Z

QT

|µn−µm|dxdt+c13T Z

RN|νn−νm|dx. (2.33) Therefore, {u_n}n is a Cauchy sequence in L¹(Q_T) and its limit u is a weak solution of (2.14). Lettingn→ ∞, (2.20) and (2.19) follow by (2.30) and (2.32), respectively.

The proof of (2.21) is similar.

Remark 2.2 Other classes of uniqueness of solutions of the fractional heat equa- tions exist. In [2] it is proved that any positive strong solution u ∈C([0, T)×R^N) can be represented by the convolution integral defined by

u(t, x) = Z

RN

P_t(y)u(0, y)dy where

P_t(x) = 1 t^2αN e^it−

2α1 x.ξ−|ξ|^2α.

However the fact that u ∈ L¹(Q_T) is a part of the definition of strong solution therein. Furthermore, the notion of weak solution used in this paper differs from ours.

3 Proof of Theorem 1.1

If h(t, .) is monotone nondecreasing, for any λ >0, I +λh(t, .) is an homeorphism of R and the inverse function J_λ(t, .) = (I +λh(t, .))⁻¹ is a contraction. We define the Yosida approximation by

h_λ(t, .) = I−J_λ(t, .)

λ . (3.1)

The function h_λ(t, .) is monotone nondecreasing, vanishes at 0 ash does it and it is

1

λ-Lipschitz continuous. Furthermore

rh_λ(t, r)↑rh(t, r) asλ→0, ∀r∈R, (3.2) see [5, Chap 2, Prop. 2.6]. Ifu is a real valued function we will denote byh◦uand h_λ◦u respectively the functions (t, x)7→h(t, u(t, x)) and (t, x)7→h_λ(t, u(t, x)).

Lemma 3.1 Assume that h satisfies (H)-(i), λ > 0 and φ∈L¹(R^N). Then there exists a unique solution u_φ of

∂_tu+ (−∆)^αu+h_λ◦u= 0 in Q_∞,

u(0,·) =φ in R^N. (3.3)

Moreover,

H_α[φ]− Hα[h_λ◦H_α[φ₊])]≤u_φ≤H_α[φ]− Hα[h_λ◦(−H_α[φ₋])] in Q_T, (3.4) where φ_±= max{0,±φ} and

ku_φ(t, .)−u_ψ(t, .)kL¹ ≤ kφ−ψkL¹, ∀1≤p≤ ∞. (3.5) (i) u_φ≥0 if φ≥0 in Ω;

(ii) the mapping φ7→u_φ is increasing.

Proof. Existence is a consequence of the Cauchy-Lipschitz-Picard theorem (see [11, Chap 4]): we write (3.3) under the integral formu=T[u] =H_α[φ]− Hα[h_λ◦u], i.e.

T[u](t, .) =H_α[φ](t, .)− Z _t

0

H_α[h_λ◦u](t−s, .)ds. (3.6) The space C([0,∞);L¹(R^N)) endowed with the norm

kwkC−L¹ = supn

e^−ktkw(t, .)kL¹ :t≥0o ,