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BY 4.0 license Open Access Published online by De Gruyter April 20, 2022

Harnack inequality for parabolic equations with coefficients depending on time

Fabio Paronetto

Abstract

We define a homogeneous De Giorgi class of order p = 2 that contains the solutions of evolution equations of the types ξ ( x , t ) u t + A u = 0 and ( ξ ( x , t ) u ) t + A u = 0 , where ξ > 0 almost everywhere and A is a suitable elliptic operator. For functions belonging to this class, we prove a Harnack inequality. As a byproduct, one can obtain Hölder continuity for solutions of a subclass of the first equation.

1 Introduction

In this paper, we consider the two parabolic equations

(1.1) ξ ( x , t ) u t - div ( A ( x , t , u , D u ) ) + B ( x , t , u , D u ) + C ( x , t , u ) = 0 in  Ω × ( 0 , T ) ,
(1.2) t ( ξ ( x , t ) u ) - div ( A ( x , t , u , D u ) ) + B ( x , t , u , D u ) + C ( x , t , u ) = 0 in  Ω × ( 0 , T ) ,

where the coefficient ξ is in L and ξ > 0 almost everywhere, and therefore it can degenerate to zero. The functions

A : Ω × ( 0 , T ) × × n n ,
B : Ω × ( 0 , T ) × × n ,
C : Ω × ( 0 , T ) ×

are Carathéodory functions with A , B , C L ( Ω × ( 0 , T ) × × n ) satisfying

( A ( x , t , u , ξ ) , ξ ) λ | ξ | 2 ,
| A ( x , t , u , ξ ) | Λ | ξ | ,
| B ( x , t , u , ξ ) | M | ξ | ,
| C ( x , t , u ) | N | u | ,

for some given positive constants λ , Λ , M , N and for every u , ξ n and a.e. ( x , t ) Ω × ( 0 , T ) .

We show that functions belonging to a suitable De Giorgi class containing the solutions of equations (1.1) and (1.2) satisfy a Harnack inequality. In particular, we show the following result, but we refer to Theorem 5.1 and Theorem 5.2 for the precise statements.

Theorem.

There is a constant c > 0 such that for every ( x o , t o ) Ω × ( 0 , T ) and r > 0 such that

B 5 r ( x o ) × ( t o - ( 5 r ) 2 𝗁 ( x o , t o , 5 r ) , t o + ( 5 r ) 2 𝗁 ( x o , t o , 5 r ) ) Ω × ( 0 , T )

and

B r ( x o ) × ( t o , t o + 5 r 2 𝗁 ( x o , t o , r ) ) Ω × ( 0 , T ) ,

if u 0 is a solution of (1.1) or (1.2), then

sup B r ( x o ) u ( x , t o - r 2 𝗁 ( x o , t o , r ) ) c inf B r ( x o ) u ( x , t o + r 2 𝗁 ( x o , t o , r ) ) ,

where

𝗁 ( x o , t o , r ) = 1 r 2 | B r ( x o ) | t o t o + r 2 B r ( x o ) ξ ( x , t ) 𝑑 x 𝑑 t ,
𝗁 ( x o , t o , r ) = 1 r 2 | B r ( x o ) | t o - r 2 t o B r ( x o ) ξ ( x , t ) 𝑑 x 𝑑 t .

Moreover, adapting to the parabolic case the classical simple argument due to Moser (see [18], but also, e.g., [12, Section 7.9]), one can show that solutions of (1.1) with C = 0 are locally Hölder continuous (see also Remark 3.3).

Equations like (1.1) and (1.2) arise as natural generalizations of the case ξ = ξ ( x ) to the time dependent case ξ = ξ ( x , t ) . Applications can be found in some diffusion and in fluid flow problems (see [2, Chapter 3], in particular [2, Example 5.14]). In particular, this occurs in hydrology when dealing with transport of contaminants in unsaturated or variably saturated media (see [7]) or in density dependent flow in porous media (see, for instance, [17]).

There is a wide literature regarding the Harnack inequality for parabolic equations. Starting from the very first results due to Hadamard and Pini (see [16, 25]), the next important step due to Moser (see [19]) was to consider linear parabolic equations in divergence form. Due to the nature of this parabolic operator (in particular, its invariance with respect to the scaling x h x , t h 2 t ), the natural Harnack inequality for a solution u is

(1.3) sup B r ( x o ) u ( x , t o - r 2 ) C inf B r ( x o ) u ( x , t o + r 2 ) .

We also mention the papers [1, 26], where operators with linear growth, but possibly nonlinear, are considered and an inequality of the type (1.3) is derived.

Nevertheless, unlike in the elliptic case, for parabolic operators with different growth the problem is more delicate. In 1986, DiBenedetto showed that a Harnack inequality analogous to (1.3), i.e. substituting 2 with p, cannot hold for solutions of the equation

u t - div ( | D u | p - 2 D u ) = 0 , p > 2 ,

even if this equation is invariant with respect to the scaling x h x , t h p t (see [8]).

A modified Harnack inequality for this equation was proved by DiBenedetto, Gianazza and Vespri only in 2008 (see [9]).

Arriving to parabolic equations with degenerate coefficients of the type

v ( x ) u t - i , j = 1 n x i ( a i j ( x , t ) u x j ) = 0

with

w 1 ( x , t ) | ξ | 2 ( a ( x , t ) ξ , ξ ) L w 2 ( x , t ) | ξ | 2 , L 1 ,

and v , w 1 , w 2 > 0 almost everywhere, we recall the papers [4, 6, 5], where Chiarenza and Serapioni considered

v 1 , w 1 = w 2 = w

with w L loc 1 , w > 0 a.e., satisfying a Muckenhoupt condition A 2 (a subclass of A defined below in Definition 2.1) in the variable x uniformly with respect to time, i.e. there is C > 0 such that

1 | B | B w ( x , t ) 𝑑 x 1 | B | B w - 1 ( x , t ) 𝑑 x C for every ball  B  and every  t ,

or equations with

v ( x ) = w 1 ( x ) = w 2 ( x )

satisfying the A 2 condition. In [13, 14], a more general situation is considered:

v = v ( x ) , w 1 = w 1 ( x , t ) , w 2 = w 2 ( x , t ) ,

with w 1 possibly different from w 2 .

In the present paper, we focus our attention on the coefficient in front of u, taking for this reason w 1 = w 2 1 and v = ξ ( x , t ) , ξ > 0 , almost everywhere and bounded, so possibly degenerating only to zero (and not to + ).

We conclude saying that we use a technique due to DiBenedetto, Gianazza and Vespri, which adapts in some sense the technique of De Giorgi (to prove boundedness and regularity of the solutions) for the elliptic case to the parabolic case. This is mainly contained in [11].

2 Preliminaries

From now on, we will consider an open set Ω n , T > 0 and a function

ξ L ( Ω × ( 0 , T ) ) , ξ > 0  almost everywhere.

We define for each t ( 0 , T ) ,

L 2 ( Ω , ξ ( t ) ) = { u L loc 1 ( Ω ) u ξ 1 / 2 ( t ) L 2 ( Ω ) }

and

L 2 ( Ω × ( 0 , T ) , ξ ) = { u L loc 1 ( Ω × ( 0 , T ) ) u ξ 1 / 2 L 2 ( Ω × ( 0 , T ) ) } .

Moreover, we define for each t ( 0 , T ) ,

H 1 ( Ω , ξ ( t ) ) := { u L 2 ( Ω , ξ ( t ) ) D u L 2 ( Ω ) } .

By Du we will always denote the vector of derivatives of a function u with respect to the variables x 1 , , x n , ( x 1 , , x n ) Ω , even if u depends also on t. Then we define

𝒱 := { u L 2 ( Ω × ( 0 , T ) , ξ ) D u L 2 ( Ω × ( 0 , T ) ) }

and denote by 𝒱 the dual space of 𝒱 . We also define

𝒱 loc := { u L loc 2 ( Ω × ( 0 , T ) , ξ ) D u L loc 2 ( Ω × ( 0 , T ) ) } .

In the following, we will sometimes write

ξ ( E ) instead of E ξ ( x , t ) 𝑑 x 𝑑 t ,    E Ω × ( 0 , T ) ,
ξ ( t ) ( A ) instead of A ξ ( x , t ) 𝑑 x ,    A Ω .

In [20, 23], some existence results are proved (where ξ may also be zero and negative). As byproducts, we get the existence of solutions for the following equations (A, B and C introduced in the previous section):

ξ ( x , t ) u t - div ( A ( x , t , u , D u ) ) + B ( x , t , u , D u ) + C ( x , t , u ) = 0 ,
t ( ξ ( x , t ) u ) - div ( A ( x , t , u , D u ) ) + B ( x , t , u , D u ) + C ( x , t , u ) = 0 .

About ξ we will need some assumptions; before listing them, we recall some definitions.

Definition 2.1.

Given a ω L ( n ) , ω > 0 a.e., K > 0 and q > 2 , we say that

ω B 2 , q 1 ( K )

if

(2.1) r ρ ( ω ( B r ( x ¯ ) ) ω ( B ρ ( x ¯ ) ) ) 1 / q ( | B r ( x ¯ ) | | B ρ ( x ¯ ) | ) - 1 / 2 K

for every pair of concentric balls B r and B ρ with 0 < r < ρ .

We say that ω is doubling if for every t > 0 there is a positive constant c ω ( t ) such that

(2.2) t B ω ( x ) 𝑑 x c ω ( t ) B ω ( x ) 𝑑 x ,

where B is a generic ball B r ( x ¯ ) and tB denotes the ball B t r ( x ¯ ) .

We say that ω belongs to the class A ( K , ς ) if

ω ( S ) ω ( B ) K ( | S | | B | ) ς

for every ball B n and every measurable set S B .

We will denote by A ( K ) the set ς > 0 A ( K , ς ) , and A = K > 0 A ( K ) .

Remark 2.2.

If ω A , then ω is doubling (see [10]).

Remark 2.3.

If ω A ( K ) , then there is p 1 such that

( | S | | B | ) p K ω ( S ) ω ( B )

for every measurable S B and every B ball of n (see [10]).

About the function ξ, we will assume that there are some positive constants K 1 , K 2 , K 3 , ς , σ and L 0 such that the following conditions (H) hold:

(H1) { [ 0 , T ] t Ω v ( x ) w ( x ) ξ ( x , t ) 𝑑 x is absolutely continuous, | Ω v ( x ) w ( x ) ξ ( x , t ) 𝑑 x | L v H 1 ( Ω ) w H 1 ( Ω ) for every  v , w H 1 ( Ω ) ,
(H2) ξ ( , t ) B 2 , q 1 ( K 1 ) for almost every  t ( 0 , T ) ,
(H3) ξ ( , t ) A ( K 2 , ς ) for almost every  t ( 0 , T ) ,
(H4) ξ A ( K 3 , σ ) .

Comments on the assumptions.

(H1) This assumption is needed in [20, 23] to obtain the results about existence and uniqueness of the solutions of (1.1) and (1.2) with suitable boundary conditions.

Notice that this requirement is done for every v , w H 1 ( Ω ) , which turns out to be a dense subset of H 1 ( Ω , ξ ( t ) ) for each t ( 0 , T ) .

This assumption is clearly satisfied if

ξ , ξ t L ( Ω × ( 0 , T ) )

since in this case we get

| d d t Ω v ( x ) w ( x ) ξ ( x , t ) 𝑑 x | = | Ω v ( x ) w ( x ) ξ t ( x , t ) 𝑑 x | ξ t v L 2 ( Ω ) w L 2 ( Ω ) .

But more interesting are the cases when t ξ ( x , t ) is not absolutely continuous; it may be only continuous (see an example in [22]) and even discontinuous for every x Ω . The simplest example is the following: consider ( Ω = ( 0 , 1 ) , T = 1 )

ξ : ( 0 , 1 ) × ( 0 , 1 ) such that ξ ( x , t ) := { 2 for  x < t , 1 for  x > t .

Then

d d t Ω v ( x ) w ( x ) ξ ( x , t ) 𝑑 x = d d t ( 2 0 t v ( x ) w ( x ) 𝑑 x + t 1 v ( x ) w ( x ) 𝑑 x ) = v ( x ) w ( x ) .

Since H 1 ( 0 , 1 ) C 0 ( [ 0 , 1 ] ) , one can estimate the left-hand side in the previous equality by the H 1 -norm of v and w (in this example, H 1 ( Ω , ξ ( t ) ) = H 1 ( Ω ) for every t).

Analogous examples may be considered in higher dimensions, for which we refer to [20, 23].

Assumptions (H2) and (H3) are needed for Theorem 2.8 to hold. Notice moreover that in fact they hold for every, and not for almost every, t [ 0 , T ] , thanks to assumption (H1).

Finally, assumption (H4) (more precisely, its consequence (C3) stated below) is needed in Lemma 4.6 and consequently for the expansion of positivity (Theorem 4.8).

Some consequences of the assumptions.

  1. From assumption (H1), one can derive that for every ball B Ω there exists a constant γ B , depending only on B, such that

    sup t [ 0 , T ] B ξ ( x , t ) 𝑑 x γ B inf t [ 0 , T ] B ξ ( x , t ) 𝑑 x .

    By that, one also derives that

    sup t [ 0 , T ] [ B ξ ( x , t ) 𝑑 x ] - 1 γ B inf t [ 0 , T ] [ B ξ ( x , t ) 𝑑 x ] - 1 .

    This is actually a consequence of Lemma 2.4 stated below, which can be proved starting from (H1).

  2. By Remark (2.2) and since, by (H3), ξ ( t ) A ( K 2 ) uniformly in t, we obtain that for every θ > 0 there is a constant c ξ ( θ ) (depending on K 2 and θ) such that

    θ B ξ ( x , t ) 𝑑 x c ξ ( θ ) B ξ ( x , t ) 𝑑 x for every  t ( 0 , T ) .

    Similarly, by (H4) we also have that for every θ > 0 there is a constant C ξ ( θ ) such that

    t o - θ δ t o + θ δ θ B ξ ( x , t ) 𝑑 x 𝑑 t C ξ ( θ ) t o - δ t o + δ B ξ ( x , t ) 𝑑 x 𝑑 t .

    We will simply write c ξ or C ξ if θ = 2 .

  3. Assumptions (H3) and (H4) imply that there is p 1 such that

    ( | S | | B | ) p K 2 ξ ( t ) ( S ) ξ ( t ) ( B ) , ( | S ~ | | B × I | ) p K 3 ξ ( S ~ ) ξ ( B × I )

    for almost every t ( 0 , T ) , every ball B n , every interval I , every measurable set S B , and every measurable set S ~ B × I .

Lemma 2.4.

If ξ satisfies (H1), then

[ 0 , T ] t B ξ ( x , t ) 𝑑 x is continuous

for every ball B = B r ( x o ) Ω .

Proof.

Consider a function ζ such that

ζ 1 in  B r ( x o ) ,
ζ 0 outside of  B R ( x o ) ,
| D ζ | 1 R - r .

Then, by the assumptions, the function

[ 0 , T ] t B R ζ n ( x ) ξ ( x , t ) 𝑑 x

is continuous for every n , n 1 . Consider now t , s [ 0 , T ] and estimate as follows:

| ξ ( t ) ( B r ) - ξ ( s ) ( B r ) | = | B R ζ n ( x ) ξ ( x , t ) 𝑑 x - B R B r ζ n ( x ) ξ ( x , t ) 𝑑 x - B R ζ n ( x ) ξ ( x , s ) 𝑑 x + B R B r ζ n ( x ) ξ ( x , s ) 𝑑 x |
| B R ζ n ( x ) ξ ( x , t ) 𝑑 x - B R ζ n ( x ) ξ ( x , s ) 𝑑 x | + | B R B r ζ n ( x ) ξ ( x , t ) 𝑑 x - B R B r ζ n ( x ) ξ ( x , s ) 𝑑 x | .

Now fix ε > 0 . There is δ > 0 such that if | t - s | < δ , then

| B R ζ n ( x ) ξ ( x , t ) 𝑑 x - B R ζ n ( x ) ξ ( x , s ) 𝑑 x | < ε 3 .

Moreover, we can also chose n great enough in such a way that

B R B r ζ n ( x ) ξ ( x , t ) 𝑑 x < ε 3 and B R B r ζ n ( x ) ξ ( x , s ) 𝑑 x < ε 3 .

Then we conclude that

| ξ ( t ) ( B r ) - ξ ( s ) ( B r ) | < ε

for | t - s | < δ . ∎

For the following result, see, e.g., [21, Lemma 2.14]. This lemma is needed to prove the main result (see in particular the fourth step).

Lemma 2.5.

Consider x o Ω and r > 0 such that B 2 r ( x o ) Ω , σ ( 0 , r ) , ω B 2 , q 1 ( K ) for some q > 2 , ω A , and α , β > 0 . Consider an open and non-empty subset B of B r ( x o ) such that B σ = { x Ω dist ( x , B ) < σ } is a subset of B r ( x o ) . Then, for every a > 0 and ε , δ ( 0 , 1 ) , there exists η ( 0 , 1 ) such that for every u H 1 ( Ω , ω ) satisfying

σ | D u | 2 𝑑 x α | B r ( x o ) | r 2

and

ω ( { u > a } ) β ω ( B r ( x o ) ) ,

there exists x * B with B η ρ ( x * ) B such that

ω ( { u > ε a } B η r ( x * ) ) > ( 1 - δ ) ω ( B η r ( x * ) ) .

For the following result, see [3].

Theorem 2.6.

Consider q > 2 , r > 0 , x o R n , and ω L ( R n ) , ω > 0 a.e., ω B 2 , q 1 ( K ) and doubling, i.e. satisfying (2.1) and (2.2). Then there is a constant Γ depending (only) on n , q , K , c ω such that

[ 1 ω ( B r ) B r | u ( x ) | q ω ( x ) 𝑑 x ] 1 / q Γ r [ 1 | B r | B r | D u ( x ) | 2 𝑑 x ] 1 / 2

for every Lipschitz continuous function u defined in B r = B r ( x 0 ) , with either support contained in B r ( x 0 ) or with null mean value.

By Theorem 2.6 and one of its direct consequences (a result contained in [15, 14], but see also [21, Theorem 2.9]), one gets the following result.

Remark 2.7.

The assumptions required in the result in [15] involve the ratio between two weights. In our case (in Theorem 2.8) this assumption is simply

(2.3) ( ξ ( t ) ) - 1 A ( ξ ( t ) ) , where  A ( ξ ( t ) )  is the  A  class with respect to the measure  ξ ( t ) d x ) .

If ξ satisfies (H3), this request is satisfied.

Instead of (H3), since we suppose ξ L , another condition which implies (2.3) is that there is a constant C > 0 such that

ξ ( t ) L ( B ) C 1 | B | B ξ ( x , t ) 𝑑 x = C ξ ( t ) ( B ) | B |

for every ball B and for a.e. t ( 0 , T ) .

Theorem 2.8.

Consider q > 2 and ξ L ( R n + 1 ) , ξ > 0 a.e., satisfying (H2) and (H3) for some K 1 , K 2 > 0 . Then there is κ ( 1 , q 2 ) and Γ depending (only) on n , q , K 1 , K 2 , ς such that

s 1 s 2 1 η ( t ) ( B r ) B r | u | 2 κ ( x , t ) η ( x , t ) 𝑑 x 𝑑 t
Γ 2 r 2 ( max s 1 t s 2 1 ξ ( t ) ( B r ) B r u 2 ( x , t ) ξ ( x , t ) 𝑑 x ) κ - 1 1 | B r | s 1 s 2 B r | D u | 2 ( x , t ) 𝑑 x 𝑑 t

for every B r ( x o ) R n , every ( s 1 , s 2 ) ( 0 , T ) , every Lipschitz continuous function u defined in B r × ( s 1 , s 2 ) , u ( , t ) with either support contained in B r ( x o ) or with null mean value, and where the inequality holds both with η = ξ and η 1 .

Proof.

Here we do not present the proof, since it can be derived adapting easily that of [21, Theorem 2.9]. We only stress that (2.3) is needed, but this is a consequence of (H3) as observed in Remark 2.7. ∎

Remark 2.9.

About the proof of the previous theorem, we want to consider the following question: whether η = ξ requiring ξ to be doubling (together to ξ ( t ) B 2 , q 1 ( K 1 ) uniformly in t) is in fact sufficient to get the thesis of Theorem 2.8. The assumption ξ ( t ) A ( K 2 ) is needed just to get the thesis with η 1 (and, more general, with η ξ ).

In this regard, see [21, Remark 2.3 and Theorem 2.9].

Other results useful in the sequel are the following two lemmas, for which we refer to [21].

Lemma 2.10.

Consider ω as in Theorem 2.6, k , l R with k < l , and p ( 1 , 2 ] . Then

( l - k ) ω ( { v < k } ) ω ( { v > l } ) 2 Γ r ( ω ( B r ) ) 2 ( 1 | B r | B r { k < v < l } | D v | p d x ) 1 / p

for every Lipschitz continuous function v defined in the ball B r .

We conclude stating a standard lemma (see, for instance, [12, Lemma 7.1]) needed later.

Lemma 2.11.

Let ( y h ) h be a sequence of positive real numbers such that

y h + 1 c b h y h 1 + α

with c , α > 0 and b > 1 . If y 0 c - 1 / α b - 1 / α 2 , then

lim h + y h = 0 .

3 DG classes

We are going now to introduce a De Giorgi class suited to the parabolic equations (1.1) and (1.2) of order 2; before, we need to we consider a suitable class of decay functions, namely the set

𝒳 c := { ζ Lip ( Ω × ( 0 , T ) ) ζ ( , t ) Lip c ( Ω )  for each  t ( 0 , T ) , ζ 0 , ζ t 0 } .

By v + we will denote the positive part of a function v, and by v - we will denote the negative part of a function v.

Definition 3.1.

Given γ > 0 , we say that a function u 𝒱 loc , such that

( 0 , T ) t Ω u 2 ( x , t ) ξ ( x , t ) 𝑑 x C loc 0 ( 0 , T ) ,

belongs to D G ( Ω , T , ξ , γ ) if u satisfies for every function ζ 𝒳 c , every t 1 , t 2 ( 0 , T ) and every k the inequality

Ω ( u - k ) + 2 ζ 2 ξ ( x , t 2 ) 𝑑 x + t 1 t 2 Ω | D ( u - k ) + | 2 ζ 2 𝑑 x 𝑑 t
Ω ( u - k ) + 2 ζ 2 ξ ( x , t 1 ) d x + γ [ t 1 t 2 Ω ( u - k ) + 2 | D ζ | 2 d x d t + t 1 t 2 Ω ( u - k ) + 2 ζ ζ t ξ d x d t
(3.1)     + t 1 t 2 Ω ( u - k ) + 2 ζ 2 d x d t + k 2 t 1 t 2 { u ( t ) > k } ( ζ 2 + | D ζ | 2 ) d x d t ] .

and if u satisfies an analogous inequality for every function ζ 𝒳 c , every t 1 , t 2 ( 0 , T ) , and every k , with ( u - k ) - instead of ( u - k ) + and with

k 2 t 1 t 2 { u ( t ) < k } ( ζ 2 + | D ζ | 2 ) 𝑑 x 𝑑 t

instead of the last term in inequality (3.1).

Remark 3.2 (Comments on Definition 3.1).

  1. The energy inequality (3.1) can be derived as done in [22] starting from both equations (1.1) and (1.2). Notice that in [22] only k 0 is considered to derive (3.1) (and only k 0 to derive the analogous one for ( u - k ) - ): indeed, these restrictions are due to the fact that in [22] a wider class was considered and can be removed in our case.

  2. A solution of (1.1) belongs to { v 𝒱 ξ v 𝒱 } and a solution of (1.2) belongs to { v 𝒱 ( ξ v ) 𝒱 } , but in fact these two spaces, under assumption (H1), turn out to be the same space.

  3. A function belonging to D G ( Ω , T , ξ , γ ) is locally bounded (see, for instance, [24]).

Remark 3.3.

The solutions of equation (1.1) with C 0 satisfy

Ω ( u - k ) + 2 ζ 2 ξ ( x , t 2 ) 𝑑 x + t 1 t 2 Ω | D ( u - k ) + | 2 ζ 2 𝑑 x 𝑑 t
Ω ( u - k ) + 2 ζ 2 ξ ( x , t 1 ) d x + γ [ t 1 t 2 Ω ( u - k ) + 2 | D ζ | 2 d x d t
(3.2)     + t 1 t 2 Ω ( u - k ) + 2 ζ ζ t ξ d x d t + t 1 t 2 Ω ( u - k ) + 2 ζ 2 d x d t ]

and the analogous one for ( u - k ) - . Notice that if u satisfies these inequalities, also u + c satisfies them, where c is an arbitrary constant (in particular, this holds for solutions of (1.1) with C 0 ).

This is the key point to use the argument due to Moser, by which one can derive the local Hölder continuity from the Harnack inequality for functions satisfying (3.2) and the analogous estimate with ( u - k ) - .

For fixed ( x o , t o ) Ω × ( 0 , T ) and r > 0 , we define the quantities

𝗁 ( x o , t o , r ) := 1 r 2 | B r ( x o ) | t o - r 2 t o B r ( x o ) ξ ( x , t ) 𝑑 x 𝑑 t ,
𝗁 ( x o , t o , r ) := 1 r 2 | B r ( x o ) | t o t o + r 2 B r ( x o ) ξ ( x , t ) 𝑑 x 𝑑 t .

Notice that

(3.3) { 𝗁 ( x o , t o - r 2 , r ) = 𝗁 ( x o , t o , r ) , 𝗁 ( x o , t o , r ) = 𝗁 ( x o , t o + r 2 , r ) .

Notice that for 0 < r < r ~ one has

(3.4) 𝗁 ( x o , t o , r ) ( r ~ r ) n + 2 𝗁 ( x o , t o , r ~ ) , 𝗁 ( x o , t o , r ) ( r ~ r ) n + 2 𝗁 ( x o , t o , r ~ ) .

Once we fixed ( x o , t o ) Ω × ( 0 , T ) , for a generic R > 0 we will set

Q R ( x o , t o ) := B R ( x o ) × ( t o - R 2 𝗁 ( x o , t o , R ) , t o ) ,
Q R ( x o , t o ) := B R ( x o ) × ( t o , t o + R 2 𝗁 ( x o , t o , R ) ) .

Given r ( 0 , R ] and θ > 0 (possibly also greater than 1), we define

(3.5) { Q r , θ ( x o , t o ) := B r ( x o ) × ( t o - θ r 2 𝗁 ( x o , t o , R ) , t o ) , Q r , θ ( x o , t o ) := B r ( x o ) × ( t o , t o + θ r 2 𝗁 ( x o , t o , R ) ) ,

where 𝗁 and 𝗁 always refer to a fixed radius R bigger than or equal to r (only for θ = 1 and r = R the cylinder Q r , θ coincides with Q R ). To lighten the notation, we will omit writing ( x o , t o ) if not strictly necessary and simply write

Q R , Q R , Q r , θ , Q r , θ

if it is clear which point we are referring to.

4 Expansion of positivity

We recall that u D G ( Ω , T , ξ , γ ) is locally bounded, as mentioned in Remark 3.2.

Proposition 4.1.

Consider ( x ¯ , t ¯ ) Ω × ( 0 , T ) and R > 0 such that

Q R = Q R ( x ¯ , t ¯ ) Ω × ( 0 , T ) .

Let h = h ( x ¯ , t ¯ , R ) . Consider r , r ~ , θ , θ ~ satisfying

0 < r < r ~ R , 0 < θ < θ ~    such that    Q r ~ , θ ~ ( x ¯ , t ¯ ) Ω × ( 0 , T ) .

Then, for every choice of a , σ ( 0 , 1 ) , every u D G ( Ω , T , ξ , γ ) and every μ + , ω satisfying

μ + sup Q r ~ , θ ~ ( x ¯ , t ¯ ) u , ω osc Q r ~ , θ ~ ( x ¯ , t ¯ ) u ,

there is ν, depending (only) on γ, 1 - a , κ, γ B r ~ ( x ¯ ) , Γ, r ~ - 1 , θ ~ - 1 , R / r ~ , c ξ ( R / r ~ ) , C ξ ( θ ~ ) , r ~ - r , θ ~ - θ , and μ + σ ω , such that if

| { ( x , t ) Q r ~ , θ ~ ( x ¯ , t ¯ ) u ( x , t ) > μ + - σ ω } | | Q r ~ , θ ~ ( x ¯ , t ¯ ) | + ξ ( { ( x , t ) Q r ~ , θ ~ ( x ¯ , t ¯ ) u ( x , t ) > μ + - σ ω } ) ξ ( Q r ~ , θ ~ ( x ¯ , t ¯ ) ) ν ,

then

u ( x , t ) μ + - a σ ω for a.e.  ( x , t ) Q r , θ ( x ¯ , t ¯ ) .

Remark 4.2.

This is the only result (indeed, also Proposition 4.3) stated for each function belonging to D G ( Ω , T , ξ , γ ) , while in all of the remainder we consider nonnegative functions.

In this way, the constant ν may depend on the oscillation of a specific function in the class D G ( Ω , T , ξ , γ ) , but in fact it depends on the ratio

μ + ω ,

as highlighted in the statement. In particular, if one confines to consider a function u 0 (as we will do) and choose

μ + = sup Q r ~ , θ ~ ( x ¯ , t ¯ ) u , ω = osc Q r ~ , θ ~ ( x ¯ , t ¯ ) u ,

in fact ν does not depend anymore on μ + or on ω since μ + ω 1 (see in particular the last lines of the following proof and also step 1 of the proof of Theorem 5.2).

Proof.

Consider for h ,

r h = r + r ~ - r 2 h ,
θ h = θ + θ ~ - θ 4 h ,
Q h := Q r h , θ h ( x ¯ , t ¯ ) ,
σ h = a σ + 1 - a 2 h σ ,
k h = μ + - σ h ω .

With these choices, we have that

θ h - θ h + 1 = 3 θ ~ - θ ( r ~ - r ) 2 ( r h - r h + 1 ) 2

and that

Q h + 1 Q h .

Consider now a sequence of functions satisfying

ζ h 1 in  Q h + 1 ,
ζ h 0 outside of  Q h for  t t ¯ ,
0 ζ h 1 ,
| D ζ h | 1 r h - r h + 1

and

0 ( ζ h ) t 1 ( θ h r h 2 - θ h + 1 r h + 1 2 ) 𝗁
1 ( θ h r 2 - θ h + 1 r 2 ) 𝗁
= 4 h + 1 3 r 2 𝗁 1 θ ~ - θ
= 1 3 r 2 𝗁 ( r h - r h + 1 ) 2 ( r ~ - r ) 2 θ ~ - θ
< 1 r 2 𝗁 ( r h - r h + 1 ) 2 ( r ~ - r ) 2 θ ~ - θ ,

and finally define

A h = { ( x , t ) Q h u ( x , t ) > k h } .

First, we have (κ is the value in Theorem 2.8 and γ B is defined in (C1))

( ( 1 - a ) σ ω ) 2 2 2 h + 2 | A h + 1 | | Q r ~ , θ ~ | = ( k h + 1 - k h ) 2 | A h + 1 | | Q r ~ , θ ~ |
1 | Q r ~ , θ ~ | A h + 1 ( u - k h ) + 2 𝑑 x 𝑑 t
1 | Q r ~ , θ ~ | Q h + 1 ( u - k h ) + 2 𝑑 x 𝑑 t
1 | Q r ~ , θ ~ | Q h ( u - k h ) + 2 ζ h 2 𝑑 x 𝑑 t
( | A h | | Q r ~ , θ ~ | ) κ - 1 κ ( 1 | Q r ~ , θ ~ | Q h ( u - k h ) + 2 κ ζ h 2 κ 𝑑 x 𝑑 t ) 1 κ
(4.1) = ( | A h | | Q r ~ , θ ~ | ) κ - 1 κ ( 1 θ ~ r ~ 2 𝗁 ) 1 κ ( 1 | B r ~ | Q h ( u - k h ) + 2 κ ζ h 2 κ 𝑑 x 𝑑 t ) 1 κ .

Similarly,

( ( 1 - a ) σ ω ) 2 2 2 h + 2 ξ ( A h + 1 ) inf t ( t ¯ - θ h + 1 r h + 1 2 𝗁 , t ¯ ) 1 ξ ( t ) ( B r ~ )
= ( k h + 1 - k h ) 2 ξ ( A h + 1 ) inf t ( t ¯ - θ h + 1 r h + 1 2 𝗁 , t ¯ ) 1 ξ ( t ) ( B r ~ )
inf t ( t ¯ - θ h + 1 r h + 1 2 𝗁 , t ¯ ) 1 ξ ( t ) ( B r ~ ) A h + 1 ( u - k h ) + 2 ξ 𝑑 x 𝑑 t
A h 1 ξ ( t ) ( B r ~ ) ( u - k h ) + 2 ζ h 2 ξ 𝑑 x 𝑑 t
( A h 1 ξ ( t ) ( B r ~ ) ξ 𝑑 x 𝑑 t ) κ - 1 κ ( Q h 1 ξ ( t ) ( B r ~ ) ( u - k h ) + 2 κ ζ h 2 κ ξ 𝑑 x 𝑑 t ) 1 κ ,

by which

(4.2) ( ( 1 - a ) σ ω ) 2 2 2 h + 2 1 γ B r ~ ξ ( A h + 1 ) ξ ( Q r ~ , θ ~ ) γ B r ~ κ - 1 κ ( ξ ( A h ) ξ ( Q r ~ , θ ~ ) ) κ - 1 κ ( 1 θ ~ r ~ 2 𝗁 ) 1 κ ( Q h 1 ξ ( t ) ( B r ~ ) ( u - k h ) + 2 κ ζ h 2 κ ξ 𝑑 x 𝑑 t ) 1 κ .

Now we use Theorem 2.8 to estimate the last factors in (4.1) and (4.2). In the following, η may denote the function ξ or the constant function 1. We get

( Q h 1 η ( t ) ( B r ~ ) ( u - k h ) + 2 κ ζ h 2 κ η ( x , t ) 𝑑 x 𝑑 t ) 1 κ
Γ 2 κ r ~ 2 κ ( max t ¯ - θ h r h 2 𝗁 t t ¯ 1 ξ ( t ) ( B r ~ ) B r h ( u - k h ) 2 ζ h 2 ξ ( x , t ) 𝑑 x ) κ - 1 κ ( 1 | B r ~ | Q h | D ( ( u - k h ) + ζ h ) | 2 𝑑 x 𝑑 t ) 1 κ
Γ 2 κ r ~ 2 κ ( max t ¯ - θ h r h 2 𝗁 t t ¯ 1 ξ ( t ) ( B r ~ ) ) κ - 1 κ ( 1 | B r ~ | ) 1 κ
    [ max t ¯ - θ h r h 2 𝗁 t t ¯ B r h ( u - k h ) 2 ζ h 2 ξ ( x , t ) d x + 2 Q h | D ( u - k h ) + | 2 ζ h 2 d x d t
(4.3)        + 2 Q h | D ζ h | 2 ( u - k h ) + 2 d x d t ] .

Now since

(4.4) ( u - k h ) + σ h ω ,

we can estimate the last term in the right-hand side as

(4.5) 2 Q h | D ζ h | 2 ( u - k h ) + 2 𝑑 x 𝑑 t 2 | A h | ( r h - r h + 1 ) 2 ( σ h ω ) 2 .

Now we estimate the first two of the three addends of the last factor in the right-hand side using (3.1):

max t ¯ - θ h r h 2 𝗁 t t ¯ B r h ( u - k h ) 2 ζ h 2 ξ ( x , t ) 𝑑 x + 2 Q h | D ( u - k h ) + | 2 ζ h 2 𝑑 x 𝑑 t
2 γ [ Q h ( u - k h ) + 2 | D ζ h | 2 d x d t + Q h ( u - k h ) + 2 ζ h ( ζ h ) t ξ d x d t
    + Q h ( u - k h ) + 2 ζ h 2 d x d t + k h 2 A h ( ζ h 2 + | D ζ h | 2 ) d x d t ]
2 γ [ 1 ( r h - r h + 1