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Mathematische Nachrichten, 16 May 2017

On a generalized nonlinear heat equation in Besov and Triebel-Lizorkin spaces

Franka Baaske1,∗ and Hans-Jürgen Schmeißer1,∗∗

1 Institute of Mathematics, Friedrich-Schiller-University Jena, 07737 Jena, Germany

Received XXXX, revised XXXX, accepted XXXXPublished online XXXX

Key words Generalized nonlinear heat equation, function spaces, molecular and wavelet decompositions, mildand strong solutionsMSC (2010) 46E35, 35K25, 35K55, 35Q35

The paper deals with solutions of the Cauchy problem of a nonlinear generalized heat equation in the contextof Besov and Triebel-Lizorkin spaces Bs

p,q(Rn) and F sp,q(Rn) where 1 ≤ p, q ≤ ∞ and s > n/p with initial

data belonging to some spaces As0p,q(Rn) where A ∈ B,F and s− α < s0 ≤ s.

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1 Introduction

We deal with a generalized nonlinear heat equation

∂

∂tu(x, t) + (−∆x)αu(x, t)−Du2(x, t) = 0, x ∈ Rn, 0 < t < T, (1.1)

u(x, 0) = u0(x), x ∈ Rn (1.2)

where 0 < T ≤ ∞, 2 ≤ n ∈ N, α ∈ N and Du2 =n∑i=1

∂∂xi

u2. The case α = 1 corresponds to a classical

nonlinear heat equation. Our motivation to consider this nonlinearity comes from the interest in generalizedNavier-Stokes equations. For this purpose, one transfers the results obtained in this paper to the vector-valuedcase, using (1.1), (1.2) as a scalar model. This method has already been applied for heat and Navier-Stokesequations, that is for α = 1, in [18, Chapter 6] and [1].

Concerning initial value problems for inhomogeneous parabolic differential equations with constant coefficientelliptic operators we refer, for instance, to [4], [5], [21], [8], [9], [10]. In particular, in [21] generalized Navier-Stokes equations in distinguished homogeneous and inhomogeneous Besov spaces are studied, replacing−∆x inthe classical case by (−∆x)α with α > 0. In [8], [9], [10], mild solutions of generalized nonlinear heat equationswith a real number α > 0 and a particular class of nonlinearities are determined in appropriate function spaces.In dependence on α and the nonlinearity this evolution equation models for example a generalized convection-diffusion equation or a quasi-geostrophic dissipative equation.

In our approach, a solution of (1.1), (1.2) is considered as fixed point of the operator Tu0given as

Tu0u(x, t) := Wαt u0(x) +

∫ t

0

Wαt−τDu

2(x, τ)dτ, x ∈ Rn, 0 < t < T (1.3)

in some weighted Lebesgue spaces Lv((0, T ), b, X), see (1.7) and (1.8) below. Here Wαt ω with ω ∈ S′(Rn) is

defined on the Fourier side as

(Wαt ω)

∧(ξ) := e−t|ξ|

2α

ω(ξ), ξ ∈ Rn, t > 0, α ∈ N. (1.4)

∗ Corresponding author E-mail: [email protected], Phone: +49 3641 946120, Fax: +49 3641 946102∗∗ E-mail: [email protected], Phone: +49 3641 946125, Fax: +49 3641 946102


2 F. Baaske and H. J. Schmeißer: On a generalized nonlinear heat equation in Besov and Triebel-Lizorkin spaces

This can be reformulated as

Wαt ω(x) = (Kα

t ∗ ω)(x), x ∈ Rn (1.5)

where

Kαt (x) = (2π)−n/2

(e−t|ξ|

2α)∨

(x). (1.6)

Here ∧ and ∨ stand for the Fourier transform and its inverse in S′(Rn), respectively. More detailed explanationshow to understand the rather formal expression (1.3) and the concept of solutions of (1.1), (1.2) may be found inSection 3.

We are interested in solutions u of (1.1), (1.2) belonging to some Besov or Triebel-Lizorkin spaces Asp,q(Rn),A ∈ B,F, with respect to the space variable. Because of the structure of the nonlinearity these spaces haveto fulfill certain multiplication properties. We assume that they are multiplication algebras which holds if s > n

p

and in some limiting cases s = np . Concerning the initial data, we suppose u0 ∈ As0p,q(Rn) with s− α < s0 ≤ s.

We ask for solutions u belonging to some spaces Lv((0, T ), b, X) defined as the collection of all functionsu : (0, T )→ X such that∫ T

0

tbv‖u(·, t)|X ‖vdt <∞, if v <∞ (1.7)

and

sup0<t<T

tb‖u(·, t)|X ‖ <∞, if v =∞, (1.8)

respectively, with 1 ≤ v ≤ ∞, b ∈ R, 0 < T ≤ ∞ and X = Asp,q(Rn) is a Banach space. It turns out that thesolution is also a C∞- function with respect to space and time.

Our method is as follows. First we transfer (1.1), (1.2) by means of the Fourier transform and Duhamel’sprinciple into a corresponding fixed point problem for the operator Tu0 in our solution space. To solve the fixedpoint problem, we need a-priori estimates in some spaces Asp,q(Rn) for both terms of the right hand side of (1.3).The key estimate, formulated in Theorem 3.5, reads as

td/2α‖Wαt ω|As+dp,q (Rn)‖ ≤ c‖ω|Asp,q(Rn)‖, 0 < t ≤ 1 (1.9)

if 1 ≤ p, q ≤ ∞ (p < ∞ in case of F - spaces), s ∈ R, d ≥ 0, and α ∈ N. To achieve (1.9), we apply decom-position methods by means of wavelets and molecules to these spaces. More precisely, having an appropriaterepresentation of ω ∈ Asp,q(Rn) by means of sufficiently smooth Daubechies wavelets, we show that Wα

t ω canbe characterized by means of molecules in As+dp,q (Rn).

The main result is contained in Theorem 3.8, where existence and uniqueness of a local mild solution of (1.1),(1.2) in the sense of (1.3) is proved for arbitrary initial values u0 ∈ As0p,q(Rn) with s − α < s0 ≤ s. Moreoverwe show that the solution is strong.

The paper is organized as follows. In Section 2 we fix notation and define function and related sequencespaces. Then we introduce the concepts of Daubechies wavelets and molecules (as far as we need them for ourconsiderations). Furthermore, we recall in Proposition 2.4 the characterization of Asp,q(Rn) via wavelets and inProposition 2.8 via molecules. The core of the paper is Section 3. Proposition 3.1 and Theorem 3.2 deal with socalled α-caloric wavelets which are of interest of their own and apply also to parameters p, q < 1. Subsection3.2 is devoted to the study of mapping properties of the operator Wα

t in Besov and Triebel-Lizorkin spaces, seeTheorem 3.5. This is related to the solution of the linear homogeneous generalized heat equation. In Subsection3.3 we prove in Theorem 3.8 our main result showing that the operator Tu0 has a unique fixed point in our solutionspace.

2 Preliminaries

2.1 Function spaces

Let Rn be Euclidean n - space with n ∈ N where N indicates the collection of all natural numbers, N0 = N∪0,

Rn+1+ = (x, t) ∈ Rn+1 : x ∈ Rn, t > 0


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and Rn+1+ its closure. Put R = R1, whereas C is the complex plane. As usual, Z is the collection of all integers

and Zn, n ∈ N, the lattice of all points m = (m1, . . . ,mn) with mj ∈ Z, j = 1, . . . , n. Let Nn0 , where n ∈ N,be the set of all multi-indices

γ = (γ1, . . . , γn) with γj ∈ N0 and |γ| =n∑j=1

γj .

For x ∈ Rn and γ ∈ Nn0 we write

xγ = xγ11 · · ·xγnn and Dγ =∂|γ|

∂xγ11 . . . ∂xγnn.

Cu(Rn) denotes the space of all u- times continuously differentiable functions in Rn. S(Rn) denotes theSchwartz space of all complex-valued infinitely differentiable rapidly decreasing functions on Rn and S′(Rn)its dual, the space of all tempered distributions. Furthermore, Lp(Rn) with 0 < p ≤ ∞ is the space of all p -integrable complex-valued functions with respect to the Lebesgue measure, quasi-normed by

‖f |Lp(Rn)‖ =

(∫Rn|f(x)|pdx

)1/p

with the usual modification if p =∞. If φ ∈ S(Rn) then

φ(ξ) = (Fφ)(ξ) = (2π)−n/2∫Rne−ixξφ(x)dx, ξ ∈ Rn (2.1)

denotes the Fourier transform of φ. F−1φ and φ∨ stand for the inverse Fourier transform, given by the right handside of (2.1) with i in place of −i and xξ for the scalar product in Rn. F and F−1 are extended in the usual wayto S′(Rn). Let φ0 ∈ S(Rn) with

φ0(x) = 1 if |x| ≤ 1 and φ0(x) = 0 if |x| ≥ 3/2. (2.2)

We define the sequence

φk(x) = φ0(2−kx)− φ0(2−k+1x) for x ∈ Rn, k ∈ N. (2.3)

Then∞∑k=0

φk(x) = 1 for all x ∈ Rn (2.4)

and φk∞k=0 is called a smooth dyadic resolution of unity. Because of the Paley-Wiener-Schwartz theoremF−1[φjFf ] are entire analytic functions for all f ∈ S′(Rn) and hence make sense pointwise. The spaces we areinterested in are Besov and Triebel-Lizorkin spaces, that is Bsp,q(Rn) and F sp,q(Rn), respectively.

Definition 2.1 Let φ = φk∞k=0 be the above dyadic resolution of unity.

(i) For 0 < p, q ≤ ∞, s ∈ R we define Bsp,q(Rn) as the collection of all f ∈ S′(Rn) such that

‖f |Bsp,q(Rn)‖φ =

( ∞∑k=0

2ksq‖F−1φkFf |Lp(Rn)‖q)1/q

<∞

with the usual modification if q =∞.

(ii) For 0 < p < ∞, 0 < q ≤ ∞, and s ∈ R the space F sp,q(Rn) is defined as the collection of all f ∈ S′(Rn)such that

‖f |F sp,q(Rn)‖φ = ‖

( ∞∑k=0

2ksq|F−1φkFf(·)|q)1/q

|Lp(Rn)‖ <∞

with the usual modification if q =∞.



A detailed study of these spaces including their history and properties can be found in [13], [14] and [16]. Inparticular they are independent (in the sense of equivalent quasi-norms) of the chosen resolution of unity as longas it fulfills (2.2) - (2.4). Therefore we will omit the subscript φ in Definition 2.1 in the sequel.

Let δ ∈ R. Then

Iδ : f 7→ F−1((1 + |ξ|)−δ/2Ff) (2.5)

is a one-to-one map onto itself both in S(Rn) and in S′(Rn). Furthermore, Iδ is a lift for the spaces Asp,q(Rn),with A ∈ B,F, s ∈ R, 0 < p, q ≤ ∞ (p <∞ for F−spaces), i.e. we have

IδAsp,q(Rn) = As+δp,q (Rn) (2.6)

in the sense of equivalent quasi-norms. For more details we refer to [13, Section 2.3.8].In order to describe our solution spaces we need to consider vector-valued Lv- spaces with respect to the

Bochner integral. Let X be a Banach space with X ⊂ S′(Rn), 0 < T ≤ ∞, b ∈ R and 1 ≤ v ≤ ∞. ThenLv((0, T ), b, X) contains all f : (0, T )→ X such that

‖f |Lv((0, T ), b, X)‖ =

(∫ T

0

tbv‖f(·, t)|X‖vdt

)1/v

<∞ (2.7)

(with the usual modification if v =∞). We deal with Banach spaces X = Asp,q(Rn) with 1 ≤ p, q ≤ ∞ (p <∞for F -spaces) and s ∈ R such that Asp,q(Rn) is a multiplication algebra (cf. the explanations in Subsection 3.3).After extending functions satisfying (2.7) from Rn × (0, T ) to Rn+1 by zero, one has always

Lv((0, T ), b, Asp,q(Rn)) ⊂ S′(Rn+1)

if s > 0 and b < 1− 1v , cf. [18, p. 160, p. 183] and the arguments in [1, Sec. 3].

2.2 Wavelets and Molecules

In what follows, we deal with (linear and nonlinear) generalized heat equations in terms of wavelet expansions.This requires some preparations. We recall what we need later on. Let

ψF ∈ Cu(R), ψM ∈ Cu(R), u ∈ N, (2.8)

be real-valued compactly supported Daubechies wavelets with ψF (0) = (2π)−1 and∫R

xvψM (x)dx = 0 for all v ∈ 0, . . . , u− 1. (2.9)

We always assume that ψF and ψM have L2-norm 1. Then

ψF (x−m), 2j/2 ψM (2jx−m) : j ∈ N0, m ∈ Z (2.10)

is an orthonormal basis in L2(R) for any u ∈ N. We extend these wavelets from R to Rn by the usual multireso-lution procedure. Let

G = (G1, . . . , Gn) ∈ G0 = F, Mn, (2.11)

which means that Gr is either F or M or

G = (G1, . . . , Gn) ∈ Gj = F, Mn∗, j ∈ N. (2.12)

Here ∗ indicates that at least one of the components of G must be an M . In the sequel, we denote such a set Gj

with G∗. We put

ΨjG,m(x) = 2jn/2

n∏r=1

ψGr (2jxr −mr), G ∈ Gj , m ∈ Zn, (2.13)



x ∈ Rn, now with j ∈ N0. Then for any u ∈ N

Ψ = ΨjG,m : j ∈ N0, G ∈ Gj , m ∈ Zn (2.14)

is an orthonormal basis in L2(Rn) and

f =

∞∑j=0

∑G∈Gj

∑m∈Zn

λj,Gm 2−jn/2 ΨjG,m =:

∑j,G,m

λj,Gm 2−jn/2 ΨjG,m

with

λj,Gm = λj,Gm (f) = 2jn/2∫Rnf(x)Ψj

G,m(x)dx = 2jn/2⟨f,Ψj

G,m

⟩is the corresponding expansion, adapted to our needs. For more detailed explanations cf. [17, Subsection 1.2.1].

Remark 2.2 In particular, ΨjG,m satisfies the moment conditions if G ∈ G∗ since∫

RxνψM (2jx−m)dx = 2−j(ν+1)

∫R(y +m)νψM (y)dy

= 2−j(ν+1)ν∑k=0

cν,k

∫Ryν−kmkψM (y)dy (2.15)

= 0, for all ν < u.

Let Qj,m with j ∈ N0 and m ∈ Zn be cubes with side length 2−j+1 parallel to the coordinate axis and 2−jmas left corner. Let χj,m denote their characteristic functions.

Definition 2.3 Let 0 < p, q ≤ ∞, s ∈ R and λ := λj,Gm ∈ C : j ∈ N0, G ∈ Gj , m ∈ Zn. Then

bsp,q(Rn) = λ : ‖λ|bsp,q(Rn)‖ <∞ (2.16)

with

‖λ|bsp,q(Rn)‖ =

∞∑j=0

2j(s−np )q

∑G∈Gj

( ∑m∈Zn

|λj,Gm |p) qp

1q

(2.17)

and

fsp,q(Rn) = λ : ‖λ|fsp,q(Rn)‖ <∞ (2.18)

with

‖λ|fsp,q(Rn)‖ =∥∥∥ ∑j,G,m

2jsq|λj,Gm χj,m|q 1

q

|Lp(Rn)∥∥∥ (2.19)

(usual modifications if max(p, q) =∞).

For fixed n ∈ N we define

σp = n

(max

(1

p, 1

)− 1

), σp,q = n

(max

(1

p,

1

q, 1

)− 1

), (2.20)

where 0 < p, q ≤ ∞. We recall the wavelet decomposition theorem according to [17, Theorem 1.20, pp.15-17].

Proposition 2.4 (i) Let 0 < p ≤ ∞, 0 < q ≤ ∞, s ∈ R and ΨjG,m be the wavelet system in (2.14) based

on (2.8) and (2.9) with

u > max(s, σp − s). (2.21)



Let f ∈ S′(Rn). Then f ∈ Bsp,q(Rn) if, and only if, it can be represented as

f =∑j, G,m

λj,Gm 2−jn/2ΨjG,m, λ ∈ bsp,q(Rn), (2.22)

unconditional convergence being in S′(Rn). The representation is unique,

λj,Gm = λj,Gm (f) = 2jn/2⟨f,Ψj

G,m

⟩(2.23)

and

I : f 7→ λj,Gm (f) (2.24)

is an isomorphic map fromBsp,q(Rn) onto bsp,q(Rn). In particular, it holds ‖f |Bsp,q(Rn)‖ ∼ ‖λ |bsp,q(Rn)‖.

(ii) Let 0 < p <∞, 0 < q ≤ ∞, s ∈ R, and

u > max(s, σp,q − s). (2.25)

Let f ∈ S′(Rn). Then f ∈ F sp,q(Rn) if, and only if, it can be represented as

f =∑j, G,m

λj,Gm 2−jn/2ΨjG,m, λ ∈ fsp,q(Rn), (2.26)

unconditional convergence being in S′(Rn). The representation (2.26) is unique with (2.23). FurthermoreI in (2.24) is an isomorphic map from F sp,q(Rn) onto fsp,q(Rn) and ‖f |F sp,q(Rn)‖ ∼ ‖λ |fsp,q(Rn)‖.

Remark 2.5 For a detailed discussion how to understand⟨f,Ψj

G,m

⟩as a dual pairing of f ∈ Asp,q(Rn) and

ΨjG,m ∈ Cu(Rn) we refer to [16, Section 3.1.3] and for further reading to [13, Section 2.11.1 - 2.11.3].

Next we collect necessary facts about molecules.

Definition 2.6 Let 0 < p, q ≤ ∞, s ∈ R, and µ := µjm ∈ C : j ∈ N0, m ∈ Zn. Then

bsp,q(Rn) = µ : ‖µ | bsp,q(Rn)‖ <∞ (2.27)

with

‖µ | bsp,q(Rn)‖ =

∞∑j=0

2j(s−np )q

( ∑m∈Zn

|µjm|p) qp

1q

<∞ (2.28)

and

fsp,q(Rn) = µ : ‖µ|fsp,q(Rn)‖ <∞ (2.29)

with

‖µ | fsp,q(Rn)‖ =∥∥∥ ∑j∈N0,m∈Zn

2jsq|µjmχj,m(·)|q 1

q

|Lp(Rn)∥∥∥ (2.30)

(usual modifications if max(p, q) =∞).



Definition 2.7 Let K ∈ N0, N ∈ N0, and L > N + n − 1. Then L∞-functions bj,m : Rn 7→ C are called(K,N,L)-molecules, related to Qj,m, if

|Dγbj,m(x)| ≤ 2j|γ|(1 + 2j |x− 2−jm|)−L, |γ| ≤ K, j ∈ N0, m ∈ Zn, (2.31)

and ∫Rnxβbj,m(x) dx = 0, |β| < N, j ∈ N, m ∈ Zn. (2.32)

If N = 0 then (2.32) is empty (no condition). The following analogue of Proposition 2.4 for molecules can befound in [12].

Proposition 2.8 Let 0 < p, q ≤ ∞ (p <∞ for the F-spaces), s ≥ 0. Let K ∈ N0, N ∈ N0 with

K > s, N >

σp − s, B-spaces,σp,q − s, F -spaces,

(2.33)

and

L > N + n− 1, L > n+

σp, B-spaces,σp,q, F -spaces.

(2.34)

Let f ∈ S′(Rn). Then f ∈ Asp,q(Rn) if, and only if, it can be represented as

f =

∞∑j=0

∑m∈Zn

µjmbj,m, µ ∈ asp,q(Rn), (2.35)

unconditional convergence being in S′(Rn), where bj,m are (N,K,L)-molecules. Furthermore

‖f |Asp,q(Rn)‖ ∼ inf ‖µ | asp,q(Rn)‖, (2.36)

where the infimum is taken over all admissible representations.

3 Generalized heat equations

In this section, we deal with linear and nonlinear generalized heat equations. Let ω ∈ D′(Rn) and δ be the δ-distribution in R. Then ω⊗ δ = ω(x)⊗ δ(t) is the usual tensor product of the distributions ω and δ in D′(Rn+1).Further, let f ∈ D′(Rn+1) with supp f ⊂ Rn+1

+ . Then W ∈ D′(Rn+1) with suppW ⊂ Rn+1+ and

∂

∂tW + (−∆x)αW = f + ω(x)⊗ δ(t), α ∈ N (3.1)

is called a solution of the Cauchy problem in Rn+1 with the initial data ω. Let ω, W and f in (3.1) be regulardistributions. Furthermore, we assume that ω is continuous on Rn, f is continuous on the half space Rn+1

+ , and

W is 2α-times continuously differentiable on the domain Rn+1+ and continuous on the half space Rn+1

+ . Underthese assumptions one can show, similar to [15, Lemma 3.3.4], that W is a solution of the classical Cauchyproblem

∂

∂tW (x, t) + (−∆x)αW (x, t) = f(x, t), x ∈ Rn, t > 0, (3.2)

W (x, 0) = ω(x), x ∈ Rn. (3.3)

Conversely, every classical solution is also a solution of (3.1) if the functions f and W are extended by zero intothe domain (x, t) : (x, t) ∈ Rn+1, t < 0. We ask for solutions of (3.2) in the distributional sense which areregular and belong to some space Lv((0, T ), b, X) as defined in (2.7). Assume that f ∈ Lv((0, T ), b, X) and



let us denote by F its extension by zero to Rn × (R \ (0, T )). Suppose that F ∈ Lloc1 (Rn+1) ⊂ D′(Rn+1). We

put

Gα(x, t) =

Kαt (x), t > 0, x ∈ Rn,

0, t ≤ 0, x ∈ Rn(3.4)

with Kαt as defined in (1.6). Similar to [15, Theorem 3.2.4/1] or [20, §10.6] one can show that Gα is a funda-

mental solution of the generalized heat equation, i.e. ∂Gα

∂t + (−∆x)αGα = δ in D′(Rn+1). If there exists theconvolution Gα ∗ F in D′(Rn+1) in the sense of [20, §7, formula (17)] then it holds(

∂

∂t+ (−∆x)α

)(Gα ∗ F ) = F in D′(Rn+1). (3.5)

Because of F |Rn×(0,T )= f and supp F ⊂ Rn × [0, T ] we have(

∂

∂t+ (−∆x)α

)(Gα ∗ F ) = f in D′(Rn × (0, T )). (3.6)

On the other hand, we find for any ω ∈ S′(Rn) that(∂

∂t+ (−∆x)α

)(Kα

t ∗ ω)(x) =

(∂

∂t+ (−∆x)α

)Wαt ω(x) = 0, x ∈ Rn, t > 0 (3.7)

in the classical sense. This yields(∂

∂t+ (−∆x)α

)(Wα

t ω +Gα ∗ F ) = f in D′(Rn × (0, T )). (3.8)

Suppose that

H(x, t) =

∫ ∞−∞

∫RnGα(t− τ, x− y)F (y, τ) dy dτ (3.9)

=

h(x, t), x ∈ Rn, t > 0,

0, x ∈ Rn, t ≤ 0(3.10)

with

h(x, t) =

∫ t

0

∫RnGα(t− τ, x− y)f(y, τ) dy dτ =

∫ t

0

Wαt−τf(x, τ)dτ (3.11)

belongs to Lloc1 (Rn+1). Then there exists the convolution Gα ∗ F ∈ D′(Rn+1) and it holds Gα ∗ F = H . For a

proof we refer to [20, §7.5]. According to (3.6) and(3.8) we obtain(∂

∂t+ (−∆x)α

)(Wα

t ω + h) = f in D′(Rn × (0, T )). (3.12)

Thus, under the above assumptions,

W (x, t) = Wαt ω(x) + h(x, t) (3.13)

is a distributional solution of the inhomogeneous generalized heat equation (3.2).

3.1 α-caloric wavelets

From now on we assume that ω ∈ Asp,q(Rn) with A ∈ B,F, respectively. Under the conditions of Proposition2.4, we can represent

ω =∑j, G,m

λj,Gm 2−jn/2ΨjG,m, λ ∈ asp,q(Rn), (3.14)



with

λj,Gm = λj,Gm (ω) = 2jn/2⟨ω,Ψj

G,m

⟩. (3.15)

We are interested in a similar decomposition of Wαt ω. We split

ω = ω0 + ωu, u ∈ N, (3.16)

where

ω0 =∑m∈Zn

λmΨm, λm = λm(ω) = 〈ω,Ψm〉 , (3.17)

with

Ψm(x) =

n∏l=1

ψF (xl −ml), m ∈ Zn (3.18)

and

ωu =

∞∑j=0

∑G∈G∗

∑m∈Zn

λj,Gm 2−jn/2ΨjG,m, (3.19)

with (3.15) and G∗ = F,Mn∗, i.e., at least one of the Gl, l = 1, . . . , n is an M . It will be essential thatu ∈ N in (2.8) can be chosen arbitrarily large. That is the reason why we indicated the chosen u ∈ N in (3.19).Moreover, any Ψj

G,m in (3.19) satisfies moment conditions up to u as in (2.15). With Wαt as in (1.4) we obtain

for t > 0

Wαt ω(x) = Wα

t ω0(x) +Wα

t ωu(x)

=∑m∈Zn

λmWαt Ψm(x) +

∞∑j=0

∑G∈G∗

∑m∈Zn

λj,Gm 2−jn/2Wαt Ψj

G,m(x). (3.20)

As we will see below, (3.20) is for each fixed t > 0 a representation ofWαt ω by molecules belonging toAs+dp,q (Rn)

with some d ≥ 0. Thus the unconditional convergence of the series is preserved and the interchange of summation

and integration is justified at least as limits in S′(Rn) with respect to the dual pairing of(e−t|ξ|

2α)∨∈ S(Rn)

and the regular distribution ω ∈ S′(Rn). Let

bjG,m(x, t) := 2−jn/2Wαt Ψj

G,m(x) =

∫Rn

Kαt (x− y)2−jn/2Ψj

G,m(y)dy

=1

(2π)n/2

∫Rn

(e−t|ξ|

2α)∨

(x− y)

n∏l=1

ψGl(2jyl −ml)dy. (3.21)

According to the case α = 1, cf. [18, Subsection 2.4.2], the functions bjG,m(x, t) are called α-caloric wavelets.As already mentioned, we show that after a slight modification they are molecules in the sense of Definition 2.7for appropriately chosen parameters N, K, L.

Proposition 3.1 Let α ∈ N and let bjG,m(x, t) have the meaning of (3.21) based on (2.13), where ψM , ψF ∈Cu(R) are Daubechies wavelets with a given u ∈ N. We put

bjG,m(x, t)d = C 2jd td/2α bjG,m(x, t), j ∈ N0, G ∈ G∗, m ∈ Zn. (3.22)

Then there exists C > 0 such that the functions bjG,m(x, t)d are (N,K,L)- molecules according to Definition2.7 for any fixed t with 2jt1/2α ≥ 1, provided that N ≤ u, K ≤ u, N + n − 1 < L < u + n − d and0 ≤ d < u−N + 1.



P r o o f. Step 1. We start with the moment conditions. It is sufficient to consider bjG,m(x, t). Let |β| < u.Then

∫Rn

xβbjG,m(x, t)dx =

∫Rn

xβ

∫Rn

Kαt (x− y)

n∏l=1

ψGl(2jyl −ml)dy

dx

=

∫Rn

∫Rn

Kαt (z)(y + z)βdz

n∏l=1

ψGl(2jyl −ml)dy

=

∫Rn

Kαt (z)

n∏l=1

∫R

(yl + zl)βlψGl(2

jyl −ml)dyl

dz

= 0,

according to (2.15), since at least one of the Gl = M . So the moment conditions (2.32) hold for bjG,m(·, t)d ifN ≤ u.Step 2. We prove (2.31) with |γ| = 0, where we may assume m = 0, i.e., we wish to estimate

bjG(x, t) = bjG,0(x, t) =

∫Rn

Kαt (x− y)

n∏l=1

ψGl(2jyl)dy, (3.23)

where j ∈ N0, G ∈ G∗, 2jt1/2α ≥ 1. We rewrite

bjG(x, t) =

∫Rn

t−n/2αKα

(x− yt1/2α

) n∏l=1

ψGl(2jyl)dy, (3.24)

where Kα(x) = (2π)−n/2(e−|η|

2α)∨

(x). Apparently,

bjG(t1/2α x, t) =

∫Rn

Kα(x− y)

n∏l=1

ψGl(2j t1/2α yl)dy. (3.25)

Next we expand Kα at the origin in a Taylor polynomial with remainder term of order u and substitute it into(3.25). Because of the moment conditions of Ψj

G,m terms of order less than u vanish and we only have to estimate

∣∣bjG(t1/2α x, t)∣∣ .

∫Rn

∑|β|=u

∣∣(DβKα)(x− ξ) yβn∏l=1

ψGl(2j t1/2α yl)

∣∣dy.

∫Rn

∑|β|=u

∣∣yβ∣∣ ∣∣∣∣∣n∏l=1

ψGl(2j t1/2α yl)

∣∣∣∣∣ dy (3.26)

.∫Rn

|y|u∣∣∣∣∣n∏l=1

ψGl(2j t1/2αyl)

∣∣∣∣∣ dy,

where we used the boundedness of the derivatives of Kα in (3.26). Since the integrand is zero outside a ball ofradius c 2−j t−1/2α centered at the origin, we obtain

|bjG(t1/2α x, t)| .∫

|y|<c 2−jt−1/2α

|y|udy ≤ c(

2−jt−1/2α)u+n

, ∀x ∈ Rn. (3.27)



On the other hand, it follows from (3.25) and Kα ∈ S(Rn)

|bjG(t1/2α x, t)| .∫

|y|<c 2−jt−1/2α

∣∣∣ (e−|η|2α)∨ (x− y)∣∣∣dy

≤∫

|y|<c 2−jt−1/2α

cL1 + |x− y|L

dy

.cL

(1 + |x|)L

∫|y|<c 2−jt−1/2α

(1 + |y|)Ldy

.cL

(1 + |x|)L

∫|y|<c 2−jt−1/2α

dy

∼ cL(1 + |x|)L

(2−jt−1/2α)n (3.28)

for all L > 0 and 2jt1/2α ≥ 1. Let now 0 < ε < 1. Using both estimates, (3.27) and (3.28), we find

|bjG(t1/2α x, t)| = |bjG(t1/2α x, t)|ε|bjG(t1/2α x, t)|(1−ε)

≤ cε,L(1 + |x|)L

(2−jt−1/2α

)εn (2−jt−1/2α

)(1−ε)(u+n)

(3.29)

=cε,L

(1 + |x|)L

(2−jt−1/2α

)u−εu+n−L(2−jt−1/2α

)−L≤ cε,L

(1 + 2jt1/2α|x|)L(

2−jt−1/2α)u−εu+n−L

since 2jt1/2α ≥ 1. Replacing t1/2αx 7→ x in (3.29) yields

|bjG(x, t)| ≤ cε,L(1 + 2j |x|)L

(2−jt−1/2α

)u−εu+n−L. (3.30)

Let now 0 ≤ d < u−N + 1. We choose L > 0 and 0 < ε < 1 such that

N + n− 1 < L ≤ (1− ε)u+ n− d.

Thus, it follows(2−jt−1/2α

)(1−ε)u+n−L≤(

2−jt−1/2α)d. (3.31)

Hence, we have shown

|bjG(x, t)| ≤ cε,L(1 + 2j |x|)L

(2−jt−1/2α

)dfor any d ≥ 0 with d < u−N + 1.Step 3. Let |γ| ≥ 1. Then,

|DγxbjG(t1/2α x, t)| ≤ 2|γ|jt|γ|/2α

∫Rn|Kα(x− y)(Dγ

n∏l=1

ψGl)(2jt|γ|/2αyl)|dy

in the classical sense if |γ| ≤ K ≤ u. Because of the compactness of supp ψGl , the moment conditions also holdfor the derivatives of ψM up to order u, which is seen by integration by parts. Hence, we obtain

|DγxbjG,m(t1/2α x, t)| = t|γ|/2α|(DγbjG,m)(t1/2α x, t)| . 2j|γ|t|γ|/2α

(2−jt−1/2α

)u+n

. (3.32)



A similar calculation as in Step 2 yields

|DγxbjG,m(x, t)| ≤ c 2−jdt−d/2α

2j|γ|

(1 + 2j |x|)L(3.33)

with N, K, L as above. Thus, the functions in (3.22) are (N,K,L) -molecules for 2jt1/2α ≥ 1.

Theorem 3.2 Let 0 < p, q ≤ ∞ (p < ∞ for the F-spaces), s ∈ R. Let d ≥ 0 such that s + d ≥ 0, α ∈ N,and u ∈ N with

u > d+

max(s, σp), for B-spacesmax(s, σp,q), for F -spaces.

(3.34)

Then, the numbersN ,K,L in Proposition 3.1 can be chosen such that for someC > 0 and any twith 2jt1/2α ≥ 1

bjG,m(x, t)d = C 2jd td/2α bjG,m(x, t), j ∈ N0, G ∈ G∗, m ∈ Zn

are molecules for As+dp,q (Rn) in the sense of Proposition 2.8.

P r o o f. We restrict ourselves to the B-spaces. First we choose N := [σp] + 1. Then N > σp − (s + d).Since u > σp = [σp] + σp it follows that u ≥ N . So the moment conditions hold for Wα

t ΨjG,m with the

above chosen N . Let now n+ σp < L < u+ n− d. Because of the choice of N we have L > N + n− 1, too.From u > d + σp it follows that d < u − σp < u −N + 1. Regarding the derivatives of bjG,m(x, t)d, we claims + d < K ≤ u. Hence, the conditions of both, Proposition 3.1 and Proposition 2.8 are satisfied. Replacing σpby σp,q leads to similar results for the F -spaces.

Later on we only need spaces Asp,q(Rn) with p, q ≥ 1. Hence, the considered spaces are Banach spaces. Thisreduces the assumptions with respect to u, N , K and L as follows.

Corollary 3.3 Let 1 ≤ p, q ≤ ∞ (p < ∞ for the F-spaces), s ∈ R. Let d ≥ 0 such that s + d ≥ 0, α ∈ N,and u ∈ N with

u > d+ max(s, 0). (3.35)

Then there exists C > 0 such that

bjG,m(x, t)d = C 2jd td/2α bjG,m(x, t), j ∈ N0, G ∈ G∗, m ∈ Zn

for any t with 2jt1/2α ≥ 1 are molecules for As+dp,q (Rn) in the sense of Proposition 2.8 if N = 1, K = u, andn < L < u+ n− d.

3.2 Mapping properties related to the generalized heat equation

In order to prove the above mentioned a-priori estimate (1.9), we first derive a corresponding result for ωu definedin (3.19).

Proposition 3.4 Let 1 ≤ p, q ≤ ∞ (p <∞ for the F-spaces), s ∈ R. Let d ≥ 0 such that s+ d ≥ 0, α ∈ N,and u ∈ N with

u > d+ max(s, 0) (3.36)

Then there exists a constant c > 0 such that for any t ≥ 1 and all ω ∈ Asp,q(Rn)

td/2α‖Wαt ω

u|As+dp,q (Rn)‖ ≤ c‖ω|Asp,q(Rn)‖, (3.37)

whereas ωu is given by (3.19).



P r o o f. t ≥ 1 ensures 2jt1/2α ≥ 1 for j ∈ N0. Consequently, we can apply Corollary 3.3. Taking intoaccount the molecular representation of Wα

t ωu, we write

Wαt ω

u =

∞∑j=0

∑G∈G∗

∑m∈Zn

λj,Gm 2−jn/2Wαt Ψj

G,m

=

∞∑j=0

∑G∈G∗

∑m∈Zn

λj,Gm bjG,m(·, t) (3.38)

=

∞∑j=0

∑G∈G∗

∑m∈Zn

µj,Gm bjG,m(·, t)d

with

C µj,Gm = 2−jdt−d/2αλj,Gm , (3.39)

where C > 0 has the same meaning as in (3.22). Let

µ∗ = µj,Gm : j ∈ N0, G ∈ G∗, m ∈ Zn (3.40)

and similarly λ∗. By Proposition 2.8, it remains to show that µ∗ ∈ as+dp,q (Rn) if A = B. It holds

‖µ∗|bs+dp,q (Rn)‖ =

∞∑j=0

2j(s+d−np )q

∑G∈G∗

( ∑m∈Zn

|µj,Gm |p) qp

1q

∼

∞∑j=0

2jdq 2j(s−np )q

∑G∈G∗

( ∑m∈Zn

2−jdpt−dp/2α|λj,Gm |p) qp

1q

= t−d/2α

∞∑j=0

2j(s−np )q

∑G∈G∗

( ∑m∈Zn

|λj,Gm |p) qp

1q

= t−d/2α‖λ∗|bsp,q(Rn)‖.

Thus,

‖Wαt ω

u|Bs+dp,q (Rn)‖ ≤ c ‖µ∗|bs+dp,q (Rn)‖ ∼ t−d/2α‖λ∗|bsp,q(Rn)‖

≤ c t−d/2α‖λ |bsp,q(Rn)‖ ∼ t−d/2α‖ω|Bsp,q(Rn)‖, (3.41)

where we used Proposition 2.4. The proof for the F -spaces is similar.

Theorem 3.5 Let 1 ≤ p, q ≤ ∞ (p <∞ for the F-spaces), s ∈ R, d ≥ 0 and α ∈ N. Then there is a constantc > 0 such that

td/2α‖Wαt ω|As+dp,q (Rn)‖ ≤ c‖ω|Asp,q(Rn)‖ (3.42)

for all t with 0 < t ≤ 1 and ω ∈ Asp,q(Rn).

P r o o f. Step 1. We prove (3.42) for s+ d ≥ 0 and take advantage of the representation of ω ∈ Asp,q(Rn) asin Proposition 2.4. Let 2−2αk < t ≤ 2−2α(k−1), k ∈ N. Instead of (3.16)-(3.19) we split ω into

ω = ω0k + ωk =

∑j<k

∑G,m

λj,Gm 2−jn/2ΨjG,m +

∑j≥k

∑G,m

λj,Gm 2−jn/2ΨjG,m. (3.43)

We do not indicate u ∈ N chosen according to 3.35. If j ≥ k then 2jt1/2α > 2j2−k = 2j−k ≥ 1. ApplyingCorollary 3.3 and Proposition 2.8, we find as in Proposition 3.4

td/2α ‖Wαt ωk|As+dp,q (Rn)‖ ≤ c ‖ω|Asp,q(Rn)‖, (3.44)



where c > 0 is independent of k ∈ N, 2−2αk < t ≤ 2−2α(k−1) and ω ∈ Asp,q(Rn).Let now j < k and A = B. Then,

‖ω0k|Bs+dp,q (Rn)‖ ∼

∑j<k

2j(s+d−np )q

∑G∈Gj

( ∑m∈Zn

|λj,Gm |p) qp

1q

≤ 2kd

∑j<k

2j(s−np )q

∑G∈Gj

( ∑m∈Zn

|λj,Gm |p) qp

1q

. 2kd‖λ |bsp,q(Rn)‖. (3.45)

Similarly for the F -spaces. By means of the generalized Minkowski’s inequality and (3.45) we obtain

‖Wαt ω

0k|As+dp,q (Rn)‖ ≤

∫Rn

Kαt (y)‖ω0

k(x− y)|As+dp,q (Rn)‖dy (3.46)

≤ ‖ω0k|As+dp,q (Rn)‖

∫Rn

Kαt (y)dy ≤ c 2kd‖λ|asp,q(Rn)‖ (3.47)

∼ c 2kd‖ω|Asp,q(Rn)‖. (3.48)

Because of the assumption 2−2αk < t ≤ 2−2α(k−1), it follows that t−d/2α ∼ 2kd. Consequently,

td/2α ‖Wαt ω

0k|As+dp,q (Rn)‖ ≤ c ‖ω|Asp,q(Rn)‖. (3.49)

Together with (3.44) the assertion follows for s+ d ≥ 0.Step 2. It remains to be proved the case s+d < 0. For this purpose consider the lift operator Iδ , δ ∈ R introducedin (2.5). Using (1.4) we see that

F(Wαt (Iδω))(ξ) = F(Wα

t (F−1[(1 + | · |2)−δ2Fω])(ξ)

= e−t|ξ|2α

(1 + |ξ|2)−δ2Fω(ξ)

and

F(Iδ(Wαt ω))(ξ) = (1 + |ξ|2)−

δ2F(Wα

t ω)(ξ)

= e−t|ξ|2α

(1 + |ξ|2)−δ2Fω(ξ).

Thus, Wαt = I−δW

αt Iδ . Let s+ d < 0 and δ ∈ R such that s+ d− δ ≥ 0. Then we have by (2.6)

td/2α‖Wαt ω|As+dp,q (Rn)‖ = td/2α‖I−δWα

t Iδω|As+dp,q (Rn)‖

∼ td/2α‖Wαt Iδω|As+d−δp,q ‖ ≤ c ‖Iδω|As−δp,q ‖ (3.50)

∼ c ‖ω|Asp,q(Rn)‖

which completes the proof.

Now we come to the corresponding estimate for W (x, t), x ∈ Rn, t > 0, as given in (3.13) and h(x, t)corresponding to (3.11).

Proposition 3.6 Let 1 ≤ p, q ≤ ∞ (p <∞ for the F-spaces), s ∈ R and α ∈ N. Let T > 0 and

1

α< v ≤ ∞, −∞ < a < α− 1

v, 0 ≤ d < 2

(α− 1

v

), −∞ < αg ≤ d. (3.51)

Let

u0 ∈ As+αgp,q (Rn) and f ∈ Lαv((0, T ),a

α, Asp,q(Rn)). (3.52)



Then there is a constant c > 0, independent of u0 and f , such that

‖W (·, t)|As+dp,q (Rn)‖ ≤ c t−d−αg2α ‖u0|As+αgp,q (Rn)‖ (3.53)

+ c t1−1αv−

d2α−

aα

t∫0

τav‖f(·, τ)|Asp,q(Rn)‖αvdτ

1/αv

(3.54)

for all t with 0 < t < T (with the usual modification if v =∞).

P r o o f. We apply Theorem 3.5 with s+ αg in place of s and d− αg in place of d. Thus,

‖Wαt u0|As+dp,q (Rn)‖ ≤ c t−

d−αg2α ‖u0|As+αgp,q (Rn)‖. (3.55)

For the second summand in (3.13) we obtain

‖t∫

0

Wαt−τf(·, τ)dτ |As+dp,q (Rn)‖ ≤

t∫0

(t− τ)−d2α τ−

aα τ

aα ‖f(·, τ)|Asp,q(Rn)‖dτ

≤ c

t∫0

(t− τ)−d(αv)′

2α τ−aα (αv)′dτ

1/(αv)′ t∫0


1/αv

≤ c t1−1αv−

d2α−

aα

t∫0


1/αv

using the assumptions a < α − 1/v and d < 2(α − 1/v), Theorem 3.5 and Hölder’s inequality with exponentαv.

Note that it was immaterial to replace 1 in Theorem 3.5 by any fixed T > 0.

3.3 Nonlinear generalized heat equation

In this subsection we prove our main result, that is the existence and uniqueness of mild and strong solutions of

∂

∂tu(x, t) + (−∆x)αu(x, t)−Du2(x, t) = 0, x ∈ Rn, 0 < t < T, (3.56)

u(x, 0) = u0(x), x ∈ Rn (3.57)

where 0 < T ≤ ∞, 2 ≤ n ∈ N, α ∈ N and Du2 =n∑i=1

∂∂xi

u2. Interpretation of the nonlinear term Du2

as inhomogeneity allows us to apply Proposition 3.6 to the operator Tu0defined in (1.3). Then we combine

the previous estimates with a fixed point argument for Tu0. We call solutions u ∈ S′(Rn+1) of (3.56), (3.57),

which are a fixed point of Tu0 , mild. In addition to uniqueness (local in time, 0 < t < T ) we look for strongsolutions. A solution u is called strong if it is mild and belongs to the spaceC([0, T ), As0p,q(Rn)) for all initial datau0 ∈ As0p,q(Rn). For more detailed explanations in this context we refer to [3] and [7]. This is the understandingof Theorem 3.8 below.

To deal with the nonlinearity Du2, we assume that the considered solution spaces with respect to the spacevariable are multiplication algebras. Then ‖f1f2|Asp,q(Rn)‖ ≤ c‖f1|Asp,q(Rn)‖‖f2|Asp,q(Rn)‖ holds for someconstant c > 0 and all f1, f2 ∈ Asp,q(Rn). Recall that Asp,q(Rn) is a multiplication algebra if, and only if,

Asp,q(Rn) = Bsp,q(Rn) with

s > n/p where 0 < p, q ≤ ∞,s = n/p where 0 < p <∞, 0 < q ≤ 1,

or

Asp,q(Rn) = F sp,q(Rn) with

s > n/p where 0 < p <∞, 0 < q ≤ ∞,s = n/p where 0 < p ≤ 1, 0 < q ≤ ∞.



This result may be found for instance in [11].Remark 3.7 We have to clarify first the question if H(x, t) as defined in (3.9) - (3.11) with f = Du2 is

locally integrable in Rn+1. The answer to that will be crucial for proving that our solution is a C∞- functionwith respect to space and time. Therefor let u ∈ L2αv((0, T ), a

2α , Asp,q(Rn)), with the same assumptions on the

parameters as formulated in Theorem 3.8 below, be a mild solution of (3.56), (3.57), i.e. a fixed point of Tu0

defined in (1.3). Then it holds

u(x, t) = Wαt u0(x) +

∫ t

0

Wαt−τDu

2(y, τ) dτ, x ∈ Rn, t > 0. (3.58)

Here

Wαt−τDu

2(·, τ) = (Gαt−τ ∗Du2)(·, τ), 0 < τ < t

stands for the convolution of Gαt−τ ∈ S(Rn) and Du2 ∈ S′(Rn).It follows from Theorem 3.5 with −s+ α− αg < d < 2α whereas α ∈ N and 0 < g ≤ 1 that

Wα(x, t) =

Wαt u0(x), x ∈ Rn, t ∈ (0, T ),

0, x ∈ Rn, t ∈ R \ (0, T )

belongs to Lloc1 (Rn+1) for all u0 ∈ As−α+αg

p,q (Rn) and it holds(∂

∂t+ (−∆x)α

)Wαt u0(x) = 0, x ∈ Rn, t > 0.

Let

U(x, t) =

u(x, t), x ∈ Rn, t ∈ (0, T ),

0, x ∈ Rn, t ∈ R \ (0, T ).

Then U2 ∈ Lloc1 (Rn+1). Defining

h(x, t) :=

t∫0

(Gαt−τ ∗ u2)(x, τ)dτ =

t∫0

∫RnGα(x− y, t− τ)u2(y, τ) dy dτ

and using again Theorem 3.5 we find similar to the proof of Proposition 3.6 that

H(x, t) =

h(x, t), x ∈ Rn, t > 0,

0, x ∈ Rn, t ≤ 0

belongs to Lloc1 (Rn+1), too.

Hence, the convolution of Gα and U2 and its distributional derivative DH = D(Gα ∗ U2) = (DGα) ∗ U2

exist in D′(Rn+1). Since Gα is a fundamental solution of the generalized heat equation this yields(∂

∂t+ (−∆x)α

)DH = D

(∂

∂t+ (−∆x)α

)(Gα ∗ U2) = DU2 in D′(Rn+1).

On the other hand, we can show that DGα ∗ U2 ∈ Lloc1 (Rn+1). Its restriction to Rn+1

+ coincides with

k(x, t) :=

∫ t

0

Wαt−τDu

2(x, τ) dτ, x ∈ Rn, 0 < t < T.

Combining these arguments we obtain using (3.58)(∂

∂t+ (−∆x)α

)u =

(∂

∂t+ (−∆x)α

)k = Du2 in D′(Rn × (0, T )). (3.59)

This means that any mild solution is also a solution in the distributional sense.



Theorem 3.8 Let n ≥ 2, α ∈ N, 1 ≤ p, q ≤ ∞ (p < ∞ for F - spaces) and s such that Asp,q(Rn) is amultiplication algebra.

(i) Let

0 < λ < g ≤ 1,2

α< v ≤ ∞, a = α− 1

v− αλ (3.60)

and let u0 ∈ As−α+αgp,q (Rn) for the initial data. Then there exists a number T > 0 such that

∂∂tu(x, t) + (−∆x)αu(x, t)−Du2(x, t) = 0, x ∈ Rn, 0 < t < T,

u(x, 0) = u0(x), x ∈ Rn (3.61)

has a unique mild solution

u ∈ L2αv((0, T ),a

2α, Asp,q(Rn)) ∩ C∞(Rn × (0, T )).

(ii) If, in addition, p <∞, q <∞ and

g

2≤ λ < g ≤ 1 if v <∞ and

g

2< λ < g ≤ 1 if v =∞ (3.62)

then the above solution is strong, that means u ∈ C([0, T ), As−α+αgp,q (Rn)).

P r o o f. Step 1. We assume v <∞. The main idea is to apply the estimates of Proposition 3.6 with s− α inplace of s, d = α and

f = Du2 ∈ Lαv((0, T ),a

α, As−1

p,q (Rn)). (3.63)

We ask for a fixed point of (1.3), i.e.

Tu0u(x, t) = Wα

t u0(x) +

∫ t

0

Wαt−τDu

2(x, τ)dτ (3.64)

in L2αv((0, T ), a2α , A

sp,q(Rn)). Then we obtain for 0 < t ≤ T

‖Tu0u(·, t)|Asp,q(Rn)‖ ≤ c t−

α−αg2α ‖u0|As−α+αg

p,q (Rn)‖

+ c t1−1αv−

α2α−

aα

(∫ t

0

τav‖Du2(·, τ)|As−αp,q (Rn)‖αvdτ)1/αv

. (3.65)

We enlarge the integral extending the limits from (0, t) to (0, T ) and multiply both sides with ta2α . Raising to the

power of 2αv and integrating over (0, T ) yields∫ T

0

tav‖Tu0u(·, t)|Asp,q(Rn)‖2αvdt ≤ c

∫ T

0

t(−α+αg+a)v dt ‖u0|As−α+αgp,q (Rn)‖2αv

+ c

∫ T

0

tαv−2−av dt

(∫ T

0

τav‖Du2(·, τ)|As−αp,q (Rn)‖αvdτ

)2

(3.66)

≤ c T δ‖u0|As−α+αgp,q (Rn)‖2αv + c Tκ

(∫ T

0

τav‖Du2(·, τ)|As−αp,q (Rn)‖αvdτ

)2

where

δ = (−α+ αg + a)v + 1 = α(g − λ)v > 0 (3.67)



since g > λ and

κ = αv − 1− av = αλv > 0 (3.68)

since λ > 0. Now we use the embedding As−1p,q (Rn) → As−αp,q (Rn) with α ≥ 1, that D is a bounded mapping

from Asp,q(Rn) to As−1p,q (Rn) and the fact that Asp,q(Rn) is assumed to be a multiplication algebra. It follows that∫ T

0

tav‖Tu0u(·, t)|Asp,q(Rn)‖2αvdt

≤ c T δ‖u0|As−α+αgp,q (Rn)‖2αv + c Tκ

(∫ T

0

τav‖u(·, τ)|Asp,q(Rn)‖2αvdτ

)2

. (3.69)

Thus, we see that if T > 0 for given u0 is chosen sufficiently small then Tu0maps the unit ball UT in

L2αv((0, T ), a2α , A

sp,q(Rn)) into itself. To show that Tu0

: UT 7→ UT is a contraction, consider u, v ∈ UT . Asimilar calculation as above gives

‖Tu0u(·, t)− Tu0

v(·, t)|Asp,q(Rn)‖

≤ c t 12−

aα−

1αv

(∫ t

0

τav‖u2(·, τ)− v2(·, τ)|Asp,q(Rn)‖αvdτ)1/αv

(3.70)

≤ c t 12−

aα−

1αv

(∫ t

0

τav‖u(·, τ)− v(·, τ)|Asp,q(Rn)‖αv‖u(·, τ) + v(·, τ)|Asp,q(Rn)‖αvdτ)1/αv

.

Let temporarily XsT = L2αv((0, T ), a

2α , Asp,q(Rn)). Applying Hölder’s inequality with exponent 2 and using

u, v ∈ UT this yields

‖Tu0u− Tu0

v|XsT ‖ ≤ c T

κ2αv ‖u− v|Xs

T ‖‖u+ v|XsT ‖ (3.71)

with the same κ as is (3.68). The second norm can be estimated by means of Minkowski’s inequality. Thus,Tu0

: UT 7→ UT is a contraction if T > 0 is small enough. Since we deal with Banach spaces we have shownthat Tu0

has a unique fixed point in UT which is according to Lemma 3.7 a solution of (3.56).

Step 2. So far we know that (3.56) has a unique mild solution in UT . To extend this assertion to the wholespace, we require a certain amount of preparation. In particular, we show first that the solution u is aC∞-functionwith respect to space and time. Let again be Xs

T = L2αv((0, T ), a2α , A

sp,q(Rn)). We start with the smoothness

with respect to the space variable. To this end, we apply Proposition 3.6 with s− α in place of s and d = α+ ηwith some η > 0 such that α+ η < 2(α− 1

v ), which is possible since 2 < αv, and obtain

‖Tu0u(·, t)|As+ηp,q (Rn)‖ ≤ c t−

(α+η)−αg2α ‖u0|As−α+αg

p,q (Rn)‖

+ c t12−

1αv−

η2α−

aα

(∫ t

0

τav‖u(·, τ)|Asp,q(Rn)‖2αvdτ)1/αv

. (3.72)

The idea is to iterate this argument whereas each iteration step generates a solution of (3.56) which is smootherthan the previous one. Therefore we need smoother initial data. Thus, we apply (3.65) - (3.69) with initial datauε(x) = u(x, ε) at some ε > 0. Since Tu0

u = u for a solution u of (3.56) we have uε(x) ∈ As+ηp,q (Rn) →As+η−α+αgp,q (Rn). Replacing s− α by s− α+ η yields

‖Tuεu(·, t)|As+2ηp,q (Rn)‖ ≤ c t−

(α+η)−αg2α ‖uε|As+η−α+αg

p,q (Rn)‖

+ c t12−

1αv−

η2α−

aα

(∫ t

0

τav‖Du2(·, τ)|As+η−αp,q (Rn)‖αvdτ)1/αv

≤ c t−(α+η)−αg

2α ‖uε|As+η−α+αgp,q (Rn)‖

+ c t12−

1αv−

η2α−

aα

(∫ t

0

τav‖u(·, τ)|As+ηp,q (Rn)‖2αvdτ)1/αv

(3.73)



since As+ηp,q (Rn) is a multiplication algebra, too. In particular, we see that the exponents remain the same. Weproceed as in Step 1 and enlarge the integral extending the limits to (0, T ). Raising to the power of 2αv andintegrating over (0, T ) leads after the k-th iteration to

‖Tuεu|Xs+kηT ‖ ≤ c T δ′

2αv ‖uε|As+(k−1)η−α+αgp,q (Rn)‖+ c T

κ2αv ‖u|Xs+(k−1)η

T ‖

where

δ′ = δ − ηv > 0 and κ = κ − ηv > 0

with δ, κ as in (3.67), (3.68) and a sufficiently small η > 0. In doing so we obtain after the k-th step a so-lution u of (3.56) which belongs to some function space As+kηp,q (Rn). Now we conclude with the embedding

As+kηp,q (Rn) → Bs+kη−np∞,∞ = Cs+kη−

np (Rn) for any k ∈ N that u is a C∞-function with respect to the space

variable.

Step 3. We show that the solution u is a C∞-function with respect to t in any time interval (t0, t1) with0 < t0 < t1 < T . From Step 1 we know that u ∈ L∞((0, T ), a

2α , Asp,q(Rn)), i.e.

supt∈(t0,t1)

ta2α ‖u(·, t)|Asp,q(Rn)‖ <∞.

From this it follows

supt∈(t0,t1)

‖u(·, t)|Asp,q(Rn)‖ = supt∈(t0,t1)

t−a2α t

a2α ‖u(·, t)|Asp,q(Rn)‖

≤ t−a2α

0 supt∈(t0,t1)

ta2α ‖u(·, t)|Asp,q(Rn)‖ <∞

independent of s. In combination with Step 2 this leads to

supx∈Ω, t∈(t0,t1)

|Dβxu(x, t)| ≤ cβ for all β ∈ Nn0 .

where Ω ⊂ Rn is a bounded domain. Since u satisfies

(∂

∂t+ (−∆α

x)u = Du2, (3.74)

as we have shown in Lemma 3.7, this is also true for the distributional derivative ∂∂tu. Thus we obtain iteratively

supx∈Ω, t∈(t0,t1)

| ∂k

∂tkDβxu(x, t)| ≤ cβ , for all β ∈ Nn0 ,

hence, ∂k

∂tkDβxu ∈ L∞(Ω× (t0, t1)). Application of Sobolevs embedding theorem leads to the desired result for

any k ∈ N, β ∈ N0.

Step 4. To extend the uniqueness to the whole space let u ∈ UT be the above solution and v ∈ XsT a second

solution. From (3.70) - (3.71) we know that

‖Tu0u(·, t)− Tu0v(·, t)|Asp,q(Rn)‖ ≤ c t 12−

1αv−

aα ‖u− v|Xs

T0‖‖u+ v|Xs

T0‖ (3.75)

for any 0 < t ≤ T0 ≤ T . We apply Minkowski’s inequality and use u ∈ UT . Then we obtain for 0 < T0 ≤ Tand the same κ > 0 as in (3.68)

‖u− v|XsT0‖ ≤ c Tκ

0 (1 + ‖v|XsT ‖)‖u− v|Xs

T0‖. (3.76)

If we choose T0 > 0 small enough such that c Tκ0 (1 + ‖v|Xs

T ‖) < 1 it follows that u(·, t) = v(·, t) for anyt ∈ (0, T0]. Denote

T1 := supt ∈ (0, T ] : u(·, t) = v(·, t)



and assume T1 < T . Because of the continuity of u and v, T1 is the maximum with this property. Now wetake u(·, T1) ∈ Asp,q(Rn) → As−α+αg

p,q as new initial value and proceed as in the previous steps until (3.76)inclusively. There exists a unique solution u in a neighbourhood Uδ(T1) with u(·, T1) = u(·, T1). Since it holdsthat u(·, t) = u(·, t) for all t ∈ (0, T1]∩Uδ(T1) we have extended u to some interval (0, T2] with T1 < T2. Thus,we have prolongated u(·, t) − v(·, t) = 0 to some interval (0, T2] where T1 < T2 ≤ T which contradicts theassumption. This proves the uniqueness in L2αv((0, T ), a

2α , Asp,q(Rn)).

Step 5. We show first, that the continuity up to t = 0 depends only on Wαt u0 and u0.

‖u(·, t)− u0|As−α+αgp,q (Rn)‖

≤ c ‖Wαt u0 − u0|As−α+αg

p,q (Rn)‖+

∫ t

0

‖Wαt−τDu

2(·, τ)|As−α+αgp,q (Rn)‖dτ (3.77)

To the second summand, which we denote by (I), we apply Theorem 3.5 with d = αg, s − α in place of s andfixed t ∈ (0, T ) as follows

(I) ≤ c∫ t

0

(t− τ)−αg2α ‖Du2(·, τ)|As−αp,q (Rn)‖dτ

≤ c∫ t

0

(t− τ)−g2 ‖u2(·, τ)|Asp,q(Rn)‖dτ

≤ c∫ t

0

(t− τ)−g2 τ−

aα τ

aα ‖u(·, τ)|Asp,q(Rn)‖2dτ

≤ c(∫ t

0

(t− τ)−g2 (av)′τ−

a(αv)′α dτ

) 1(αv)′

(∫ t

0

τav‖u(·, τ)|Asp,q(Rn)‖2αvdτ) 1αv

= t−g2−

aα+1− 1

αv

(∫ t

0

τav‖u(·, τ)|Asp,q(Rn)‖2αvdτ) 1αv

. (3.78)

Assuming v <∞ and letting t tend to zero then (3.78) tends to zero if − g2 −aα + 1− 1

αv ≥ 0. That means, onecan consider solutions u ∈ L2αv((0, T ), a

2α , Asp,q(Rn)) with parameters a ≤ α− αg

2 −1v . If v =∞ one has to

choose a < α− αg2 −

1v since the integral is substituted by the supremum. This is satisfied if λ ≥ g

2 and λ > g2 ,

respectively.Concerning the first summand in (3.77) we obtain

Wαt u0(x)− u0(x) =

1

(2π)n/2

∫Rn

t−n/2α(e−|ξ|

2α)∨(x− y

t1/2α

)u0(y)dy − u0(x)

=1

(2π)n/2

∫Rn

(e−|ξ|

2α)∨

(z)[u0(x− t1/2αz)− u0(x)]dz. (3.79)

Thus, since p, q ≥ 1

‖Wαt u0 − u0|As−α+αg

p,q (Rn)‖

.∫

|z|≤N

(e−|ξ|

2α)∨

(z)‖u0(x− t1/2αz)− u0(x)|As−α+αgp,q (Rn)‖dz

+

∫|z|>N

(e−|ξ|

2α)∨

(z)‖u0(x− t1/2αz)− u0(x)|As−α+αgp,q (Rn)‖dz. (3.80)

Since u0 ∈ As−α+αgp,q (Rn) the second integral is smaller than ε, independent of x and t, if we choose a sufficiently

large N > 0. Fixing this N , also the first integral is smaller than ε, provided that t is appropriately small,0 < t ≤ t0(ε), and max(p, q) <∞. Thus,

‖Wαt u0 − u0|As−α+αg

p,q (Rn)‖ ≤ 2ε if t ≤ t0(ε). (3.81)



Since ε can be choosen arbitrarily small we have shown the continuity of u(·, t) up to t = 0 in the Banach spaceAs−α+αgp,q (Rn). Hence the solution u is strong in the prescribed sense.

Remark 3.9 The classical case α = 1 of Theorem 3.8 is essentially covered by [19, Theorems 4.10, 4.14, pp.119, 121]

Acknowledgements We would like to thank Hans Triebel for useful comments and suggestions. Furthermore we thank thereferees for careful reading and valuable comments.

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