A weak Harnack inequality for fractional evolution equations with discontinuous coefficients

A weak Harnack inequality for fractional evolution equations with discontinuous coefficients

RICOZACHER

Abstract. We study linear time fractional diffusion equations in divergence form of time order less than one. It is merely assumed that the coefficients are mea- surable and bounded, and that they satisfy a uniform parabolicity condition. As a main result we establish for nonnegative weak supersolutions of such problems a weak Harnack inequality with optimal critical exponent. The proof relies on new a priori estimates for time fractional problems and uses Moser’s iteration tech- nique and an abstract lemma of Bombieri and Giusti, the latter allowing to avoid the rather technically involved approach viaB M O. As applications of the weak Harnack inequality we establish the strong maximum principle, the continuity of weak solutions att = 0, and a uniqueness theorem for global bounded weak solutions.

Mathematics Subject Classification (2010): 35R09 (primary); 45K05 (sec- ondary).

1. Introduction and main result

LetT >0 andbe a bounded domain inR^N. In this paper we are concerned with linear partial integro-differential equations of the form

@_t^↵(u u0) div A(t,x)Du =0, t 2(0,T), x 2. (1.1) Here u0 = u0(x) is a given initial data foru, A = (ai j)is R^{N⇥N-valued, Du} denotes the spatial gradient ofu, and@_t^↵stands for the Riemann-Liouville fractional derivation operator with respect to time of order↵2(0,1); it is defined by

@_t^↵v(t,x)=@_t Z t

0 g1 ↵(t ⌧)v(⌧,x)d⌧, t >0, x2,

Work partially supported by the European Community’s Human Potential Programme [Evolution Equations for Deterministic and Stochastic Systems], contract code HPRN-CT-2002-00281, and by the Deutsche Forschungsgemeinschaft (DFG), Bonn, Germany.

Received July 23, 2011; accepted December 12, 2011.

whereg denotes the Riemann-Liouville kernel g (t)= t ¹

0( ), t >0, >0.

As to applications, equation (1.1) is a special case of problems arising in mathe- matical physics when describing dynamic processes in materials with memory,e.g.

in the theory of heat conduction with memory, see [24] and the references therein.

Time fractional diffusion equations are also used to model anomalous diffusion, see e.g.[20]. In this context, equations of the type (1.1) are termedsubdiffusion equa- tions(the time order↵lies in(0,1); in the case↵2(1,2), which is not considered here, one speaks ofsuperdiffusion equations. Time fractional diffusion equations of time order↵ 2 (0,1)are closely related to a class of Montroll-Weiss continu- ous time random walk models where the waiting time density behaves ast ^{↵ 1}for t ! 1, seee.g.[14, 15, 20]. Problems of the type (1.1) are further used to describe diffusion on fractals ([20, 25]), and they also appear in mathematical finance, see e.g.[27].

Letting_T =(0,T)⇥we will assume that (H1) A2L₁(_T;R^{N⇥N), and}

XN i,j=1

|ai j(t,x)|²3², for a.a.(t,x)2_T. (H2) There exists⌫ >0 such that

A(t,x)⇠|⇠ ⌫|⇠|², for a.a.(t,x)2_T, and all⇠ 2R^N. (H3) u02L2().

We say that a functionuis aweak solution (subsolution, supersolution)of (1.1) in

_T, ifubelongs to the space Z↵:= {v2L ²

1 ↵, w([0,T];L2())\L2([0,T];H₂¹())such that g1 ↵⇤v2C([0,T];L2()), and(g1 ↵⇤v)|t=0=0}, and for any nonnegative test function

⌘2H^1,1₂ (_T):=H₂¹([0,T];L2())\L2([0,T];H¹₂()) ⇣

H¹₂():=C₀¹()^H21()⌘

with⌘|t=T =0 there holds Z T

0

Z



⇣

⌘_t[g1 ↵⇤(u u0)] +(ADu|D⌘)⌘

dxdt = (, )0. (1.2)

HereLp, wdenotes the weakLpspace and f1⇤ f2the convolution on the positive halfline with respect to time, that is(f₁⇤ f₂)(t)=Rt

0 f₁(t ⌧)f₂(⌧)d⌧,t 0.

Weak solutions of (1.1) in the class Z↵have been constructed in [37]. Notice also that the function u₀ plays the role of the initial data foru, at least in a weak sense. In case of sufficiently smooth functionsuandg1 ↵⇤(u u0)the condition (g1 ↵⇤u)|^t=0=0 impliesu|^t=0=u0, see [37].

To formulate our main result, letB(x,r)denote the open ball with radiusr >0 centered atx 2R^{N. ByµN} we mean the Lebesgue measure inR^{N. For} 2(0,1), t0 0,⌧ >0, and a ballB(x0,r), define the boxes

Q (t0,x0,r)=(t0,t0+ ⌧r^2/↵)⇥B(x0, r),

Q₊(t0,x0,r)=(t0+(2 )⌧r^2/↵,t0+2⌧r^2/↵)⇥B(x0, r).

Theorem 1.1. Let↵2(0,1),T >0, and⇢R^Nbe a bounded domain. Suppose the assumptions(H1)-(H3)are satisfied. Let further 2(0,1),⌘>1, and⌧ >0be fixed. Then for anyt0 0andr >0witht0+2⌧r^2/↵ T, any ballB(x0,⌘r)⇢, any 0 < p < ₂₊²_N⁺_↵^N↵_2↵, and any nonnegative weak supersolution u of (1.1)in (0,t0+2⌧r^2/↵)⇥B(x0,⌘r)withu0 0inB(x0,⌘r), there holds

1

µ_N+1 Q (t0,x0,r) Z

Q (t0,x0,r)

u^pdµN+1

!1/p

C ess inf

Q₊(t0,x0,r)u, (1.3) where the constantC=C(⌫,3, ,⌧,⌘,↵,N,p).

Theorem 1.1 states that nonnegative weak supersolutions of (1.1) satisfy a weak form of Harnack inequality in the sense that we do not have an estimate for the supremum of u on Q (t₀,x₀,r)but only an L_p estimate. We also show that the critical exponent ₂₊²_N↵^+N↵_2↵ is optimal, i.e. the inequality fails to hold for

p ₂₊²_N↵^+N↵_2↵.

Theorem 1.1 can be viewed as the time fractional analogue of the correspond- ing result in the classical parabolic case↵=1, seee.g.[19, Theorem 6.18] and [29].

Sending↵ ! 1 in the expression for the critical exponent yields 1+2/N, which is the well-known critical exponent for the heat equation. We would like to point out that the statement of Theorem 1.1 remains valid for (appropriately defined) weak supersolutions of (1.1) on(t0,t0+2⌧r^2/↵)⇥ B(x0,⌘r)which are nonneg- ative on (0,t0 + 2⌧r^2/↵)⇥ B(x0,⌘r). To avoid further technicalities we have confined ourselves to consider only weak supersolutions on the full time-interval (0,t0+2⌧r^2/↵). Note that the global positivity assumption cannot be replaced by a local one, as simple examples show,cf.[36]. This significant difference to the case

↵ = 1 is due to the non-local nature of@_t^↵. The same phenomenon is known for integro-differential operators like( 1)^↵with↵2(0,1), seee.g.[16].

As a simple consequence of the weak Harnack inequality we derive the strong maximum principle for weak subsolutions of (1.1), see Theorem 5.1 below. The weak maximum principle has been proven in [33], even in a more general setting.

As a further consequence of the weak Harnack inequality we obtain a unique- ness theorem for global bounded weak solutions, see Corollary 5.3 below. It states that any bounded weak solution of (1.1) onR+⇥R^{N withu}0=0 vanishes a.e. on R+⇥R^N.

The proof of Theorem 1.1 relies on new a priori estimates for time fractional problems, which are derived by means of the fundamental identity (2.6) (see below) for the regularized fractional derivative. It further uses Moser’s iteration technique and an elementary but subtle lemma of Bombieri and Giusti [2], which allows to avoid the rather technically involved approach via B M O-functions. This simplifi- cation is already of great significance in the classical parabolic case, see Moser [23]

and Saloff-Coste [26].

One of the technical difficulties in deriving the desired estimates in the weak setting is to find an appropriate time regularization of the problem. In the case

↵ = 1 this can be achieved by means of Steklov averages in time. In the time fractional case this method does not work anymore, since Steklov average operators and convolution do not commute. It turns out that instead one can use the Yosida approximation of the fractional derivative, which leads to a regularization of the kernelg1 ↵. This method has already been used in [12, 30, 37], and [33].

We point out that the results obtained in this paper can be generalized to quasi- linear equations of the form

@_t^↵(u u0) diva(t,x,u,Du)=b(t,x,u,Du), t 2(0,T),x 2, (1.4) with suitable structure conditions on the functions a and b. This is possible, as also known from the elliptic and the classical parabolic case, since the test function method used in the proof of Theorem 1.1 does not depend so much on the linearity of the differential operator w.r.t. the spatial variables but on a certain nonlinear structure,cf.[11, 19, 29], and [33].

In the literature there exist many papers where equations of the type (1.1), as well as nonlinear or abstract variants of them are studied in a strong setting, assuming more smoothness on the coefficients and nonlinearities, see e.g.[1, 4, 7, 10,12,24,34,35]. Concerning theweaksetting described above one finds only a few results. Existence of weak solutions has been shown in [37] in an abstract setting for a more general class of kernels. Boundedness of weak solutions has been obtained in [33] in the quasilinear case by means of the De Giorgi technique. In the very recent work [31], it is proved that any weak solution of (1.1) is H¨older continuous in the interior of the parabolic domain. The proof is quite involved and relies on the De Giorgi technique and the method of non-local growth lemmata (cf.[28]).

With the weak Harnack inequality, the present paper establishes another key result towards a De Giorgi-Nash-Moser theory for time fractional evolution equations in divergence form of order↵2(0,1).

In the classical parabolic case boundedness and the weak (or full) Harnack in- equality imply an H¨older estimate for weak solutions,cf. [9, 18, 19, 22]. We also refer to [11] and [21] for the elliptic case. In the present situation one cannot argue anymore as in the classical parabolic case, due to the global positivity assumption

in Theorem 1.1. The same problem arises for the fractional Laplacian, see [28].

However, in our case it is possible to establish without much effort at least continu- ity att =0. This is done in Theorem 5.2 in the caseu₀=0. It is shown that in this case any bounded weak solutionu of (1.1) is continuous at(0,x₀)for allx₀ 2 and lim_(t,x)!(0,x0)u(t,x) = 0. Thus for such weak solutions the initial condition u|^t=0=0 is satisfied in the classical sense.

We further remark that in the purely time-dependent case, that is for scalar equations of the form

@_t^↵(u u0)+ u=0, t 2(0,T),

with 0, a weak Harnack inequality with optimal exponent 1/(1 ↵)has been proven in [32] for nonnegative supersolutions. Recently, the full Harnack inequality for nonnegative solutions has been established in [36]. This, together with the above results, indicates that the full Harnack inequality should also hold for nonnegative solutions of (1.1), which is still an open problem, even in the case A(t,x)⌘ I d.

The paper is organized as follows. In Section 2 we collect the basic tools needed for the proof of Theorem 1.1. These include two abstract lemmas on Moser iterations and the lemma of Bombieri and Giusti. We further explain the approxima- tion method for the fractional derivation operator and state the fundamental identity (2.6), which is frequently used in Section 3, where we give the proof of the main result. In Section 4 we show that the critical exponent in Theorem 1.1 is optimal.

Finally, Section 5 is devoted to applications of the weak Harnack inequality.

ACKNOWLEDGEMENTS. This paper was initiated while the author was visiting the Technical University Delft (NL) in 2003/2004. The author is greatly indebted to Philippe Cl´ement for many fruitful discussions and valuable suggestions.

2. Preliminaries

2.1. Moser iterations and an abstract lemma of Bombieri and Giusti

Throughout this subsectionU , 0< 1, will denote a collection of measurable subsets of a fixed finite measure space endowed with a measureµ, such thatU 0⇢ U if ⁰  . For p 2(0,1)and 0 < 1, Lp(U )stands for the Lebesgue space Lp(U ,dµ)of allµ-measurable functions f :U ! R^with|f|Lp(U ) :=

(R

U |f|^pdµ)^1/p<1.

The following two lemmas are basic to Moser’s iteration technique. The argu- ments in their proofs have been repeatedly used in the literature (seee.g.[11,19,21, 22, 26, 29]), so it is worthwhile to formulate them as lemmas in abstract form, also for future reference. We provide proofs for the sake of completeness.

The first Moser iteration result reads as follows, see also [8, Lemma 2.3].

Lemma 2.1. Let >1, p¯ 1,C 1, and > 0. Suppose f is aµ-measurable function onU1such that

|f|L (U ₀) 

✓C(1+ )

( ⁰)

◆1/

|f|L (U ), 0< ⁰< 1, >0. (2.1) Then there exist constantsM =M(C, ,,p)¯ and 0= 0( ,)such that

ess sup

U |f|

✓ M

(1 ) ⁰

◆1/p

|f|Lp(U1) for all 2(0,1), p2(0,p¯]. Proof. Forq>0 and 0< 1, let

8(q, )=

✓Z

U |f|^qdµ

◆1/q

.

Let 0< p p¯and 2(0,1). Set pi = pⁱ,i =0,1, . . .and define the sequence { i},i =0,1, . . ., by 0=1 and i =1 Pi

j=12 ^j(1 ),i =1,2, . . .; observe that 1= 0> ₁> . . . > _i > _i+1> as well as i 1 i =2 ⁱ(1 ),i 1.

Suppose nown2N. By using (2.1) with = pi,i =0,1, . . . ,n 1, we obtain 8(pn, )8(pn, _n)=8(pn 1, _n)

 C(1+ p^{n 1}) [2 ⁿ(1 )]

!¹_p ^{(n 1)}

8(p_{n 1}, _{n 1})

 C(2p¯ ^{n 1}) [2 ⁿ(1 )]

!¹_p ^{(n 1)}

8(pn 1, _{n 1})

 C(C,˜ p, )¯ ⁿ ^{(n 1)}

(1 )

!¹_p ^{(n 1)}

8(pn 1, _{n 1}). . .

⇣

C˜^{Pnj=01(j+1) j}

Pn 1 j=0j ^j

(1 )

Pn 1 j=0 ^j⌘^1/p

8(p0, ₀)

 M(C,p, ,¯ ) (1 )^1

!1/p

8(p,1).

We let nowntend to1and use the fact that

nlim!18(p_n, )=ess sup

U |f| to get

ess sup

U |f| M(C,p, ,¯ ) (1 )^1

!1/p

|f|Lp(U1). Hence the proof is complete.

The second Moser iteration result is the following, see also [8, Lemma 2.5].

Lemma 2.2. Assume thatµ(U1)1. Let >1,0< p0<, andC 1, >0.

Suppose f is aµ-measurable function onU1such that

|f|L (U0)

✓ C

( ⁰)

◆1/

|f|L (U ), 0< ⁰ < 1, 0<  p₀

 <1.

(2.2)

Then there exist constantsM =M(C, ,)and 0= 0( ,)such that

|f|Lp0(U)

✓ M

(1 ) ⁰

◆1/p 1/p0

|f|Lp(U1) for all 2(0,1), p2⇣ 0, p0

 i

.

Proof. Set pi = p0 ⁱ,i =1,2, . . .Given 2(0,1)we take again the sequence { i},i = 0,1,2, . . ., defined by 0 = 1 and i = 1 Pi

j=12 ^j(1 ),i 1.

Suppose nown2N. By using (2.2) with = pi,i =1, . . . ,n, we obtain 8(p0, ) 8(p0, _n) = 8(p1, _n)  C^/p0

[2 ⁿ(1 )] ^/p0 8(p1, _{n 1})

 C^/p0 [2 ⁿ(1 )] ^/p0

C^2/p0

[2 ^{(n 1)}(1 )] ^2/p0 8(p2, _{n 2})  . . .

 C^{p10(+2+...+n)}

2 ^{p0(n+(n 1)2+...+2n 1+n)}(1 )^{p0(+2+...+n)}

8(pn, ₀).

Since pi = p0 ⁱ, we have 1

p0

Xn j=1

^j = (ⁿ 1)

p0( 1) =  p0( 1)

✓p0

pn 1

◆

= 

 1

✓ 1 pn

1 p0

◆ .

Employing the formula Xn

j=1

j^{j 1}= 1 (n+1)ⁿ+nⁿ⁺¹ ( 1)² we have further

Xn j=1

(n+1 j)^j =(n+1) Xn

j=1

^j Xn

j=1

j^j

=(n+1)ⁿ 1

 1 1 (n+1)ⁿ+nⁿ⁺¹ ( 1)²

=ⁿ⁺¹ (n+1)+n

( 1)²  

( 1)²ⁿ⁺¹

 ³

( 1)³(ⁿ 1) ³ ( 1)³

✓p0

pn 1

◆ ,

which yields 1 p0

Xn j=1

(n+1 j)^j  ³ ( 1)³

✓ 1 pn

1 p0

◆ .

Therefore

8(p₀, ) 2 642

3 ( 1)3C^1 (1 )^1

3 75

1 pn 1

p0

8(p_n, ₀).

Given p2(0,p0/]there existsn 2 such that pn< p pn 1. We then have 1

pn

1

p0 = ⁿ 1

p0  ⁿ+^{n 1}  1

p0 = (1+)(^{n 1} 1) p0

=(1+)

✓ 1 pn 1

1 p0

◆

(1+)

✓1 p

1 p0

◆ ,

as well as

8(pn, ₀)=8(pn,1)8(p,1),

by H¨older’s inequality and the assumptionµ(U1)1. All in all, we obtain

8(p0, ) 2 642

3 ( 1)3

C^1 (1 )^1

3 75

(1+)(¹_{p p}¹ 0)

8(p,1),

which proves the lemma.

The following abstract lemma is due to Bombieri and Giusti [2]. For a proof we also refer to [26, Lemma 2.2.6] and [8, Lemma 2.6].

Lemma 2.3. Let ,⌘2(0,1), and let ,Cbe positive constants and0< ₀ 1. Suppose f is a positiveµ-measurable function onU1which satisfies the following two conditions:

(i) |f|L ₀(U 0)[C( ⁰) µ(U1) ¹]^{1/ 1/ 0}|f|L (U ), for all , ⁰, such that0<  ⁰< 1and0< min{1,⌘ ₀}.

(ii) µ({log f > })Cµ(U1) ¹ for all >0.

Then

|f|L₀(U) Mµ(U₁)^{1/ 0}, whereM depends only on ,⌘, ,C, and ₀.

2.2. The Yosida approximation of the fractional derivation operator

Let 0 < ↵ < 1, 1  p < 1,T > 0, and X be a real Banach space. Then the fractional derivation operator Bdefined by

Bu= d

dt (g1 ↵⇤u), D(B)= {u2Lp([0,T];X): g1 ↵⇤u20H_p¹([0,T];X)}, where the zero means vanishing att=0, is known to bem-accretive inLp([0,T];X), cf. [3, 6], and [12]. Its Yosida approximations Bn, defined by Bn = n B(n + B) ¹,n 2 N, enjoy the property that for anyu 2 D(B), one has Bnu ! Bu inLp([0,T];X)asn! 1. Further, one has the representation

Bnu= d

dt (g1 ↵,n⇤u), u2Lp([0,T];X), n2N^{, (2.3)} whereg1 ↵,n =ns↵,n, ands↵,n is the unique solution of the scalar-valued Volterra equation

s↵,n(t)+n(s↵,n⇤g↵)(t)=1, t >0, n2N^,

seee.g.[30]. Leth↵,n 2 L1,loc(R+)be the resolvent kernel associated withng↵, that is

h↵,n(t)+n(h↵,n⇤g↵)(t)=ng↵(t), t >0, n2N^{. (2.4)} Convolving (2.4) withg_{1 ↵}and usingg_↵⇤g_{1 ↵}=1, we obtain

(g1 ↵⇤h↵,n)(t)+n([g1 ↵⇤h↵,n]⇤g↵)(t)=n, t >0, n2N^. Hence

g1 ↵,n=ns↵,n =g1 ↵⇤h↵,n, n2N^{. (2.5)} The kernelsg_{1 ↵,n} are nonnegative and nonincreasing for alln 2N, and they be- long toH₁¹([0,T]),cf.[24] and [30]. Note that for any function f 2Lp([0,T];X), 1 p<1, there holdsh↵,n⇤ f ! f inLp([0,T];X)asn! 1. In fact, setting u=g↵⇤ f, we haveu2D(B), and

Bnu= d

dt (g1 ↵,n⇤u)= d

dt (g1 ↵⇤g↵⇤h↵,n⇤ f)=h↵,n⇤ f !Bu

= f inLp([0,T];X)

asn! 1. In particular,g1 ↵,n !g1 ↵inL1([0,T])asn! 1.

We next state a fundamental identity for integro-differential operators of the form_dt^d(k⇤u),cf.also [33]. Supposek2 H₁¹([0,T])andH 2C¹(R). Then it fol- lows from a straightforward computation that for any sufficiently smooth function uon(0,T)one has for a.a. t 2(0,T),

H⁰(u(t))d

dt(k⇤u)(t)= d

dt (k⇤H(u))(t)+ H(u(t))+H⁰(u(t))u(t) k(t) +

Z t 0

H(u(t s)) H(u(t)) H⁰(u(t))[u(t s) u(t)] [ ˙k(s)]ds, (2.6)

wherek˙denotes the derivative ofk. In particular this identity applies to the Yosida approximations of the fractional derivation operator. We remark that an integrated version of (2.6) can be found in [13, Lemma 18.4.1].

Observe that the last term in (2.6) is nonnegative in case H is convex andk is nonincreasing. This fact is frequently used in the estimates below. However we would like to point out that for the delicate logarithmic estimates in Section 3.3 one really needs the full identity (2.6), even though we work all the time with convex or concave functions. In particular, the crucial fractional differential inequality (3.40) cannot be obtained by using merely convexity inequalities.

The subsequent two lemmas are also obtained by simple algebra.

Lemma 2.4. LetT > 0and↵ 2 (0,1). Suppose thatv 2 0H₁¹([0,T])and' 2 C¹([0,T]). Then

g↵⇤('v))(t˙ )='(t)(g↵⇤v)(t˙ )+ Z t

0

v( )@ g↵(t )['(t) '( )] d ,

a.a.t 2(0,T).

If in additionvis nonnegative and'is nondecreasing there holds

g↵⇤('v))(t˙ ) '(t)(g↵⇤v)(t˙ ) Z t

0

g↵(t )'( )v( )˙ d , a.a.t 2(0,T).

Lemma 2.5. Let T > 0,k 2 H₁¹([0,T]), v 2 L1([0,T]), and ' 2 C¹([0,T]).

Then '(t)d

dt(k⇤v)(t)= d

dt k⇤['v] (t)+ Z t

0

k(t˙ ⌧) '(t) '(⌧) v(⌧)d⌧, a.a.t2(0,T).

2.3. An embedding result and a weighted Poincar´e inequality

LetT >0 andbe a bounded domain inR^{N. For 1< p} 1we define the space Vp:=Vp([0,T]⇥)=L2p([0,T];L2())\L2([0,T];H₂¹()), (2.7) endowed with the norm

|u|Vp([0,T]⇥):= |u|L2p([0,T];L2())+ |Du|L2([0,T];L2()). Set

:=_p := 2p+N(p 1)

2+N(p 1) (2.8)

with₁=1+2/N. ThenVp ,! L2([0,T]⇥), and

|u|L2([0,T]⇥)C(N,p)|u|Vp([0,T]⇥), (2.9)

for all u 2 Vp \L2([0,T]; H¹₂()). This is a consequence of the Gagliardo- Nirenberg and H¨older’s inequality. The case p=1is contained,e.g., in [18, page 74 and 75]. The proof given there easily extends to the general case. For a more general embedding result (without proof) we also refer to [33, Section 2].

The following result can be found in [22, Lemma 3], see also [19, Lemma 6.12].

Proposition 2.6. Let' 2 C(R^N)with non-empty compact support of diameterd and assume that0'  1. Suppose that the domains{x 2R^N : '(x) a}are convex for alla1. Then for any functionu2H₂¹(R^N),

Z

R^N u(x) u' 2

'(x)dx  2d²µ_N(supp')

|'|L1(R^N)

Z

R^N |Du(x)|²'(x)dx, where

u'=

RR^Nu(x)'(x)dx RR^N '(x)dx .

3. Proof of the main result

3.1. The regularized weak formulation, time shifts, and scalings

The following lemma is basic to derivinga prioriestimates for weak (sub-/super-) solutions of (1.1). It provides an equivalent weak formulation of (1.1) where the singular kernelg₁ ↵is replaced by the more regular kernelg_{1 ↵,n}(n2N^{) given in} (2.5). In what follows the kernelsh_n :=h_↵,n,n2N, are defined as in Section 2.2.

Lemma 3.1. Let↵ 2 (0,1), T > 0, and  ⇢ R^N be a bounded domain. Sup- pose the assumptions (H1)-(H3) are satisfied. Thenu 2 Z↵ is a weak solution (subsolution, supersolution)of(1.1)in_T if and only if for any nonnegative func- tion 2H¹₂()one has

Z



⇣

@_t[g1 ↵,n⇤(u u0)] +(hn⇤[ADu]|D )⌘

dx =(, )0, a.a.t 2(0,T),n2N^. For a proof we refer to [33, Lemma 3.1], where a more general situation is con- sidered with a slightly different function space for the solution. The proof of Lemma 3.1 is analogous.

Letu 2Z↵be a weak supersolution of (1.1) in_T and assume thatu0 0 in

. Then Lemma 3.1 and positivity ofg1 ↵,nimply that Z



⇣

@_t(g1 ↵,n⇤u)+(hn⇤[ADu]|D )⌘

dx 0, a.a.t 2(0,T),n2N^{, (3.1)} for any nonnegative function 2H¹₂().

Let nowt1 2 (0,T)be fixed. For t 2 (t1,T)we introduce the shifted time s = t t1 and set f˜(s) = f(s+t1),s 2(0,T t1), for functions f defined on (t₁,T). From the decomposition

(g1 ↵,n⇤u)(t,x)= Z t

t1

g1 ↵,n(t ⌧)u(⌧,x)d⌧+ Z t1

0 g1 ↵,n(t ⌧)u(⌧,x)d⌧, t 2(t1,T), we then deduce that

@_t(g1 ↵,n⇤u)(t,x)=@_s(g1 ↵,n⇤u)(s,˜ x)+ Z t1

0 g˙1 ↵,n(s+t1 ⌧)u(⌧,x)d⌧. (3.2) Assuming in addition thatu 0 on(0,t1)⇥it follows from (3.1), (3.2), and the positivity of and of g˙1 ↵,nthat

Z



⇣

@_s(g1 ↵,n⇤u)˜ + (hn⇤[ADu])˜ |D ⌘

dx 0, a.a.s 2(0,T t₁),n2N^,

(3.3) for any nonnegative function 2H¹₂(). This relation will be the starting point for all of the estimates below.

We conclude this section with a remark on the scaling properties of equation (1.1). Lett0,r >0 andx02R^{N. Supposeu}2Z↵is a weak solution (subsolution, supersolution) of (1.1) in(0,t0r^2/↵)⇥B(x0,r). Changing the coordinates according tos =t/r^2/↵andy=(x x0)/rand settingv(s,y)=u(sr^2/↵,x0+yr),v₀(y)= u0(x0+yr), and A(s,˜ y)= A(sr^2/↵,x0+yr), the problem foruis transformed to a problem forvin(0,t₀)⇥ B(0,1), namely there holds withD = D_y (also in the weak sense)

@_s^↵(v v₀) div A(s,˜ y)Dv = (, )0, s2(0,t0), y2 B(0,1). (3.4) 3.2. Mean value inequalities

For > 0 we put B(x,r) := B(x, r). Recall thatµ_N denotes the Lebesgue measure inR^N.

Theorem 3.2. Let↵2(0,1),T >0, and⇢R^Nbe a bounded domain. Suppose the assumptions(H1)-(H3)are satisfied. Let⌘ > 0and 2 (0,1)be fixed. Then for anyt0 2(0,T]andr > 0witht0 ⌘r^2/↵ 0, any ball B = B(x0,r)⇢ , and any weak supersolutionu " > 0of (1.1) in(0,t₀)⇥ Bwithu₀ 0in B , there holds

ess sup

U 0

u ¹ CµN+1(U1) ¹ ( ⁰)^⌧0

!1/

|u ¹|L (U ),  ⁰ < 1, 2(0,1]. HereU = (t0 ⌘r^2/↵,t0)⇥ B, 0 <  1,C = C(⌫,3, ,⌘,↵,N)and

⌧₀=⌧₀(↵,N).

Proof. We may assume thatr = 1 and x0 = 0. In fact, in the general case we change coordinates ast ! t/r^2/↵andx ! (x x0)/r, thereby transforming the equation to a problem of the same type on(0,t₀/r^2/↵)⇥B(0,1),cf.Section 3.1.

Fix ⁰and such that  ⁰ < 1 and putB₁= B. For⇢ 2(0,1]we set V⇢ = U⇢ . Given 0 < ⇢⁰ < ⇢  1, lett1 = t0 ⇢ ⌘andt2 = t0 ⇢⁰ ⌘.

Then 0  t1 < t2 < t0. We introduce further the shifted times = t t1 and set f˜(s)= f(s+t1),s2(0,t0 t1), for functions f defined on(t1,t0). Sinceu0 0 in B andu is a positive weak supersolution of (1.1) in(0,t0)⇥ B, we have (cf.

(3.3))

Z

B

⇣

v@_s(g_{1 ↵,n}⇤u)˜ + (h_n⇤[ADu])˜ |Dv ⌘

dx 0,

a.a.s 2(0,t0 t1),n2N^{, (3.5)} for any nonnegative function v 2H¹₂(B). Fors 2 (0,t0 t1)we choose the test function v = ²u˜ with < 1 and 2 C₀¹(B₁)so that 0   1, = 1 in⇢⁰B1, supp ⇢ ⇢B1, and|D |2/[ (⇢ ⇢⁰)]. By the fundamental identity (2.6) applied to k = g1 ↵,n and the convex function H(y) = (1+ ) ¹y¹⁺ , y >0, there holds for a.a.(s,x)2(0,t0 t1)⇥B

u˜ @_s(g_{1 ↵,n}⇤u)˜ 1

1+ @_s(g_{1 ↵,n}⇤u˜¹⁺ )+

⇣u˜¹⁺

1+ u˜¹⁺ ⌘ g_{1 ↵,n}

= 1

1+ @_s(g1 ↵,n⇤u˜¹⁺ )

1+ u˜¹⁺ g1 ↵,n.

(3.6)

We further have

Dv=2 D u˜ + ²u˜ ¹Du.˜

Using this and (3.6) it follows from (3.5) that for a.a.s2(0,t0 t1) 1

1+ Z

B1

2@_s(g1 ↵,n⇤u˜¹⁺ )dx+| | Z

B1

(hn⇤[ADu])˜ | ²u˜ ¹Du dx˜

2 Z

B1

(h_n⇤[ADu])˜ | D u˜ dx+ 1+ Z

B1

2u˜¹⁺ g_{1 ↵,n}dx.

(3.7)

Next, choose' 2C¹([0,t₀ t₁])such that 0 ' 1,' =0 in[0, (t₂ t₁)/2], '=1 in[t₂ t₁,t₀ t₁], and 0'˙ 4/(t₂ t₁). Multiplying (3.7) by (1+ ) >0 and by'(s), and convolving the resulting inequality withg↵yields

Z

B1

g↵⇤ '@_s(g1 ↵,n⇤[ ²u˜¹⁺ ]) dx

+ (1+ )g↵⇤ Z

B1

(hn⇤[ADu])˜ | ²u˜ ¹Du˜ 'dx

2|1+ |g↵⇤ Z

B1

(hn⇤[ADu])˜ | D u˜ 'dx

+ | |g↵⇤ Z

B1

2u˜¹⁺ g1 ↵,n'dx,

(3.8)

for a.a.s 2(0,t0 t1). By Lemma 2.4, Z

B1

g↵⇤ '@_s(g1 ↵,n⇤[ ²u˜¹⁺ ]) dx Z

B1

'g↵⇤ @_s(g1 ↵,n⇤[ ²u˜¹⁺ ]) dx Z s

0

g↵(s )'( )˙ g_{1 ↵,n}⇤ Z

B1

2u˜¹⁺ dx ( )d .

(3.9)

Furthermore, by virtue of

g1 ↵,n⇤[ ²u˜¹⁺ ]20H₁¹([0,t0 t1];L1(B1)) andg1 ↵,n =g1 ↵⇤hnas well asg↵⇤g1 ↵=1 we have

g↵⇤@_s(g_{1 ↵,n}⇤[ ²u˜¹⁺ ])=@_s(g↵⇤g_{1 ↵,n}⇤[ ²u˜¹⁺ ])=h_n⇤( ²u˜¹⁺ ). (3.10) Combining (3.8), (3.9), and (3.10), sendingn ! 1, and selecting an appropriate subsequence, if necessary, we thus obtain

Z

B1

' ²u˜¹⁺ dx+ (1+ )g↵⇤ Z

B1

AD˜ u˜| ²u˜ ¹Du˜ 'dx

2|1+ |g↵⇤ Z

B1

AD˜ u˜| D u˜ 'dx+ | |g↵⇤ Z

B1

2u˜¹⁺ g₁ ↵'dx

+ Z s

0

g↵(s )'( )˙ g1 ↵⇤ Z

B1

2u˜¹⁺ dx ( )d , a.a.s2(0,t0 t1).

(3.11)

Putw= ˜u ⁺²¹. ThenDw= ⁺₂¹u˜ ²¹Du. By assumption (H2), we have˜ (1+ )g_↵⇤

Z

B1

˜

ADu˜| ²u˜ ¹Du˜ 'dx

⌫ (1+ )g↵⇤ Z

B1

' ²u˜ ¹|Du˜|²dx

= 4⌫

1+ g↵⇤ Z

B1

' ²|Dw|²dx.

(3.12)

Using (H1) and Young’s inequality we may estimate

2 AD˜ u˜| D u˜ ' 23 |D ||Du˜| ˜u '=23 |D | |Du˜| ˜u ²¹u˜ ⁺¹² '

 ⌫| | 2

2'|Du˜|²u˜ ¹+ 2

⌫| |3²|D |²'u˜ ⁺¹ (3.13)

= 2⌫| | (1+ )²

2'|Dw|²+ 2

⌫| |3²|D |²'w².

From (3.11), (3.12), and (3.13) we conclude that Z

B1

' ²w²dx+2⌫| |

|1+ |g↵⇤ Z

B1

' ²|Dw|²dxg↵⇤F, a.a.s2(0,t0 t1), (3.14) where

F(s)=23²|1+ |

⌫| | Z

B1

|D |²'w²dx+ | |'(s)g1 ↵(s) Z

B1

2w²dx

+ ˙'(s)

✓ g1 ↵⇤

Z

B1

2w²dx

◆

(s) 0, a.a.s 2(0,t0 t1).

We may drop the second term in (3.14), which is nonnegative. By Young’s in- equality for convolutions and the properties of ' we then infer that for all p 2 (1,1/(1 ↵))

✓Z t0 t1 t2 t1

✓Z

B1

[ (x)w(s,x)]²dx

◆p

ds

◆1/p

|g↵|Lp([0,t0 t1])

Z t0 t1 0

F(s)ds, (3.15) where

|g_↵|Lp([0,t0 t1]) = (t₀ t₁)^{↵ 1+1/p} 0(↵)[(↵ 1)p+1]^1/p

 ⌘^{↵ 1+1/p}

0(↵)[(↵ 1)p+1]^1/p =:C1(↵,p,⌘).

(3.16)

We choose any of these pand fix it.

Returning to (3.14), we may also drop the first term, convolve the resulting inequality withg₁ ↵and evaluate ats=t₀ t₁, thereby obtaining

Z t0 t1 t2 t1

Z

B1

2|Dw|²dx ds |1+ | 2⌫| |

Z t0 t1 0

F(s)ds. (3.17) Using

Z t0 t1 t2 t1

Z

B1

|D( w)|²dx ds2 Z t0 t1

t2 t1

Z

B1

2|Dw|²+ |D |²w² dx ds

we infer from (3.15)-(3.17) that

| w|²_Vp([t2 t1,t0 t1]⇥B1) 2

✓

C1(↵,p,⌘)+ |1+ |

⌫| |

◆ Z t0 t1 0

F(s)ds +4

Z t0 t1 0

Z

B1

|D |²w²dx ds.

(3.18)

We will next estimate the right-hand side of (3.18). By the assumptions on and ', and since| |>1, we have

Z t0 t1 0

Z

B1

|D |²w²dx ds 4

2(⇢ ⇢⁰)² Z t0 t1

0

Z

⇢B1

w²dx ds

and

F(s) 83²|1+ |

⌫ ²(⇢ ⇢⁰)² + | |g1 ↵((t2 t1)/2)

! Z

⇢B1

w²dx

+ 4

t2 t1

✓ g_{1 ↵}⇤

Z

⇢B1

w²dx

◆

(s), a.a.s 2(0,t₀ t₁).

Recall that >0. So we have Z t0 t1

0

F(s)ds 83²|1+ |

⌫ ²(⇢ ⇢⁰)²+ 2^↵| |

0(1 ↵)(⇢ ⇢⁰)^↵( ⌘)^↵

!Z _{t0 t1}

0

Z

⇢B1

w²dx ds

+ 4

(⇢ ⇢⁰) ⌘ Z t0 t1

0

g2 ↵(t0 t1 ⌧) Z

⇢B1

w(⌧,x)²dx d⌧

C(⌫,3, ,⌘,↵)1+ |1+ | (⇢ ⇢⁰)²

Z t0 t1 0

Z

⇢B1

w²dx ds.

Combining these estimates and (3.18) yields

| w|Vp([t2 t1,t0 t1]⇥B1) C(⌫,3, ,⌘,↵,p)1+ |1+ |

⇢ ⇢⁰ |w|L2([0,t0 t1]⇥⇢B1). We apply next the interpolation inequality (2.9) to the function wand make use of =1 in⇢⁰B1to deduce that

|w|L2([t2 t1,t0 t1]⇥⇢⁰B1)

C(⌫,3, ,⌘,↵,p,N)1+ |1+ |

⇢ ⇢⁰ |w|L2([0,t0 t1]⇥⇢B1), (3.19) where the number > 1 is given in (2.8). Sincew = ˜u ⁺²¹ and by transforming back to the timet, we see that (3.19) is equivalent to

Z

V_⇢0

u ^{|1+ |}dµ_N+1

!_2¹

 C(1˜ + |1+ |)

⇢ ⇢⁰

Z

V⇢

u ^{|1+ |}dµ_N+1

!¹₂

withC˜ = ˜C(⌫,3, ,⌘,↵,p,N). Hence, with = |1+ |,

|u ¹|^L (V_⇢0) C˜²(1+ )² (⇢ ⇢⁰)²

!1/

|u ¹|^{L (V}⇢), 0<⇢⁰ <⇢1, >0.