Elsevier

Journal of Differential Equations

Volume 287, 25 June 2021, Pages 376-427
Journal of Differential Equations

An Lq(Lp)-theory for diffusion equations with space-time nonlocal operators

https://doi.org/10.1016/j.jde.2021.04.003Get rights and content

Abstract

We present an Lq(Lp)-theory for the equationtαu=ϕ(Δ)u+f,t>0,xRd;u(0,)=u0. Here p,q>1, α(0,1), tα is the Caputo fractional derivative of order α, and ϕ is a Bernstein function satisfying the following: δ0(0,1] and c>0 such that(0.1)c(Rr)δ0ϕ(R)ϕ(r),0<r<R<. We prove uniqueness and existence results in Sobolev spaces, and obtain maximal regularity results of the solution. In particular, we prove|tαu|+|u|+|ϕ(Δ)u|Lq([0,T];Lp)N(fLq([0,T];Lp)+u0Bp,qϕ,22/αq), where Bp,qϕ,22/αq is a modified Besov space on Rd related to ϕ.

Our approach is based on BMO estimate for p=q and vector-valued Calderón-Zygmund theorem for pq. The Littlewood-Paley theory is also used to treat the non-zero initial data problem. Our proofs rely on the derivative estimates of the fundamental solution, which are obtained in this article based on the probability theory.

Introduction

Many types of diffusion equations have been used to describe diverse phenomena in various fields including mathematics, engineering, biology, hydrology, finance, and chemistry. The classical heat equation tu=Δu describes the heat propagation in homogeneous media. When α(0,1), the equation tαu=Δu describes the anomalous diffusion exhibiting subdiffusive behavior caused by particle sticking and trapping effects (e.g. [30], [31]). On the other hand, the spatial nonlocal operator ϕ(Δ) describes long range jumps of particles, diffusions on fractal structures, and long time behavior of particles moving in space with quenched and disordered force field (e.g. [5], [13]).

The space-time fractional diffusion equation can be used to describe the combined phenomena, for instance, jump diffusions with a higher peak and heaver tails (see e.g. [10], [16]). The space-time fractional equation is also related to the scaling limit of continuous time random walk (see [11], [16], [28]).

In this article we study the space-time fractional equationtαu=ϕ(Δ)u+f,t>0;u(0,)=u0. Here, α(0,1),p,q>1, tα is the Caputo fractional derivative of order α, and ϕ is a Bernstein function satisfying ϕ(0+)=0, that is, ϕ:(0,)(0,) such that(1)nϕ(n)(λ)0,λ>0,n=1,2,. The operator ϕ(Δ) is defined byϕ(Δ)u:=ϕ(Δ)u:=Fd1[ϕ(|ξ|2)Fd(u)(ξ)],uCc. For instance, if ϕ(λ)=λν/2, ν(0,2), then ϕ(Δ)=Δν becomes the fractional Laplacian. It turns out that ϕ(Δ) is a type of integro-differential operator, and the class of ϕ(Δ) is characterized by the infinitesimal generators of subordinate Brownian motions. See Section 2 for details.

Probabilistic representation of solution to equation (1.1) has been introduced e.g. in [7], [8], [10], [28]. Actually the transition density of subordinate Brownian motion delayed by an inverse subordinator becomes the fundamental solution, and this transition density and its appropriate time-fractional derivative appear in the solution representation. See Sections 4 and 5.

The main purpose of this article is to present a Sobolev-regularity theory of equation (1.1). We prove the uniqueness and existence in Sobolev spaces and obtain the maximal Lq(Lp)-regularity of solutions. In particular, we prove for any p,q>1,|tαu|+|u|+|ϕ(Δ)u|Lq([0,T];Lp)NfLq([0,T];Lp)+Nu0Bp,qϕ,22/αq, where Bp,qϕ,22/αq is a Besov space on Rd related to ϕ. Moreover, we obtain the maximal regularity of higher order derivatives as well as negative order derivatives of solutions.

Our proof for (1.2) is mainly based on BMO-estimate if u0=0, and Littlewood-Paley theory is used to treat the case u00. Specifically speaking, we prove that if u0=0 then|(tαu)#(t,x)|+|(ϕ(Δ)u)#(t,x)|NfL,t>0,xRd. Here (tαu)# denotes the sharp function of tαu. The BMO estimate and the Marcinkiewicz interpolation theorem lead to (1.2) for p=q, and the case pq is covered based on the vector-valued Calderón-Zygmund theorem. For the implement of these procedures, we rely on sharp upper bounds of arbitrary order space-time derivatives of the fundamental solution, which are obtained in Section 3. Due to the non-integrability of derivatives of the fundamental solution, our proofs of e.g. (1.3) are much more delicate than the proofs for PDEs with local operators. Condition (0.1) is a minimal assumption on ϕ such that our derivative estimates of the fundamental solution hold for all t>0. This is essential in this article because we are aiming to prove estimates for solutions which are independent of the time intervals where the solutions are defined.

Here are some related Lq(Lp)-theories for the diffusion equations with either time fractional derivative or spatial integro-differential operators. An Lq(Lp)-theory for the time fractional equationtαu=i,j=1daij(t,x)uxixj+f was introduced in [6], [36] when aij=δij. The result of [6], [36] is based on semigroup theory, and similar approach is used in [39] to treat the equation with uniformly continuous coefficients. Recently, the continuity condition of [39] is significantly relaxed in [12], [22]. For instance, if p=q then [12] only requires that the coefficients are only measurable in t and have small mean oscillation in x. The approach in [12] is based on the level set arguments. Regarding the equations with spatial integro-differential operators, an Lp-theory of the diffusion equation the typeut=Rd(u(x+y)u(x)χ(y)yu(y))J(x,dy)+f was introduced in [34]. Here χ is a certain indicator function and the jump kernel J(x,dy) is of the type a(x,y)|y|dα, where a(x,y) is homogeneous of order zero and sufficiently smooth in y. Recently, the condition on J(x,dy) has been generalized and weaken e.g. in [20], [23], [32], [33], [40].

This article is organized as follows. In Section 2, we introduce some basic facts on the fractional calculus, integro-differential operator ϕ(Δ), and related function spaces. We also introduce our main result, Theorem 2.8, in Section 2. In Section 3 we obtain sharp upper bounds of space-time derivatives of the fundamental solution. In section 4 we study the zero initial data problem, and non-zero initial data problem is considered in Section 5. Finally we prove our main result in Section 6.

We finish the introduction with some notations. We use “:=” or “=:” to denote a definition. The symbol N denotes the set of positive integers and N0:=N{0}. Also we use Z to denote the set of integers. As usual Rd stands for the Euclidean space of points x=(x1,,xd). We setBr(x):={yR:|xy|<r},R+d+1:={(t,x)Rd+1:t>0}. For i=1,,d, multi-indices σ=(σ1,,σd), and functions u(t,x) we setxiu=uxi=Diu,Dσu=D1σ1Ddσdu,|σ|=σ1++σd. We also use the notation Dxm for arbitrary partial derivatives of order m with respect to x. For an open set O in Rd or Rd+1, Cc(O) denotes the set of infinitely differentiable functions with compact support in O. By S=S(Rd) we denote the class of Schwartz functions on Rd. For p>1, by Lp we denote the set of complex-valued Lebesgue measurable functions u on Rd satisfyinguLp:=(Rd|u(x)|pdx)1/p<. Generally, for a given measure space (X,M,μ), Lp(X,M,μ;F) denotes the space of all F-valued Mμ-measurable functions u so thatuLp(X,M,μ;F):=(Xu(x)Fpμ(dx))1/p<, where Mμ denotes the completion of M with respect to the measure μ. If there is no confusion for the given measure and σ-algebra, we usually omit the measure and the σ-algebra. We denote ab:=min{a,b} and ab:=max{a,b}. By F and F1 we denote the d-dimensional Fourier transform and the inverse Fourier transform respectively, i.e.F(f)(ξ):=fˆ(ξ):=Rdeiξxf(x)dx,F1(f)(ξ):=1(2π)dRdeiξxf(x)dx. For any a,b>0, we write ab if there is a constant c>1 independent of a,b such that c1abca. Finally if we write N=N(), this means that the constant N depends only on what are in the parentheses. The constant N can differ from line to line.

Section snippets

Main results

First we introduce some definitions and facts related to the fractional calculus. For α>0 and φL1((0,T)), the Riemann-Liouville fractional integral of the order α is defined asItαφ:=1Γ(α)0t(ts)α1φ(s)ds,0tT. We also define I0φ:=φ. By Jensen's inequality, for p[1,],ItαφLp((0,T))N(T,α)φLp((0,T)). Using Fubini's theorem, one can easily check for any α,β0,ItαItβφ=Itα+βφ,(a.e.)tT. Let α[n1,n), nN. If φ(t) is (n1)-times differentiable and (ddt)n1Itnαφ is absolutely continuous on [0

Estimates of the fundamental solution

In this section we obtain sharp bounds of arbitrary order derivatives of the fundamental solution to equation (2.17).

We first study the derivatives of the transition density of d-dimensional subordinate Brownian motion.

Lemma 3.1

Let Assumption 2.1 hold. Then there exists a constant N=N(c,δ0) such thatλ1r1ϕ(r2)drNϕ(λ2),λ>0.

Proof

Note first that Assumption 2.1 combined with the concavity of ϕ givesc(Rr)δ0ϕ(R)ϕ(r)Rr,0<r<R<.

By the change of variables and (3.2),λ1r1ϕ(r2)dr=1r1ϕ(λ2r2)dr=1r1ϕ(λ2r

Key estimates: BMO and Lq(Lp)-estimates

In this section we prove some a priori estimates for solutions to the equation with zero initial conditiontαu=ϕ(Δ)u+f,t>0;u(0,)=0.

We first present the representation formula.

Lemma 4.1

(i) Let uCc(R+d+1) and denote f:=tαuϕ(Δ)u. Thenu(t,x)=0tRdqα,1(ts,xy)f(s,y)dyds.

(ii) Let fCc(R+d+1) and define u as in (4.2). Then u satisfies equation (4.1) for each (t,x).

Proof

See [22, Lemma 3.5], which treats the case ϕ(λ)=λ. The proof for the general case is same. The only difference is one needs to use formula

Homogeneous equation

In this section we study the homogeneous equation with non-zero initial conditiontαu=ϕ(Δ)u,u(0,x)=u0(x).

We first show that q is a fundamental solution to equation (5.1).

Lemma 5.1

Let u0Cp(Rd), and define u asu(t,x):=Rdq(t,xy)u0(y)dy.

(i) As t0, u(t,) converges to u0() uniformly on Rd and also in Hpn for any nN0.

(ii) uCpα,([0,T]×Rd) and u satisfies tαu=ϕ(Δ)u for t>0.

Proof

(i) By (3.24), for any t>0,Rdq(t,y)dy=qˆ(t,0)=Eα(0)=1. For any δ>0,|Rdq(t,xy)u0(y)dyu0(x)|=|Rdq(t,y)(u0(xy)u0(x))dy||y|δ

Proof of Theorem 2.8

Due to Remark 2.4 and Lemma 2.7 (iii), we only need to prove case γ=0.

Step 1 (Uniqueness). Let uHq,pα,ϕ,2(T) be a solution to equation (2.17) with f=0 and u0=0. Then by Corollary 5.5, uHq,p,0α,ϕ,2(T). Hence, by Lemma 2.7 (ii), there exists unCc(R+d+1) such that unu in Hq,pα,ϕ,2(T). Due to Lemma 4.1, it also holds thatun(t,x)=0tRdqα,1(ts,xy)fn(s,y)dyds, where fn:=tαunϕ(Δ)un. NotefnLq,p(T)=tα(unu)ϕ(Δ)(unu)Lq,p(T)tαuntαuLq,p(T)+ϕ(Δ)unϕ(Δ)uLq,p(T)0 as n. Thus by

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    The authors were supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. NRF-2020R1A2C1A01003354).

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