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Weak Harnack inequality for a mixed local and nonlocal parabolic equation

This article proves a weak Harnack inequality with a tail term for sign changing supersolutions of a mixed local and nonlocal parabolic equation. Our argument is purely analytic. It is based on energy estimates and the Moser iteration technique. Instead of the parabolic John-Nirenberg lemma, we adopt a lemma of Bombieri to the mixed local and nonlocal parabolic case. To this end, we prove an appropriate reverse H\"older inequality and a logarithmic estimate for weak supersolutions.


Introduction
In this article, we establish a weak Harnack inequality for the mixed local and nonlocal parabolic Laplace equation where T > 0, Ω ⊂ R N with N ≥ 2 is a bounded domain (i.e. bounded, open and connected set) and L is an integro-differential operator of the form (1.2) Lu(x, t) = P.V.ˆR N u(x, t) − u(y, t) K(x, y, t) dx dy dt, where P.V. denotes the principal value and K is a symmetric kernel in x and y such that for some 0 < s < 1 and Λ ≥ 1, we have uniformly in t ∈ (0, T ). If K(x, y, t) = C |x − y| N +2s , for some constant C, then L reduces to the well known fractional Laplace operator (−∆) s and (1.2) is the mixed local and nonlocal fractional heat equation (1.4) ∂ t u + (−∆) s u = ∆u.
This kind of evolution equations arise in the study of Lévy processes, image processing etc, see Dipierro-Valdinoci [23] and the references therein for more details on the physical interpretation.
The main purpose of this article is to provide a weak Harnack inequality for (1.1) (Theorem 2.8). To the best of our knowledge, a weak Harnack inequality is unknown even for the prototype equation (1.4). Our main result is stated for sign changing weak supersolutions of (1.1). For sign changing solutions of nonlocal problems, in both the elliptic and parabolic context, an extra quantity referred to as "Tail" or "Parabolic Tail", generally appears in the Harnack estimates. This phenomenon has been first observed by Kassmann in [30] for the fractional Laplace equation (−∆) s u = 0 and further extended by Di Castro-Kuusi-Palatucci [16,17] and Brasco-Lindgren-Schikorra [8,9] to the fractional p-Laplace equation. For the parabolic nonlocal case, see Strömqvist [39], Brasco-Lindgren-Strömqvist [10], Banerjee-Garain-Kinnunen [1], Ding-Zhang-Zhou [19] and the references therein. For the mixed local and nonlocal elliptic equations, a new tail quantity appears that captures both the local and nonlocal behavior of the mixed operator as observed in [28]. For the mixed local and nonlocal parabolic problem (1.1), we introduce an appropriate tail term (Definition 2.7), which captures both local and nonlocal behavior of the mixed equation.
In contrast to the probabilistic approach in [2,15], we prove the weak Harnack estimate (Theorem 2.8) by analytic techniques. More precisely, we employ the approach of Moser [36] that uses a lemma by Bombieri-Giusti [6], further avoiding the use of technically demanding parabolic John-Nirenberg lemma, see Moser [37] and Fabes-Garofalo [25]. We discuss energy estimates for negative and positive powers of weak supersolutions for (1.1) (Lemma 3.1 and Lemma 3.2). These energy estimates together with the Sobolev inequality and the Moser iteration technique, enable us to estimate the supremum of the negative power of a weak supersolution (Lemma 4.1) of (1.1) and to prove the reverse Hölder inequality (Lemma 4.2). Finally, a logarithmic estimate (Lemma 4.3) of weak supersolutions of (1.1) is deduced that allows us to apply the Bombieri lemma (Lemma 2.13) to establish our main result (Theorem 2.8).

Preliminaries and main results
We use the following notation throughout. We denote the positive and negative parts of a ∈ R by a + = max{a, 0} and a − = max{−a, 0}, respectively. The Lebesgue outer measure of a set S is denoted by |S|. The barred integral sign denotes the corresponding integral average. We write C to denote a constant which may vary from line to line or even in the same line. If C depends on r 1 , r 2 , . . . , r k , we denote C = C(r 1 , r 2 , . . . , r k ).
We recall some known results for the fractional Sobolev spaces, see Di Nezza-Palatucci-Valdinoci [18] for more details. Definition 2.1. Let 0 < s < 1 and assume that Ω ⊂ R N is an open and connected subset of R N . The fractional Sobolev space W s,2 (Ω) is defined by and it is endowed with the norm The fractional Sobolev space with zero boundary values is defined by Both W s,2 (Ω) and W s,2 0 (Ω) are reflexive Banach spaces, see [18]. We denote the classical Sobolev space by W 1,2 (Ω). The parabolic Sobolev space L 2 (0, T ; W 1,2 (Ω)), T > 0, consists of measurable functions u on Ω × (0, T ) such that The space L 2 loc (0, T ; W 1,2 loc (Ω)) is defined by requiring the conditions above for every The next result asserts that the classical Sobolev space is continuously embedded in the fractional Sobolev space, see [18,Proposition 2.2]. The argument applies an extension property of Ω so that we can extend functions from W 1,2 (Ω) to W 1,2 (R N ) and that the extension operator is bounded.
The following result for the fractional Sobolev spaces with zero boundary value follows from [11, Lemma 2.1]. The main difference compared to Lemma 2.2 is that the result holds for any bounded domain, since for the Sobolev spaces with zero boundary value, we always have a zero extension to the complement.
Let Ω be a bounded domain in R N and 0 < s < 1. There exists a positive constant for every u ∈ W 1,2 0 (Ω). Here we consider the zero extension of u to the complement of Ω. The notion of weak supersolutions for (1.1) is defined as follows.
Remark 2.5. By Lemma 2.2 and Lemma 2.3, we observe that the Definition 2.4 well stated. Note that if u is a weak supersolution of (1.1), so is u + c for any scalar c.
Remark 2.6. Below we obtain energy estimates where the test functions depend on the supersolution itself. The admissibility of these test functions can be justified by using the mollification in time defined for f ∈ L 1 (Ω × (0, T )) by See [5,34] for more details on f h .
Next, we define the parabolic tail which appears in estimates throughout the article.
Definition 2.7. Let x 0 ∈ R N , t 1 , t 2 ∈ (0, T ) and r > 0. The parabolic tail of a weak supersolution u of (1.1) (Definition 2.4) is Now we are ready to state our main result, which asserts that a weak Harnack inequality holds for weak supersolutions of (1.1).
We state some useful results that are needed to prove our main result (Theorem 2.8). The following inequalities follows from [26,Lemma 3.3].
Lemma 2.11. Let φ : B r (x 0 ) → [0, ∞) be a radially decreasing function. There exists a positive constant C = C(N, φ) such that The following version of the Gagliardo-Nirenberg-Sobolev inequality will be useful for us, see [35,Corollary 1.57]. There exists a positive constant C = C(N ) such that for every u ∈ W 1,2 0 (Ω). Our final auxiliary result is a lemma of Bombieri-Giusti [6], which can be proved with the same arguments as in the proof of [ Lemma 2.13. Assume that ν is a Borel measure on R N +1 and let θ, A and γ be positive Let f be a positive measurable function on U (1) which satisfies the reverse Hölder inequality for some constant C = C(θ, δ, α, γ, A) > 0.

Energy estimates
In this section, we establish energy estimates for weak supersolutions of (1.1). The first one is the following lemma that helps us to estimate the supremum of the inverse of weak supersolutions.
Let 0 < r ≤ 1 be such that r < R and denote v = u + l, l > 0. Then for any m > 0 there exists a positive constant C(m) ≈ 1 + m such that and ess sup is an admissible test function in (2.2). Indeed, following [5,34], since v is a weak supersolution of (1.1), noting the definition of (·) h from (2.3), we have Estimate of I h : We observe that Since . Therefore, the first integral in the above estimate of I h is nonpositive. As a consequence, we obtain (3.4) Now passing the limit as h → 0, we have where Estimate of A h : Following the same arguments as in the proof of [1, Lemma 3.1], we obtain Therefore, using (3.5), (3.7), (3.8) in (3.3), we have ∇v∇φ dx dt (3.9) Estimate of I 1 : Using [(2.7), Lemma 2.10] for C(ǫ) = max 4, 6ǫ−5 2 , we have (3.10) Using these estimates and the fact that Using Young's inequality (3.13) ab for any a, b ≥ 0 and δ > 0, we have Choosing δ = ǫ 2(ǫ+1) and employing the estimate (3.14) in (3.12) we obtain where we have also used the fact that By combining the estimates (3.6), (3.10), (3.11) and (3.15) in (3.9), we obtain (3.16) Letting t 2 → τ 2 in (3.16), we havê we obtain from (3.16) ess sup The following energy estimate is useful to obtain reverse Hölder inequality for weak supersolutions.

Estimates for weak supersolutions
We establish an estimate of supremum for weak supersolutions of (1.1).
We obtain a reverse Hölder inequality for weak supersolutions of the problem (1.1).
Proof. We divide the interval (σ, σ ′ ) into k parts by setting We shall choose k below. Denote r j = σ j r, B j = B r j (x 0 ), Γ j = (t 0 , t 0 + r 2 j ) and Let 0 < ǫ < 1 and α = 1 − ǫ. Using Lemma 2.12 along with Hölder's inequality with exponents t = κ κ−2 and t ′ = κ 2 , for some positive constant C = C(N ), we have Setting r = r j , τ 1 = t 0 , τ 2 = t 0 + r 2 j+1 and τ = r 2 j − r 2 j+1 in Lemma 3.1, for some constant C = C(Λ) > 0, we have Proceeding as in the proof of Lemma 4.1, we obtain and (4.23) for some positive constant C = C(N, s, Λ). Using the properties of ψ j , we obtain for some positive constant C = C(N, s, Λ). Note that C is independent of ǫ as long as α is away from 1. We use the Moser iteration technique to conclude the result. Fix q and q such that q > q and k such that qγ k−1 ≤ q ≤ qγ k . Let t 0 be such that t 0 ≤ q and q = γ k t 0 . Let t j = γ j t 0 , j = 0, 1, · · · , k. By iterating (4.26) and Hölder's inequality, we arrive at Note that, due to the singularity of ǫ at 0 in the estimates (4.24) and (4.25), the constant C in (4.27) depends on q. It is easy to observe that C * and β are uniformly bounded over k. Now, since qγ k−1 ≤ t 0 γ k , we have t 0 ≥ q γ . Hence, the result follows from (4.27), with θ = 2γ 2 γ−1 . Next, we prove the following logarithmic estimate for weak supersolutions of (1.1).
where Tail ∞ is defined by (2.4). Then there exists a constant C = C(N, s, Λ) > 0 such that where U + (r) = B r (x 0 ) × (t 0 , t 0 + r 2 ). Moreover, there exists a constant C = C(N, s, Λ) > 0 such that Proof. We only prove the estimate (4.28), since the proof of (4.29) follows similarly. Without loss of generality, we assume that x 0 = 0 and denote by B r = B r (0), B 3r 2 = B 3r 2 (0). Since v is a weak supersolution of (1.1), choosing φ(x, t) = ψ(x) 2 v(x, t) −1 as a test function in (2.2) (which can again be justified as in the proof of Lemma 3.1), we get (4.31) where for any t 0 ≤ t 1 < t 2 ≤ t 0 + r 2 , we have (4.32) which gives the estimate, Let w(x, t) = − log v(x, t) and Since 0 ≤ ψ ≤ 1 in B 3r 2 and ψ ≡ 1 in B r , we obtain´B 3r 2 ψ(x) 2 dx ≈ r N . Then by Lemma 2.11, for some positive constant C 1 > 0 (independent of r), we obtain By dividing through by´B 3r 2 ψ 2 dx on both sides of (4.37) and using (4.38) together with the fact that ψ ≡ 1 in B r , we get This shows that W (t) is a monotone decreasing function in (t 1 , t 2 ). Hence W (t) is differentiable almost everywhere with respect to t. Hence, from (4.39) for almost every t such that t 1 < t < t 2 , we obtain (4.40) W ′ (t) + 1 A 1 r N +2ˆB r w(x, t) − W (t) 2 dx ≤ 0.