Local boundedness of solutions to non-local parabolic equations modeled on the fractional p-Laplacian

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Local boundedness of solutions to non-local parabolic equations modeled on the fractional p-Laplacian Martin Strömqvist Department of Mathematics, Uppsala University, S-751 06 Uppsala, Sweden Received 20 December 2017; revised 28 September 2018

Abstract We state and prove estimates for the local boundedness of subsolutions of non-local, possibly degenerate, parabolic integro-differential equations of the form ˆ K(x, y, t)|u(x, t) − u(y, t)|p−2 (u(x, t) − u(y, t)) dy = 0,

∂t u(x, t) + P.V. Rn

(x, t) ∈ Rn ×R, where P.V. means in the principle value sense, p ∈ (1, ∞) and the kernel obeys K(x, y, t) ≈ |x − y|n+ps for some s ∈ (0, 1), uniformly in (x, y, t) ∈ Rn × Rn × R. © 2018 Elsevier Inc. All rights reserved. MSC: 30L; 35R03; 35K92 Keywords: Quasilinear non-local operators; Quasilinear parabolic non-local operators; Caccioppoli estimates; Local boundedness; Intrinsic geometry

1. Introduction and statement of main results In this work we study local regularity properties of solutions to the equation ∂u(x, t) + Lu(x, t) = 0 in  × (t1 , t2 ), ∂t E-mail address: martin.strö[email protected]. https://doi.org/10.1016/j.jde.2018.12.021 0022-0396/© 2018 Elsevier Inc. All rights reserved.

(1.1)

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for a bounded domain . In (1.1), L is a nonlinear, nonlocal operator of p-Laplace type. Specifically, we assume that L is formally given by ˆ Lu(x, t) = P.V. |u(x, t) − u(y, t)|p−2 (u(x, t) − u(y, t))K(x, y, t)dy, (1.2) Rn

where P.V. means principal value and the kernel K satisfies, for some  ≥ 1 and s ∈ (0, 1), −1  ≤ K(x, y, t) ≤ . n+sp |x − y| |x − y|n+sp

(1.3)

Throughout the paper we will assume that p ≥ 2, which corresponds to equations that are possibly degenerate. Elliptic nonlocal equations of this type (Lu = 0) has received great attention in recent years. Ishii and Nakamura [16] were the first authors to study this equation, with K(x, y, t) = (1 − s)|x − y|n+sp and in a localized setting. They proved existence and uniqueness of viscosity solutions and showed that in this case L converges to the p-Laplace operator as s → 1. In [9] Di Castro, Kuusi and Palatucci studied the elliptic counterpart of (1.1) and proved local boundedness and Hölder continuity of solutions. In [8] the same authors proved a very interesting nonlocal version of the Harnack inequality for solutions u. It involves the so-called tail of the negative part of u and does not require solutions to be globally positive. For elliptic equations with a nonzero right hand side (Lu = f ), Hölder continuity of the solutions has been obtained in [19], [5]. Through the use of fractional DeGiorgi classes, M. Cozzi [7] proved the results of [9] and [8] for solutions to a more general class of equations, involving a term f (u), or solutions to associated minimum problems. When it comes to parabolic problems, an analogous theory of local boundedness, Hölder continuity and Harnack’s inequality does not exist for p = 2. In the linear case p = 2, Harnack’s inequality has been established for globally positive solutions using probabilistic methods in [2]. In the paper [4], Bonforte, Sire and Vazquez develop an optimal existence and uniqueness theory for the Cauchy problem posed in Rn . For globally positive solutions to the fractional heat equation, they prove Hölder estimates and a Harnack inequality in which the usual timelag present in parabolic Harnack inequalities does not occur. This is due to the fact that the fractional heat kernel is not of Gaussian form. Caffarelli, Chan and Vasseur [6] study parabolic nonlocal, nonlinear equations of quadratic growth in all space. They prove that solutions are bounded and Hölder continuous as soon as the initial data is in L2 . Felsinger and Kassmann [13] prove a weak Harnack inequality and Hölder continuity for weak solutions to (1.1) that are globally positive. They work with a class of kernels satisfying slightly weaker growth conditions than (1.3). Due to the assumption of global positivity, the nonlocal term involving the negative part of the solution (the tail term), that normally occur in such estimates, is not present. In [17], Schwab and Kassmann prove results similar to those in [13], but with a(t, x, y)dμ(x, y) in place of K(t, x, y)dxdy, merely assuming that μ is a measure, not necessarily absolutely continuous w.r.t. Lebesgue measure, that satisfies certain growth conditions. It should also be mentioned that the conditions on imposed on the kernels/measures in [13] and [17] are in general not sufficient to prove a Harnack inequality. This is due to a result by Bogdan and Sztonyk [3] that prove sharp conditions on the kernel for a Harnack inequality to hold (in the elliptic setting). To the authors best knowledge, there is as of yet no theory of local boundedness for equations of the type (1.1), even when p = 2.

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A homogeneous version of (1.1) (a nonlocal version of Trudinger’s equation) was studied by Lindgren and Hynd in [15] where the authors prove Hölder continuity of solutions. The equation (1.1) has been studied in [20] and [22] in the case u = 0 in Rn \ . In [20] the authors prove an estimate of Lq ()-norms of u(x, t) in terms of the initial data. In [22] one of the main results is that any solution is dominated by a special function of the form t 1/(p−2)(F (x)), independently of the initial data at t = 0. These estimates are fundamentally different from the ones in the present paper in that they are not local. The purpose of this paper is to develop a basis for further study of the regularity theory of weak solutions to equations of the type (1.1). To this end we prove Caccioppoli type inequalities and establish local boundedness of weak subsolutions. In future projects we will study Harnack/Hölder estimates for (1.1). Hölder estimates and Harnack inequalities for local equations of p-Laplace type is considerably more involved in the parabolic setting, compared to the elliptic setting, or to the parabolic setting for p = 2. This is essentially due to the inherent inhomogeneity of these equations, which leads to intrinsic Harnack/Hölder estimates that are valid only for times depending on the local size of the solution. Harnack’s inequality for local equations was proved independently by Kuusi [18] and DiBenedetto, Gianazza and Vespri [12]. The results in [18] were modified and extended to a wider class of operators in [1] by Avelin, Capogna, Citti and Nyström. For Hölder estimates we refer to [11]. Our main result is that local weak solutions to (1.1) are bounded. The estimates will depend on a nonlocal quantity called the parabolic tail of the solution. If v ∈ Lp (t0 − T0 , t0 ; W s,p (Rn )), the (parabolic) tail of v is defined by ⎛ sp

⎜r Tail(v; x0 , r, t1 − T1 , t1 ) = ⎝ T1

ˆt1

⎞

ˆ

t1 −T1 Rn \Br (x0 )

1 p−1

|v(x, t)|p−1 ⎟ dxdt ⎠ |x − x0 |n+sp

,

whenever t1 ≤ t0 and t0 − T0 ≤ t1 − T1 . If Q = Br (x0 ) × (t1 − T1 , t1 ), we set Tail(v; Q) = Tail(v; x0 , r, t1 − T1 , t1 ). At times we will use a supremum (in time) version of the tail, given by ⎛ ⎜ Tail∞ (v; x0 , r, t1 − T1 , t1 ) = ⎝r sp

ˆ sup

t1 −T1
⎞ |v(x, t)|p−1 |x − x0 |n+sp

⎟ dx ⎠

1 p−1

.

See section 4.1 for a remark on this quantity. For parabolic rescaling of cubes Q, we will use the notation λQ = Bλr (x0 ) × (t1 − λsp T1 , t1 ). In all our estimates, C ≥ 1 will denote a generic constant that depends only on n and p unless otherwise stated. The numerical value of C may change during the course of an estimate. We can now state our main theorem.

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Theorem 1.1. Let Q = BR (x0 ) × (t0 − T0 , t0 ) and suppose that u is a nonnegative subsolution in 2Q. Then, if p > 2 C sup u ≤ (1 − σ )α σQ

+





1 T0 R sp p−2 p−1 + sp Tail∞ (u+ ; x0 , σ r, t0 − T0 , t0 ) T0 R ⎞p−1 ⎛

C T0 ⎜ ⎝ sup α (1 − σ ) R sp t0 −T0
⎟ u(x, t)dx ⎠

(1.4)

,

BR

for any σ ∈ (0, 1). We remark that if Tail∞ (u+ ; x0 , σ r, t0 − T0 , t0 ) ≤ C



R sp T0

1 p−2

, then

T0 p−1 Tail∞ (u+ ; x0 , σ r, t0 − T0 , t0 ) R sp ≤ Tail∞ (u+ ; x0 , σ r, t0 − T0 , t0 ) ≤

R sp T0



1 p−2

.

Then (4.30) becomes ⎛



sup u ≤ σQ

C ⎜ ⎝ α (1 − σ )

R sp p−2 1

T0

⎛ ⎜ T0 + ⎝ sp R

⎞p−1 ⎞ sup

t0 −T0
⎟ u(x, t)dx ⎠

⎟ ⎠.

This is precisely the estimate that holds for solutions to local equations. If one follows the proof of Theorem 1.1 (until Lemma 4.3) it is easy to verify that the following version of it holds for p = 2: If R 2s ≈ T0 , then sup u ≤ σQ

C C Tail∞ (u+ ; x0 , σ r, t0 − T0 , t0 ) + α (1 − σ ) (1 − σ )α

u(x, t)dx. Q

Organization of the paper In section 2 we recall a few tools from the theory of fractional Sobolev spaces. We also prove the existence and uniqueness of weak solutions to (1.1). Section 3 is devoted to the proof of a Caccioppoli inequality for weak subsolutions, as well as other properties of subsolutions. In section 4 we use Moser’s iteration technique to prove Theorem 1.1. This is based upon the Caccioppoli inequality from section 3. Additionally, we prove an estimate for the local supremum of |u| without any assumptions on the sign of u.

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2. Weak solutions in parabolic Sobolev spaces For a domain D ⊂ Rn , the fractional Sobolev space W s,p (D) consists of all functions f ∈ such that

Lp (D)

ˆ ˆ [f ]W s,p (D) = D D

|f (x) − f (y)|p dxdy < ∞. |x − y|n+sp

The norm of f ∈ W s,p (D) is given by ⎛

ˆ ˆ

f W s,p (D) = f Lp (D) + ⎝

D D

⎞1 p |f (x) − f (y)|p ⎠ dxdy . |x − y|n+sp

We shall also need the space s,p

W0 () = {f ∈ W s,p (Rn ) : f = 0 in Rn \ }, endowed with the norm · W s,p (Rn ) . We will later use the fact that a truncation of f does not increase its norm in W s,p : [f+ ]W s,p () ≤ [f ]W s,p () , [min{f, m}]W s,p () ≤ [f ]W s,p () ,

(2.1) for any m ∈ R.

(2.2)

To prove (2.1) we need only note that |a+ − b+ | ≤ |a − b| for any a, b ∈ R. Then (2.2) is a consequence of (2.1) and the fact that min{f, m} = −(m − f )+ + m. For the fractional Sobolev embedding below we refer to [10]. Theorem 2.1 (Sobolev embedding). Suppose p ≥ 1, sp < n and let κ ∗ =   f ∈ W s,p (Rn ) and κ ∈ 1, κ ∗ , p

f Lκp (Rn )

ˆ ˆ ≤ Rn

Rn

|f (x) − f (y)|p dxdy. |x − y|n+sp

n n−sp .

Then for any

(2.3)

If f ∈ W s,p () and if  is an extension domain for W s,p , then

f Lκp () ≤ C() f W s,p () .

(2.4)

If sp = n, then (2.3) and (2.4) hold for any κ ∈ [1, ∞). If sp > n, (2.4) holds for any κ ∈ [1, ∞]. For our purposes it is enough to know that any bounded Lipschitz domain is an extension domain for W s,p .

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s,p

Lemma 2.1. Suppose f ∈ W0 (Br ), where Br = Br (x0 ) and define κ∗ =



n n−sp

if n < sp, if n ≥ sp.

2

(2.5)

Then for any κ ∈ [1, κ ∗ ], ⎞1

⎛

κ

⎟ |f | dx ⎠ ≤ Cr sp−n

⎜ ⎝

ˆ ˆ

κp

Br

Br Br

|f (x) − f (y)|p dxdy + C |x − y|n+sp

|f |p dx.

(2.6)

Br

s,p

Proof. Since f ∈ W0 (Br ) we have by definition f ∈ W s,p (Rn ) and f = 0 in Rn \ Br . Thus f ∈ W s,p (Br ). Suppose r = 1. Then (2.6) is a consequence of Theorem 2.1. For r = 1, (2.6) follows from a scaling argument. 2 Lemma 2.2. Suppose p ≥ 1, sp < n and let κ ∗ =

n n−sp

and suppose that

s,p

f ∈ Lp (t1 , t2 ; W0 (Br )). Then for any κ ∈ [1, κ ∗ ], ˆt2 |f |κp dxdt t1 Br

⎛

⎜ ≤ C ⎝r sp−n

(2.7)

ˆt2

⎞

ˆt2

p

⎟ |f (x, t)|p dxdt ⎠

[f (·, t)]W s,p (Br ) dt + t1

t1 Br

⎛

⎞

⎜ × ⎝ sup

t1
|f |

pκ ∗ (κ−1) κ ∗ −1

κ ∗ −1 κ∗

⎟ dx ⎠

.

Br

Proof. By Hölder’s inequality and Lemma 2.1 we have ˆt2

ˆt2 |f | dxdt =

|f |p |f |(κ−1)p dxdt

κp

t1 Br

t1 Br

⎛ ˆt2 ⎜ ≤⎝ t1 Br

⎞ |f |

κ∗p

⎟ dxdt ⎠

1 κ∗

⎞ κ ∗ −1 ∗

⎛ ⎜ ⎝ sup

t1
|f | Br

pκ ∗ (κ−1) κ ∗ −1

κ

⎟ dx ⎠

(2.8)

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⎛ ⎜ ≤ ⎝Cr sp−n

ˆt2 ˆ ˆ t1 Br Br

|f (x) − f (y)|p |x − y|n+sp

⎛

ˆt2 dxdydt + C

7

⎞ ⎟ |f |p dxdt ⎠

t1 Br

⎞ κ ∗ −1 ∗

⎜ × ⎝ sup

t1
|f |

pκ ∗ (κ−1) κ ∗ −1

κ

⎟ dx ⎠

.

2

Br

2.1. Weak solutions We are now in a position to define weak solutions, and will show that for any bounded domain  ⊂ Rn and T > 0, the problem ⎧ ∂u(x, t) ⎪ ⎪ + Lu(x, t) = 0, ⎪ ⎨ ∂t

in T =  × (0, T ),

u(x, t) = g(x, t), in (Rn \ ) × (0, T ), ⎪ ⎪ ⎪ ⎩ u(x, 0) = u0 (x), in Rn ,

(2.9)

has a unique solution in a suitable sense, whenever g and u0 belong to appropriate function spaces. The interval (0, T ) used here may be replaced by any other interval. Motivated by (1.1) and (1.2), we define a weak solution as follows. For the sake of brevity we will use the notation Au(x, y, t) = K(x, y, t)|u(x, t) − u(y, t)|p−2 (u(x, t) − u(y, t)), δu(x, y, t) = u(x, t) − u(y, t), dμ = dμ(x, y, t) = K(x, y, t)dxdydt. Definition 2.1. Suppose g ∈ Lp (0, T ; W s,p (Rn )),

∂t g ∈ Lp (0, T ; (W s,p (Rn ))∗ ), u0 ∈ L2 (). We say that u ∈ Lp ((0, T ); W s,p (R)) is a weak solution to (2.9) if

∂t u ∈ Lp (0, T ; (W s,p (Rn ))∗ ), s,p

u − g ∈ Lp (0, T ; W0 ()) and

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ˆT ˆ ˆ

ˆT ˆ Au(x, y, t)(η(x, t) − η(y, t))dxdydt −

0

Rn

Rn

u∂t ηdxdt 0

ˆ

=

(2.10)

Rn

u0 η(0, x)dx,  s,p

for any η ∈ Lp ((0, T ); W0 ()) such that

∂t η ∈ Lp (0, T ; (W s,p (Rn ))∗ ) and

η(x, T ) = 0. s,p

Let w = u − g. Then u solves (2.10) if and only if w ∈ Lp (0, T ; W0 ()) solves ˆT ˆ ˆ A(w + g)(x, y, t)(η(x, t) − η(y, t))dxdydt 0

Rn

(2.11)

Rn

ˆT ˆ −

ˆT ˆ w∂t ηdxdt =

0

Rn

ˆ g∂t ηdxdt +

0

Rn

u0 η(0, x)dx, 

s,p

for any η ∈ Lp ((0, T ); W0 ()) such that

∂t η ∈ Lp (0, T ; (W s,p (Rn ))∗ ) and

η(x, T ) = 0.

2.1.1. Wellposedness The existence and uniqueness of a solution to (2.11) is a consequence of the general theory for degenerate parabolic equations in Banach spaces, see [21]. We will only briefly explain the ˜ = A(· + g). Suppose u(·, t) and v(·, t) properties of the equation that need to be verified. Let A(·) s,p belong to W0 (). Then by Hölder’s inequality and (1.3), ˆ ˆ Rn

˜ Au(x, y, t)(v(x, t) − v(y, t))dxdydt

(2.12)

Rn p−1

≤ [u(·, t) + g(·, t)]W s,p [v(·, t)]W s,p p−1

p−1

≤ 2p−1 u W s,p v W s,p + 2p−1 g W s,p v W s,p . Thus A˜ defines an operator Lt : W s,p (Rn ) → (W s,p (Rn ))∗ , with Lt u, v given by (2.12) and p−1

p−1

Lt u ≤ 2p−1  u(·, t) W s,p + 2p−1  g(·, t) W s,p . Additionally, Lt is a monotone operator, i.e. Lt u − Lt v, u − v ≥ 0, Indeed,

for all u, v ∈ W s,p (Rn ).

(2.13)

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Lt u − Lt v, u − v = Lt u − Lt v, u + g − (v + g) ˆ ˆ ˆ ˆ |δ(u + g)|p dμ + |δ(v + g)|p dμ = Rn Rn

Rn Rn

ˆ ˆ

−

|δ(u + g)|p−2 δ(u + g)δ(v + g)dμ Rn Rn

ˆ ˆ

−

|δ(v + g)|p−2 δ(v + g)δ(u + g)dμ Rn Rn

ˆ ˆ ≥

ˆ ˆ

|δ(u + g)|p dμ + Rn Rn

p−1 − p −

p−1 p

ˆ ˆ

|δ(v + g)|p dμ Rn Rn

1 |δ(u + g)| dμ − p

ˆ ˆ |δ(v + g)|p dμ

p

Rn Rn

ˆ ˆ

|δ(v + g)|p dμ − Rn Rn

1 p

Rn Rn

ˆ ˆ

|δ(u + g)|p dμ = 0, Rn Rn

where we used Young’s inequality. The existence of a unique weak solution now follows from Proposition 4.1, in [21] if, in addition to (2.13) and the monotonicity, we prove that [u]s,p ≥ α u W s,p (Rn ) , p

(2.14) p

Lt u, u ≥ α[u]W s,p − C[g]W s,p , for some α > 0.

(2.15)

The Sobolev inequality guarantees that (2.14) holds. Let us prove (2.15). By Young’s inequality with ε and (1.3), Lt u, u = Lt u, u + g − g ˆ ˆ ˆ ˆ p = |δ(u + g)| dμ − |δ(u + g)|p−2 δ(u + g)δgdμ Rn Rn

Rn Rn

p ≥ −1 [u + g]W s,p

Choosing ε =

1 , 22

p

p

− ε[u + g]W s,p − C(ε)[g]W s,p

we obtain 1 p p [u + g]W s,p − C[g]W s,p 2

1 1 p p ≥ p+1 [u]W s,p − + C [g]W s,p , 2  2

Lt u, u ≥

from which (2.15) follows. The initial data u0 is assumed in the sense that

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ˆ |u(x, t) − u0 |2 dx = 0.

lim

t→0 

The reason for choosing u0 ∈ L2 () is that u0 needs to be an element of a Hilbert space H such s,p that W0 () is dense and continuously embedded into H . This is indeed true because of the s,p Sobolev embedding theorem and the fact that Cc∞ () is dense in W0 (). 3. Estimates for subsolutions Definition 3.1. We say that u is a solution to ∂t u + Lu = 0 in  × (t1 , t2 ) if ˆt2 ˆ ˆ

ˆt2 ˆ Au(x, y, t)(η(x, t) − η(y, t))dxdydt −

t1 R n

Rn

u∂t ηdxdt = 0,

(3.1)

t1 R n

for all η ∈ Lp (t1 , t2 ; W0 ()) such that ∂t η ∈ Lp (t1 , t2 ; (W s,p ())∗ ) and η(x, t1 ) = η(x, t2 ), for all x ∈ . s,p

Definition 3.2. We say that u is a subsolution to ∂t u + Lu = 0 in  × (t1 , t2 ) if ˆt2 ˆ ˆ

ˆt2 ˆ Au(x, y, t)(η(x, t) − η(y, t))dxdydt −

t1 R n R n

u∂t ηdxdt ≤ 0,

(3.2)

t1 R n

for all η as in the definition of a solution that are also non negative. We first prove that if u is a subsolution, then its positive part, u+ = max{u, 0}, is again a subsolution. Lemma 3.1. If u is a subsolution, then u+ is also a subsolution. If  is a domain such that  ⊂  and u ≤ 0 in  \  × (t1 , t2 ), then u+ is a subsolution in  × (t1 , t2 ). Proof. Let φj (τ ) be a smooth, convex approximation of τ+ such that φj (τ ) = 0 if τ ≤ 0, φj (τ ), φj (τ ) > 0 if τ > 0 and |φj | ≤ C, |φj | ≤ C(j ). Such a function may be constructed, for instance, by mollifying the function (τ − 1/j )+ , as ˆ φj (τ ) =

(s − 1/j )+ ηεj (τ − s)ds. R

Here η is a radial mollifier and the parameter εj is chosen so that φj (τ ) = 0 for τ ≤ 0 and φj (τ ) > 0 if τ > 0. Note also that mollification preserves convexity. Let v be a non negative, bounded test function for the domain  × (t1 , t2 ). That is, v satisfies the assumptions imposed on η in Definition 3.2 but with  in place of . We claim that φj (u)v is an admissible test function for  × (t1 , t2 ). Let φj (x, t) = φj (u(x, t)) and let φj (x, t) = φj (u(x, t)). We note that φj (x, t) > 0 if and only if u(x, t) > 0 and φj (x, t) = 0 if and only if u(x, t) ≤ 0. Thus φj (u)v ≡ 0

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11

in  \  × (t1 , t2 ) and φj (u)v ≡ 0 in Rn \  × (t1 , t2 ) since v is a test function for  × (t1 , t2 ). The remaining properties required of test functions can easily be deduced from the identity φj (u(x, t))v(x, t) − φj (u(y, t))v(y, t) 1 = (φj (u(x, t)) − φj (u(y, t)))(v(x, t) + v(y, t)) 2 1 + (v(x, t) − v(y, t))(φj (u(x, t)) + φj (u(y, t))), 2 together with the boundedness of φj and v. Using φj (u)v as a test function in (3.2) we obtain ˆt2 ˆ

∂t uφj (u)vdxdt

ˆt2 ˆ ˆ +

t1 R n

Au(x, y, t)δ(φj (u)v)(x, y, t)dxdydt

t1 R n R n

= I1,j + I2,j ≤ 0. We may write I1,j as ˆt2 ˆ I1,j =

ˆt2 ˆ v∂t φj (u)dxdt → I1 =

t1 R n

v∂t u+ dxdt,

as j → ∞.

(3.3)

t1 R n

We next estimate the integrand of I2,j under the assumption that u(x, t) > u(y, t). If φj (x, t) = 0, then Au(x, y, t)δ(φj (u)v)(x, y, t) = 0 since φ is monotone non decreasing. If φj (y, t) > 0, then (u(x, t) − u(y, t))p−1 (φj (x, t)v(x, t) − φj (y, t)v(y, t)) = (u+ (x, t) − u+ (y, t))p−1 (φj (x, t)v(x, t) − φj (y, t)v(y, t)) ≥ (u+ (x, t) − u+ (y, t))p−1 φj (x, t)(v(x, t) − v(y, t)). If φj (y, t) = 0 and φj (x, t) > 0, then (u(x, t) − u(y, t))p−1 (φj (x, t)v(x, t) − φj (y, t)v(y, t)) = (u(x, t) − u(y, t))p−1 φj (x, t)v(x, t) ≥ (u+ (x, t) − u+ (y, t))p−1 φj (x, t)v(x, t) ≥ (u+ (x, t) − u+ (y, t))p−1 φj (x, t)(v(x, t) − v(y, t)). We have thus shown that if u(x, t) > u(y, t), Au(x, y, t)δ(φj (u)v)(x, y, t)

(3.4)

≥ K(x, y, t)(u+ (x, t) − u+ (y, t))p−1 φj (x, t)(v(x, t) − v(y, t)). By interchanging the roles of x and y, we obtain, for u(x, t) < u(y, t), the analogous estimate

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12

Au(x, y, t)δ(φj (u)v)(x, y, t)

(3.5)

≥ K(x, y, t)(u+ (y, t) − u+ (x, t))p−1 φj (y, t)(v(y, t) − v(x, t)) = K(x, y, t)|u+ (x, t) − u+ (y, t)|p−2 × (u+ (x, t) − u+ (y, t))φj (y, t)(v(x, t) − v(y, t)). Since the expressions in (3.4) and (3.5) are L1 (Rn × Rn × (t1 , t2 )), we obtain lim inf I2,j j →∞

ˆt2 ˆ ˆ ≥

K(x, y, t)(u+ (x, t) − u+ (y, t))p−1 (v(x, t) − v(y, t))dxdydt t1 R n R n

ˆt2 ˆ ˆ =

Au+ (x, y, t)δv(x, y, t)dxdydt. t1 R n R n

In combination with (3.3), this gives ˆt2 ˆ

ˆt2 ˆ ˆ v∂t u+ dxdt +

t1 R n

Au+ (x, y, t)δv(x, y, t)dxdydt ≤ 0, t1 R n R n

for all bounded, non negative test functions v, and by a standard approximation argument, all s,p non negative test functions v ∈ Lp (t1 , t2 ; W0 ( )) such that ∂t v ∈ Lp (t1 , t2 ; (W s,p ( ))∗ ) and v(x, t1 ) = v(x, t2 ), for all x ∈  . 2 3.1. Caccioppoli estimate Let ζh (s) be a standard mollifier with support in (−h, h). Given f : Rn × R → R, we define ˆ fh (x, t) =

f (x, s)ζh (t − s)ds.

(3.6)

R

Definition 3.3. Let  ⊂ Rn be a domain, u ∈ Lp (t1 , t2 ; W s,p ()), and consider t1 < t < t2 . Then t is called a Lebesgue instant for u if ˆ |uh (x, t) − u(x, t)|2 dx = 0.

lim

h→0 

´ Since  u(x, t)dx belongs to Lp (t1 , t2 ), it follows from Lebesgue’s differentiation theorem that a.e. t ∈ (t1 , t2 ) is a Lebesgue instant.

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13

Lemma 3.2. Let p ∈ (1, ∞), s ∈ (0, 1). Let ξ ≥ 1 and assume that K satisfies the ellipticity condition (1.3). Let x0 ∈ Rn , τ1 < τ2 , Br := Br (x0 ), and assume that u is a non-negative subsolution in Br × (τ1 , τ2 ). Let t1 , t2 be Lebesgue instants for u, with τ1 < t1 < t2 < τ2 . For d > 0, let v = u + d, w = v (p−1+ξ )/p . Then ˆt2 ˆ ˆ |wφ(x, t) − wφ(y, t)|p dμ + t1 Br Br

1 ξ +1

ˆ

 t2  v(x, t)1+ξ φ p (x, t)dx 

t=t1

Br

ˆt2 ˆ ˆ ≤C

max{w(x, t), w(y, t)}p |φ(x, t) − φ(y, t)|p d μ¯ t1 Br Br



ˆ

+C

−(n+ps)

|x − y|

sup

ˆt2 ˆ

x∈supp ψ Rn \Br

ˆt2 +C t1

⎛

⎜ ⎝ sup

t1 Br

ˆ

x∈supp ψ Rn \Br

1 + (1 + ξ )

w p (x, t)φ p (x, t) dxdt

dy



ˆt2 ˆ v 1+ξ t1 Br

ˆ p−1 u(y, t)+ dy |x − y|n+sp Br ∂φ p ∂t

⎞

⎟ v ξ φ p (x, t)dx ⎠ dt

dxdt, +

for all φ(x, t) = ψ(x)ζ (t) with ζ ∈ C0∞ (τ1 , τ2 ) and ψ ∈ C0∞ (Br ). Proof. Let v = u + d,

vm = min{v, m},

m ≥ d, 1−q

and let φ be as in the statement of the theorem. Let q = 1 − ξ ≤ 0. Then η = vm φ p is an admissible test function. This is clear if q = 0. If q < 0, it is enough to note that, according to the mean value theorem, |vm (x, t) − vm (y, t)| = (1 − q)α −q |vm (x, t) − vm (y, t)|, 1−q

1−q

for some vm (y, t) < α < vm (x, t). For τ1 < t1 < t2 < τ2 , let θj (t) ∈ Cc∞ (τ1 , τ2 ) be a smooth approximation of χ(t1 ,t2 ) as j → ∞. We will test the equation (2.1) with the function

1−q ηj,h = (vm )h φ p θj , h

(3.7)

where the subscript h on the right hand side denotes mollification in the sense of (3.6). Hence we obtain

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14

ˆτ2 ˆ ˆ 0≥

Av(x, y, t)(ηj,h (x, t) − ηj,h (y, t))dxdydt

(3.8)

τ1 Br Br

ˆτ2 ˆ

ˆ Av(x, y, t)ηj,h (x, t)dxdydt

+2 τ1 Rn \Br Br

ˆτ2 ˆ −

v

∂ηj,h j,h j,h j,h dxdt = I1 + I2 + I3 . ∂t

τ1 Br j,h

For I3

we have

j,h I3

ˆτ2 ˆ =−

1−q p

(vm )h ∂t ((vm )h

(3.9)

φ θj )dxdt

τ1 Br

ˆτ2 ˆ =

1−q p

∂t (vm )h (vm )h

φ θj dxdt

τ1 Br

ˆt2 ˆ →

1−q p

∂t (vm )h (vm )h

φ dxdt = I3h ,

t1 Br

as j → ∞. Then integration by parts yields ˆt2 ˆ I3h

=

2−q

∂t t1 Br

1 = 1+ξ

→

(vm )h φ p dxdt 2−q

ˆ

(3.10)  t2

 1+ξ (vm )h (x, t)φ p (x, t)dx 

Br

1 1+ξ

ˆ

t2  1+ξ vm (x, t)φ p (x, t)dx 

t=t1

Br

j,h

1 − 1+ξ t=t1 −

1 1+ξ

ˆt2 ˆ

1+ξ

(vm )h

∂t φ p dxdt

t1 Br

ˆt2 ˆ 1+ξ vm ∂t φ p dxdt, t1 Br

j,h

as h → 0. Since I1 and I2 are finite, our taking j → ∞ in these terms simply replaces τi by ti . By standard properties of mollifiers, we may then pass to the limit h → 0 in (3.8) and obtain ˆt2 ˆ ˆ 0≥

Av(x, y, t)(η(x, t) − η(y, t))dxdydt t1 Br Br

(3.11)

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ˆt2 ˆ

15

ˆ

+2

Av(x, y, t)η(x, t)dxdydt t1 Rn \Br Br

1 + 1+ξ

ˆ

 t2 

1+ξ vm (x, t)φ p (x, t)dx 

Br

1 − 1+ξ t=t1

ˆt2 ˆ 1+ξ vm ∂t φ p dxdt t1 Br

= I 1 + I 2 + I3 . We start by estimating the integrand of I1 under the assumption that v(x, t) > v(y, t). For such x, y, t we may apply the truncation result (2.2), or rather its short proof, to Av(x, y, t), to find Av(x, y, t)(η(x, t) − η(y, t)) ≥ Avm (x, y, t)(η(x, t) − η(y, t))

(3.12)

= Avm (x, y, t)(vm φ p (x, t) − vm φ p (y, t)). In order to simplify notation, we will write v rather than vm in the estimation of I1 . We will make use of the inequality φ p (y, t) ≤ φ p (x, t) + cp εφ p (x, t) + (1 + cp ε)ε 1−p |φ(x, t) − φ(y, t)|p ,

(3.13)

valid for any ε ∈ (0, 1), see Lemma 3.1 in [9]. We let δ ∈ (0, 1) be a parameter to be chosen and set ε=δ

v(x, t) − v(y, t) . v(x, t)

Thus we obtain,

φ p (x, t) φ p (y, t) − v q−1 (x, t) v q−1 (y, t)

p v(x, t) − v(y, t) φ p (x, t) φ (x, t) − 1 + cp δ ≥ Av(x, y, t) q−1 v (x, t) v q−1 (y, t) v(x, t)



v(x, t) − v(y, t) 1−p (v(x, t) − v(y, t))1−p Av(x, y, t) 1 + cp δ δ − q−1 v (y, t) v(x, t) v 1−p (x, t)

Av(x, y, t)

(3.14)

× |φ(x, t) − φ(y, t)|p = D + E. We first estimate D and note that D = Av(x, y, t) = Av(x, y, t) ×

φ p (x, t) v q−1 (y, t)



v(x, t) − v(y, t) v q−1 (y, t) − 1 − cp δ q−1 v (y, t) v(x, t)



φ p (x, t) (v(x, t) − v(y, t)) v q (y, t)

v(y, t) v(y, t) v q (y, t) − − cp δ q−1 v(x, t) v (x, t)(v(x, t) − v(y, t)) v(x, t) − v(y, t)

(3.15)



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16

⎞ ⎛ q−1 v (y,t) q−1 (x,t) − 1 φ p (x, t) v(y, t) v ⎠. = K(x, y, t) q (v(x, t) − v(y, t))p ⎝ v(x,t) − cp δ v (y, t) v(x, t) −1 v(y,t)

For a > 1, let g(a) =

a 1−q − 1 , a−1

so that for v(x, t) > v(y, t),

v(x, t) g v(y, t)

=

v q−1 (y,t) −1 v q−1 (x,t) . v(x,t) v(y,t) − 1

Since ξ ≥ 1 we have q < 0 and hence g(a) ≥ 1. If a ≥ 2, then g(a) =

a 1−q − 1 a 1−q − a 1−q /2 1 a 1−q ≥ = . a−1 a−1 2a−1

(3.16)

Thus, for v(x, t) > 2v(y, t), we may combine (3.15) and (3.16) to obtain ⎞ ⎛ v q−1 (y,t) q−1 (x,t) φ p (x, t) v(y, t) 1 v ⎠ D ≥ K(x, y, t) q (v(x, t) − v(y, t))p ⎝ v(x,t) (3.17) − cp δ v (y, t) 2 v(x, t) − 1 v(y,t) ⎞ ⎛ v q (y,t) q−1 1 v(y, t) φ p (x, t) v (x,t) ⎠ (v(x, t) − v(y, t))p ⎝ − cp δ = K(x, y, t) q v (y, t) 2 v(x, t) − v(y, t) v(x, t) = K(x, y, t)

(v(x, t) − v(y, t))p−1 p φ (x, t) v q−1 (x, t)

1 v(y, t) v(x, t) − v(y, t) q−1 × − cp δ v (x, t) . 2 v(x, t) v q (y, t) Recalling that v(x, t) ≥ 2v(y, t), q < 0 and v(y, t) > 0 since y ∈ Br , we see that −

v(y, t) v(x, t) − v(y, t) q−1 1 v (x, t) ≥ − . q v(x, t) v (y, t) 2

Thus D ≥ K(x, y, t)



(v(x, t) − v(y, t))p−1 p 1 1 (x, t) δ . φ − c p v q−1 (x, t) 2 2

(3.18)

At this point we observe that p−q p−q (v(x, t) − v(y, t))p−1 ≥ 21−p v p−q (x, t) ≥ 21−p (v p (x, t) − v p (y, t))p . v q−1 (x, t)

(3.19)

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17

Choosing δ=

1 , 4cp

(3.20)

we arrive at D ≥ K(x, y, t)2−p−1 (v

p−q p

(x, t) − v

p−q p

(y, t))p φ p (x, t)

(3.21)

= K(x, y, t)2−p−1 (w(x, t) − w(y, t))p φ p (x, t). We now consider the remaining case v(y, t) < v(x, t) < 2v(y, t). By (3.15), the fact that g(a) ≥ 1 and the choice of δ, we have 1 φ p (x, t) D ≥ K(x, y, t) q (v(x, t) − v(y, t))p . 2 v (x, t)

(3.22)

We further estimate (w(x, t) − w(y, t))p = ≤

p p−q p p−q

p

p

⎛ ⎜ ⎝

v(x,t) ˆ

⎞p

⎟ τ −q/p dτ ⎠

(3.23)

v(y,t)

((v(x, t) − v(y, t))p . v q (y, t)

Combining (3.21), (3.22) and (3.23), we have shown that D ≥ 2−1−p K(x, y, t)(w(x, t) − w(y, t))p φ p (x, t).

(3.24)

For the estimate of E we use the facts that −v 1−q (y, t) ≥ −v 1−q (x, t) and (v(x, t) − v(y, t))/v(x, t) ≤ 1, to find that E ≥ −CK(x, y, t)w p (x, t)|φ(x, t) − φ(y, t)|p .

(3.25)

Finally, combining (3.24) and (3.25), we have shown that for v(x, t) > v(y, t), D + E ≥ CK(x, y, t)((w(x, t) − w(y, t))p φ p (x, t) − w p (x, t)|φ(x, t) − φ(y, t)|p ).

(3.26)

If v(y, t) > v(x, t), the same estimate may be deduced by interchanging the roles of x and y. If v(x, t) = v(y, t) it is sufficient to note that 0 ≥ E. Using the fact that |w(x, t)φ(x, t) − w(y, t)φ(y, t)| − c max{w p (x, t), w p (y, t)}| − φ(x, t)φ(y, t)|p ≤ c|w(x, t) − w(y, t)|p φ p (x, t),

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18

and recalling that we are actually dealing with vm rather than v, we have shown that ˆt2 ˆ

ˆ

|wm (x, t)φ(x, t) − wm (y, t)φ(y, t)| dxdydt |x − y|n+sp

I1 ≥ c t1 Rn \Br Br

ˆt2 ˆ

ˆ

−c

p

p

max{wm (x, t), wm (y, t)} t1 Rn \Br Br

(3.27)

|φ(x, t) − φ(y, t)|p dxdydt, |x − y|n+sp

p−1+ξ

where we have set wm = vm . We now turn to I2 , and first observe that ˆt2 ˆ

ˆ

I2 ≥ 2

Avm (x, y, t)χ{v(y,t)>v(x,t)} η(x, t)dxdydt.

(3.28)

t1 Rn \Br Br

We will need the following inequality to estimate I2 : If 0 ≤ a < b, then |a|p−2 a − |b|p−2 b ≤ |a − b|p−2 (a − b).

(3.29)

To prove (3.29), we make use of the fact that the l s -norm of an element (α, β) of R2 is nonincreasing in s: |(α, β)|s ≤ |(α, β)|1 ,

s ≥ 1.

α s + β s ≤ (α + β)s ,

s ≥ 1.

If α, β > 0, this means that (3.30)

Now (3.29) follows by taking α = a, β = b − a and s = p − 1 in (3.30). Using (3.29) in (3.28) gives ˆt2 ˆ

ˆ

I2 ≥ c t1 Rn \Br Br

ˆt2 ˆ

|v(x, t)|p−2 v(x, t) χ{v(y,t)>v(x,t)} η(x, t)dxdydt |x − y|n+sp ˆ

−c t1 Rn \Br Br

ˆt2 ˆ

ˆ

≥ −c t1 Rn \Br Br

|v(y, t)|p−2 v(y, t) χ{v(y,t)>v(x,t)} η(x, t)dxdydt |x − y|n+sp p−1

v(y, t)+ η(x, t)dxdydt |x − y|n+sp

(3.31)

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ˆt2 ˆ ≥ −c2

ˆ

p−1 t1 Rn \Br Br

ˆt2 ˆ − c2

p−1

u(y, t)+ η(x, t)dxdydt |x − y|n+sp

ˆ

d p−1 η(x, t)dxdydt |x − y|n+sp

p−1 t1 Rn \Br Br

ˆt2

ˆ

≥ −c

p−1

sup x∈supp ψ Rn \Br

t1

ˆ −c

sup x∈supp ψ Rn \Br

19

u(y, t)+ dy |x − y|n+sp

1 dy |x − y|n+sp

ˆ ξ p vm φ (x, t)dxdt Br

ˆt2 ˆ

p−1+ξ p

vm

φ (x, t)dxdt

t1 Br

where we used the fact that d ≤ vm in Br × (t1 , t2 ). Recalling (3.11) and collecting the estimates (4.10) and (3.31) for I1 and I2 respectively, we arrive at ˆt2 ˆ

ˆ

|wm (x, t)φ(x, t) − wm (y, t)φ(y, t)| dxdydt |x − y|n+sp

C t1 Rn \Br

+

Br

1 1+ξ

ˆ

 t2  1+ξ vm (x, t)φ p (x, t)dx 

t=t1

Br

ˆt2 ˆ

ˆ

≤C

p

p

max{wm (x, t), wm (y, t)} t1 Rn \Br Br

ˆ +C

sup x∈supp ψ Rn \Br

ˆt2

1 dy |x − y|n+sp

ˆ

+C

sup t1

x∈supp ψ Rn \Br

1 + 1+ξ

(3.32)

p−1

ˆt2 ˆ

|φ(x, t) − φ(y, t)|p dxdydt |x − y|n+sp p−1+ξ p

vm

φ (x, t)dxdt

t1 Br

u(y, t)+ dy |x − y|n+sp

ˆ ξ p vm φ (x, t)dxdt Br

ˆt2 ˆ 1+ξ vm ∂t φ p dxdt. t1 Br

Passing to the limit m → ∞, we obtain the conclusion of the lemma. 2 Lemma 3.3. Let p ∈ (1, ∞), s ∈ (0, 1). Let ξ ≥ 1 and assume that K satisfies the ellipticity condition (1.3). Let x0 ∈ Rn , τ1 < τ2 , Br := Br (x0 ), and assume that u is a non-negative sub-solution in Br × (τ1 , τ2 ). For d > 0, let v = u + d, w = v (p−1+ξ )/p . Then

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20

ˆτ2 ˆ ˆ

1 |wφ(x, t) − wφ(y, t)| dμ + sup ξ + 1 τ1
ˆ v(x, t)1+ξ φ p (x, t)dx

p

τ1 Br Br

Br

ˆτ2 ˆ ˆ ≤C

max{w(x, t), w(y, t)}p |φ(x, t) − φ(y, t)|p d μ¯ τ1 Br Br



ˆ

+C

|x − y|

sup

−(n+ps)

ˆτ2 ˆ

x∈supp ψ Rn \Br

ˆτ2 +C τ1

⎛

τ1 Br

ˆ

⎜ ⎝ sup

ˆ p−1 u(y, t)+ dy |x − y|n+sp Br

x∈supp ψ Rn \Br

1 + (1 + ξ )

w p (x, t)φ p (x, t) dxdt

dy



ˆτ2 ˆ v 1+ξ

∂φ p ∂t

⎞

⎟ v ξ φ p (x, t)dx ⎠ dt

dxdt, +

τ1 Br

for all φ(x, t) = ψ(x)ζ (t) with ζ ∈ C0∞ (τ1 , τ2 ) and ψ ∈ C0∞ (Br ). Proof. We proceed as in the proof of Lemma 3.2 but leave out θj from the test function, i.e. we use the test function

1−q ηh = (vm )h φ p . h

This leads to the desired estimate, save for the term 1 sup ξ + 1 τ1
ˆ v(x, t)1+ξ φ p (x, t)dx Br

on the left hand side. For any given ε > 0, we may choose t2 = t2 (ε) ∈ (τ1 , τ2 ) in Lemma 3.2 so that 1 sup ξ + 1 τ1
1 ξ +1

ˆ

ˆ v(x, t)1+ξ φ p (x, t)dx Br

v(x, t2 )1+ξ φ p (x, t)dx + ε. Br

Then, choosing t1 ∈ (τ1 , t2 ) outside the support of ζ and letting ε → 0, we obtain the conclusion of the lemma. 2

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21

4. Local boundedness of subsolutions Based upon the parabolic Sobolev inequality (2.2) and Lemma 3.3 (ii), we are able to prove a reverse Hölder inequality for subsolutions and do a Moser iteration to prove local boundedness. The following standard lemma, see e.g. [14] Lemma 4.3, is used in the proof. Lemma 4.1. Suppose f (s) is nonnegative and bounded in [0, 1]. If for all 0 ≤ α < β ≤ 1, 1 A f (α) ≤ f (β) + + B, 2 (α − β)γ then f (α) ≤ c(γ )

A + B . (α − β)γ

Lemma 4.2. Let 0 < r < R and let Q = Br × (t0 − T0 , t0 ). Suppose that u is a nonnegative subsolution in 2Q. Let v(x, t) = u(x, t) + d, where d = Tail∞ (u+ ; x0 , σ r, t0 − T0 , t0 ) +

r sp T0



1 p−2

.

Then for any σ ∈ (0, 1) and δ > δ0 > 0, ⎛

⎞1 δ

C ⎜ T0 sup v ≤ ⎝ sp sp r (1 − σ )α B(x0 ,σ r)×(t0 −σ T0 ,t0 ) where α =

(n+sp)2 sp

v

p−2+δ

⎟ dxdt ⎠ ,

(4.1)

Q

and C = C(n, p, δ0 ).

Proof. Let σ ∈ (0, 1) and let θ = 1 − σ ∈ (0, 1). We set r0 = r,

rj = r − θ r(1 − 2−j ),

δj = 2−j θ r,

j = 1, 2, . . .

and Uj = Bj × j = B(x0 , rj ) × (t0 − (rj /r)sp T0 , t0 ), U (λ) = B(λ) × (λ) = B(x0 , λr) × (t0 − λsp T0 , t0 ),

λ > 0.

We choose test functions ψj ∈ C ∞ (Bj ) and ζj ∈ C ∞ (j ) satisfying ψj ≡ 1 in Bj +1 , dist(supp ψj , Rn \ Bj ) ≥ such that for φj = ψj ζj we have

δj , 2

(4.2)

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22

0 ≤ φj ≤ 1,

φj = 1 in Uj +1 ,

φj = 0 on ∂p Uj ,

and    ∂φj  r sp C r sp −sp spj  ≤ 2 = Cδj .  ∂t  T (θ r)sp T0 0

C |∇φj | ≤ 2j = Cδj−1 , θr

(4.3)

Let w=v

p−1+ξ

p−1+ξ p

and ηj = φj ξ +1 .

Note that ηj satisfies the same bounds (4.3) as φj with C = C(n) p−1+ξ ξ +1 . By the Sobolev embedding theorem there holds, ˆ |wηj |κp dxdt j Bj

⎛

⎜ sp−n ≤ C ⎝rj

(4.4)

ˆ ˆ ˆ j Bj Bj

|wηj (x, t) − wηj |x − y|n+sp

(y, t)|p

⎛

⎞1

⎜ × ⎝sup j

where G =

κ∗ κ ∗ −1 .

I1 +

⎟ |wηj |p ⎠

dxdydt + j Bj

⎛

G

⎞

ˆ

⎞

ˆ

⎟ ⎜ sp−n |wηj |pG(κ−1) ⎠ = C ⎝rj I1 +

⎟ |wηj |p ⎠ ×



j Bj

Bj

I2 |Bj |

1

G

,

By Lemma 3.3,

1 sup 1 + ξ t∈j ˆ ˆ ˆ

ˆ v 1+ξ ηj (x, t)dx

(4.5)

Bj

≤C

max{w(x, t), w(y, t)}p |ηj (x, t) − ηj (y, t)|p d μ¯ j Bj Bj



+C

ˆ

|x − y|−(n+ps) dy

sup x∈supp ψj

Rn \Bj p−1

sup x∈supp ψ

t1 Rn \Br



ˆ ˆ +C

v 1+ξ ηj j Bj

For I11 we have the estimate

p

w p (x, t)ηj (x, t) dxdt j Bj

ˆt2 ˆ +C

ˆ ˆ

u(y, t)+ dy |x − y|n+sp p ∂ηj

1 1 + ξ ∂t

+

ˆ

p

v ξ ηj (x, t)dxdt Br

dxdt = I11 + I12 + I13 + I14 .

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ˆ ˆ I11 ≤ C

ˆ

|ηj (x, t) − ηj (y, t)|p dy. |x − y|n+sp

w p (x, t)dxdt sup x∈Bj

j Bj

23

Bj

Using (4.3), we see that for any x ∈ Bj , ˆ Bj

|ηj (x, t) − ηj (y, t)|p dy |x − y|n+sp ˆ ≤ Bj ∩{|x−y|≤δj } −p ≤ Cδj

ˆ

|ηj (x, t) − ηj (y, t)|p dy + |x − y|n+sp

Bj ∩{|x−y|>δj }

ˆ

ˆ

− y|p

Bj ∩{|x−y|≤δj }

|x dy + |x − y|n+sp

Bj ∩{|x−y|>δj }

|ηj (x, t) − ηj (y, t)|p dy |x − y|n+sp

2p dy |x − y|n+sp

−sp ≤ Cδj .

Since −sp δj

sp −sp rj = rj sp δj



−sp ≤ rj 2j sp

r (1 − σ )r

sp

−sp 2

= rj

j sp

θ sp

,

we get −sp 2

I11 ≤ Crj

ˆ

j sp

w p (x, t)dxdt.

θ sp

(4.6)

Uj −sp

The first factor of I12 can be estimated by C(δj This gives us −sp I12 ≤ C(δj

−sp + rj )

ˆ p

w (x, t)dxdt

−sp

+ rj

), using (4.22) and polar coordinates.

j sp −sp 2 ≤ Crj sp

Uj

ˆ w p (x, t)dxdt.

θ

Uj

We now turn to I13 and first note that if y ∈ Rn \ Bj and x ∈ sup ψj , then 1 1 |x0 − y| 1 |x − x0 | + |x − y| = ≤ |x − y| |x0 − y| |x − y| |x0 − y| |x − y| ≤ It follows that

1 + 2rj /δj Cθ −1 2j ≤ . |x0 − y| |x0 − y|

(4.7)

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24

I13 ≤

C

θ

−sp p−1 2j (n+sp) rj Tail∞ (x0 ; rj , j ) n+sp

ˆ v ξ dxdt

Uj

C −sp p−1 ≤ sp n+sp 2j (n+sp) rj Tail∞ (x0 ; σ r, 0 ) σ θ

(4.8)

ˆ v ξ dxdt

Uj p−1

≤

≤

C −sp Tail∞ (x0 ; σ r, 0 ) 2j (n+sp) rj sp n+sp σ θ d p−1 C

2j (n+sp) rj

v ξ +p−1 dxdt

Uj

ˆ

−sp

σ sp θ n+sp

ˆ

w p dxdt. Uj

Finally, by applying (4.1) and (4.3), we obtain the following estimate for I14 : ˆ ˆ I14 ≤ C

v 1+ξ j Bj

ˆ ˆ

(4.9)

w p T0 r sp −sp −sp δ dxdt ≤ Cδj r sp T0 j

≤C j Bj −sp 2

≤ Crj

r sp −sp δ T0 j

j sp

ˆ ˆ w p dxdt j Bj

ˆ ˆ w p dxdt.

θ sp j Bj

Putting together (4.5) with (4.6)–(4.9) yields 1 I1 + sup 1 + ξ t∈j

ˆ v 1+ξ ηj (x, t)dx

(4.10)

Bj

2j (n+sp) −sp ≤ C sp n+sp rj σ θ

ˆ ˆ p

w dxdt

−sp = rj cj,θ

j Bj

ˆ w p dxdt, Uj

j (n+sp)

2 where cj,θ = C (1−θ) sp θ n+sp . Note that

ˆ |ηj w| dxdt p

sp−n −sp ≤ rj rj cj,θ

j Br

Hence the term sp−n

rj

´

ffl j

Br

ˆ w p dxdt. Uj

|ηj w|p dxdt appearing in (4.4) may be absorbed in the estimate of

I1 . Let κ =1+

1+ξ . G(p − 1 + ξ )

(4.11)

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25

Note that κ ∈ (1, κ ∗ ). Then ˆ

p

I2 = sup j

In view of (4.10),

I2 1+ξ

v 1+ξ φj dx.

(4.12)

Bj

enjoys the same estimate as I1 with φj in place of ηj , i.e. −sp

ˆ

I2 ≤ (1 + ξ )cj,θ rj

w p dxdt.

(4.13)

Uj

At this point we have shown, recalling (4.4), that |wφj |κp dxdt

(4.14)

Uj

⎞⎛

⎛

⎟ ⎜ −sp w dxdt ⎠ ⎝rj cj,θ (1 + ξ )

sp ⎜ −sp ≤ Crj ⎝rj cj,θ

⎞1

ˆ

G

⎟ w dxdt ⎠

p

j Bj

j Bj

⎛ 1 1 sp ≤ Crj |j | G (1 + ξ ) G

p

⎞1+ 1

G

⎜ −sp ⎝rj cj,θ

⎟ w p dxdt ⎠

.

Uj

Let γ = 1 + 1/G. Then ⎞γ

⎛ |w|κp dxdt ≤ C Uj +1

1 1 ⎜ −sp |Uj | sp rj |j | G (1 + ξ ) G ⎝rj cj,σ |Uj +1 |

n+sp

=C

rj

1 spγ G n+sp rj (1 + ξ ) rj +1



T0 r sp

1

G

⎛ ⎜ −sp ⎝rj cj,σ

⎟ w p dxdt ⎠ Uj

(4.15)

⎞γ

⎟ w p dxdt ⎠ Uj

⎛

⎞γ



γ −1 γ γ −1 ⎜ T0 = C ⎝ sp (1 + ξ ) γ cj,σ r

⎟ w p dxdt ⎠ . Uj

Recalling the definitions of w, κ and G we may rewrite (4.15) as sp

|v|p−1+ n +γ ξ dxdt Uj +1

(4.16)

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26

⎛



⎜ T0 ≤ C ⎝ sp r

⎞γ

γ −1 γ

(1 + ξ )

γ −1 γ

⎟ |v|p−1+ξ dxdt ⎠ .

cj,σ Uj

We are now in a position to start a Moser iteration. Fix ξ0 > 1 and set ξj = γ j (ξ0 + 1) − 1, p j = p − 1 + ξj ,

j ≥ 0,

j ≥ 0.

Then we have the inductive relations ξj +1 = γ (ξj + 1) − 1, sp + γ ξj = p − 1 + ξj +1 = pj +1 . p−1+ n Hence, using ξ = ξj in (4.16) and estimating (1 + ξj ) ⎛



⎜ T0 |v|pj +1 dxdt ≤ C ⎝ sp r

γ −1 γ

(4.17) (4.18)

γ −1

≤ 2ξ0 γ γ j , we find that ⎞γ

γ −1

γ −1 γ

γ

ξ0

⎟ |v|pj dxdt ⎠ ,

γ j cj,σ

Uj +1

(4.19)

Uj

for j = 0, 1, . . .. By iterating (4.19) m times, starting at pm and taking γ m :th roots, we conclude the estimate ⎛

⎞

⎜ ⎝

⎟ |v|pm dxdt ⎠

1 γm

(4.20)

Um

≤ U (r)

|v|

p0

m−1 

−1−j γ −j Cγ cj,θ

j =0



T0 r sp

γ −1 γ −j γ

γ −1

ξ0 γ

γ −j

(γ j )γ

−j

.

The limit as m → ∞ of the product on the right hand side of (4.20) may be estimated in a standard fashion by studying its logarithm. Thus we obtain

lim

m−1 

m→∞

Since limm→∞

−1−j γ −j Cγ cj,θ

j =0

pm γm



T0 r sp

γ −1 γ −j γ

γ −1

ξ0 γ

γ −j

(γ j )γ

−j

≤ Cθ

− (n+sp) sp

= ξ0 + 1, taking m → ∞ in (4.19) results in − (n+sp) sp

sup v ξ0 +1 ≤ Cξ0 (1 − σ ) U (σ )

2

T0 r sp

v p−1+ξ0 dxdt. U (1)

2

T0 ξ0 . r sp

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27

Additionally, − (n+sp) sp

sup v ξ0 +1 ≤ sup v 2 Cξ0 (1 − σ ) U (σ )

U (1)

2

T0 r sp

v p−3+ξ0 dxdt U (1)

⎛

⎞ ξ0 +1 2 − (n+sp) sp

1 ⎜ ≤ sup v ξ0 +1 + Cξ0 ⎝(1 − σ ) 2 U (1)

where we used Young’s inequality with exponents on ξ0 − 1. By Lemma 4.1 we have

T0 r sp

v

p−3+ξ0

⎟ dxdt ⎠

ξ0 −1

,

U (1) ξ0 +1 2

and

ξ0 +1 ξ0 −1 . This means that

⎛

C will depend

⎞ ξ0 +1 2 − (n+sp) sp

⎜ sup v ξ0 +1 ≤ C ⎝(1 − σ )

U (σ r)

⎟ v p−3+ξ0 dxdt ⎠

T0 r sp

ξ0 −1

.

U (r)

Whence the result follows by choosing ξ0 = 1 + δ, with C depending only on δ0 if δ ≥ δ0 .

2

In the next lemma we extract information on u from Lemma 4.2. Lemma 4.3. Let u and Q be as in Lemma 4.2. Then for any σ ∈ (0, 1), C sup u ≤ (1 − σ )α σQ +

where α =



r sp T0

C T0 α (1 − σ ) r sp



1 p−2

 T0 p−1 + sp Tail∞ (u+ ; x0 , σ r, t0 − T0 , t0 ) r

up−1 (x, t)dxdt, Q

(n+sp)2 sp .

Proof. Choosing δ = 1 in Lemma 4.2, we get T0 C sup u ≤ sup v ≤ α r sp (1 − σ ) σQ σQ ⎛ +



r sp T0

p−1 p−2

T0 ⎜ C p−1 ⎝Tail∞ (u+ ; x0 , σ r, t0 − T0 , t0 ) + α sp (1 − σ ) r

(4.21) ⎞ ⎟ up−1 (x, t)dxdt ⎠ , Q

from which the claim easily follows. 2 Proof of Theorem 1.1. For any ε ∈ (0, 1), let r = εR and let T1 = ε sp T0 , so that εQ = Br × (t0 − T1 , t0 ).

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28

Let ψ ∈ C ∞ (Br ) and ζ ∈ C ∞ (t0 − T0 , t0 ) satisfy ψ ≡ 1 in Br ,

dist(supp ψ, Rn \ BR ) ≥

(R − r) εR = =: δ/2, 2 2

(4.22)

and be such that for φ = ψζ , we have 0 ≤ φ ≤ 1,

φ = 1 in εQ,

φ = 0 on ∂p Q,

and    ∂φ  R sp C R sp −sp  ≤ = Cδ .  ∂t  T0 (εr)sp T0

C |∇φ| ≤ = Cδ −1 , εr

(4.23)

Let v=u+



R sp T0

1 p−2

+ Tail∞ (u+ ; x0 , εr, t0 − T0 , t0 ).

By the parabolic Sobolev embedding Theorem 2.2, with f = vφ and κ = 1 + s/n, we have ˆt1 ˆ

ˆ

sp

|v|p+ n dxdt = t0 −T1 Br

sp

|v|p+ n dxdt

(4.24)

εQ

ˆt0

ˆ ˆ

≤ CR sp t0 −T0 BR BR

|vφ(x, t) − vφ(y, t)|p dxdydt |x − y|n+sp

⎛ ⎜ ×⎝

⎞ sp n

sup

t0 −T0
⎟ |v(x, t)|dx ⎠

.

The term ˆt0 ˆ ˆ R

sp t0 −T0 BR BR

|vφ(x, t) − vφ(y, t)|p dxdydt |x − y|n+sp

may be estimated precisely the way we treated the term I1 in Lemma 4.2, using the Caccioppoli inequality with ξ = 1. We thus end up with ˆ sp |v|p+ n dxdt (4.25) εQ

C ≤ (1 − ε)n+sp

ˆ Q

⎛ ⎜ |v|p dxdt ⎝

⎞ sp n

sup

t0 −T0
⎟ |v(x, t)|dx ⎠

.

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29

Using Hölder’s inequality followed by Young’s inequality, we get ˆ

⎛ ⎜ |v|p dxdt ≤ ⎝

εQ

⎞

ˆ

n n+s

sp ⎟ |v|p+ n dxdt ⎠

(4.26)

εQ

⎛ ⎜ ≤⎜ ⎝

≤

1 2

C (1 − ε)n+sp

⎛

ˆ

⎜ |v|p dxdt ⎝

Q

sup

t0 −T0
n ⎞ sp ⎞ n+s n ⎟ ⎟ |v(x, t)|dx ⎠ ⎟ ⎠

⎞p

⎛

ˆ |v|p dxdt + Q

C (1 − ε)

(n+sp)n s

⎜ ⎝

sup

t0 −T0
⎟ |v(x, t)|dx ⎠ .

An application of Lemma 4.1 gives ⎞p

⎛

ˆ |v|p dxdt ≤ εQ

C (1 − ε)

(n+sp)n s

⎜ ⎝

sup

t0 −T0
⎟ |v(x, t)|dx ⎠ .

(4.27)

Now, Lemma 4.2 with δ = 1 and σ = ε, in conjunction with Hölder’s inequality and (4.27), gives, ⎛ T1 sup v ≤ sp r 2 ε Q

C (1 − ε)

|v|

p−1

(n+sp)2 sp

C

T0 ≤ sp R

(1 − ε)

εQ

(n+sp)2 sp

⎜ ⎝

⎞ p−1 p

p⎟

|v| ⎠

εQ

⎞p−1

⎛ C

≤

(1 − ε)

C (n+sp)2 sp

(1 − ε)

(n+sp)n s

T0 ⎜ ⎝ sup R sp t0 −T0
⎟ |v(x, t)|dx ⎠

BR

Letting σ = ε 2 , as well as estimating √ 1 2 1+ σ ≤ , √ = 1−σ 1−σ 1− σ we obtain ⎞p−1

⎛ sup v ≤ σQ

(4.28)

T0 ⎜ C ⎝ sup α (1 − σ ) R sp t0 −T0
⎟ |v(x, t)|dx ⎠

,

BR

with α = (n + sp)(n + sp + sn)/sp. We complete the proof by substituting

.

(4.29)

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30



R sp v=u+ T0



1 p−2

+ Tail∞ (u+ ; x0 , σ R, t0 − T0 , t0 ).

2

Using Theorem 1.1 and Lemma 3.1 we are able to prove local boundedness of solutions to ∂t u + Lu = 0 without any assumption on the sign of u. Theorem 4.1. Let u be a solution to ∂t u + Lu = 0 in 2Q where Q = Br (x0 ) × (t0 − T0 , t0 ). Then C sup |u| ≤ (1 − σ )α σQ +





1 T0 R sp p−2 p−1 + sp Tail∞ (u; x0 , σ r, t0 − T0 , t0 ) T0 R ⎛ ⎞p−1

T0 ⎜ C ⎝ sup (1 − σ )α R sp t0 −T0
⎟ |u(x, t)|dx ⎠

(4.30)

,

BR

for any σ ∈ (0, 1). Proof. It is obvious that −u is a solution whenever u is. Thus by Lemma 3.1, both u+ and u− = (−u)+ are non negative subsolutions that Theorem 1.1 is applicable to. The result follows since |u| = u+ + u− . 2 4.1. Estimation of Tail∞ (u; x0 , r, t0 − T0 , t0 ) We end with a few remarks on the quantity Tail∞ (u; x0 , r, t0 − T0 , t0 ). If u solves ∂t u + Lu = 0 in  × (t0 − T0 , t0 ) and Br (x0 ) ⊂ , then Tail(u; x0 , r, t0 − T0 , t0 ) is bounded since u ∈ Lp (t0 − T0 , t0 ; W s,p (Rn )). On the other hand, if u is a solution in the sense of Definition 3.1, assuming (t0 − T0 , t0 ) ⊂ (t1 , t2 ), then Tail∞ (u; x0 , r, t0 − T0 , t0 ) is bounded if and only if ˆ sup

t0 −T0
and

|u|p−1 (x, t) dx < ∞ 1 + |x − x0 |n+sp

(4.31)

ˆ |u(x, t)|q dx < ∞,

sup

t0 −T0
(4.32)



for some q ≥ p − 1. While (4.31) is an assumption on the data, we can only verify (4.32) in a few cases and we´will briefly indicate how. If Br (z) is a ball such that B2r (z) ⊂ , we can prove supt0 −T0
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d=

r sp T0



1 p−2

+

31

MT0 Tailp−1 (u; z, r, t1 , t2 ), r sp

for t1 < t0 − T0 < t0 < t2 . This allows us to estimate the third term on the right hand side in Lemma 3.3 as ˆt2 C t1

≤

⎛ ⎜ ⎝ sup

ˆ

p−1

x∈supp ψ Rn \Br

C sup d M t1
u(y, t)+ dy |x − y|n+sp

ˆ

ˆ

⎟ v p−1 φ p (x, t)dx ⎠ dt

Br

v p−1 φ p (x, t)dx ≤ Br

⎞

C sup M t1
ˆ v p φ p (x, t)dx, Br

for appropriate choice of ψ . The other terms are naturally bounded. For sufficiently large M, we can move this term to the left in the Caccioppoli inequality and obtain (for appropriate choice of φ) 1 sup 2p t0 −T0
ˆ v p dx < ∞.

(4.33)

Br/2

´ To estimate supt0 −T0
g L∞ (BR ×(t1 ,t2 )) ≤ C0 , for some ball BR ⊃ . Since u −C0 and −u −C0 are also solutions, the functions v1 = (u −C0 )+ and v2 = (u + C0 )− are subsolutions in BR × (t1 , t2 ) by Lemma 3.1. This allows us to use the Caccioppoli inequality in BR × (t1 , t2 ) and obtain (4.33) for any Br ⊂ BR , for v1 and v2 . This way we obtain (4.32). References [1] Benny Avelin, Luca Capogna, Giovanna Citti, Kaj Nyström, Harnack estimates for degenerate parabolic equations modeled on the subelliptic p-Laplacian, Adv. Math. 257 (2014) 25–65, https://doi.org/10.1016/j.aim.2014.02.018, MR 3187644. [2] Martin T. Barlow, Richard F. Bass, Takashi Kumagai, Parabolic Harnack inequality and heat kernel estimates for random walks with long range jumps, Math. Z. 261 (2) (2009) 297–320, https://doi.org/10.1007/s00209-008-03265, MR 2457301. [3] Krzysztof Bogdan, Paweł Sztonyk, Harnack’s inequality for stable Lévy processes, Potential Anal. 22 (2) (2005) 133–150, https://doi.org/10.1007/s11118-004-0590-x, MR 2137058. [4] Matteo Bonforte, Yannick Sire, Juan Luis Vázquez, Optimal existence and uniqueness theory for the fractional heat equation, Nonlinear Anal. 153 (2017) 142–168, https://doi.org/10.1016/j.na.2016.08.027, MR 3614666. [5] L. Brasco, E. Lindgren, Armin Schikorra, Higher Hölder regularity for the fractional p-Laplacian in the superquadratic case, Preprint, arXiv:1711.09835, 2017. [6] Luis Caffarelli, Chi Hin Chan, Alexis Vasseur, Regularity theory for parabolic nonlinear integral operators, J. Amer. Math. Soc. 24 (3) (2011) 849–869, https://doi.org/10.1090/S0894-0347-2011-00698-X, MR 2784330.

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