Higher integrability near the initial boundary for nonhomogeneous parabolic systems of p-Laplacian type

We establish a sharp higher integrability near the initial boundary for a weak solution to the following p-Laplacian type system: { ut − divA(x, t, ∇u) = div|F|p−2F + f in ΩT , u = u0 on Ω × {0}, by proving that, for given δ ∈ (0, 1), there exists ε > 0 depending on δ and the structural data such that |∇u0| ∈ Lloc(Ω) and |F| p+ε , |f |( δp(n+2) n ) 󸀠+ε ∈ L1(0, T; Lloc(Ω)) 󳨐⇒ |∇u|p+ε ∈ L1(0, T; Lloc(Ω)). Our regularity results complement established higher regularity theories near the initial boundary for such a nonhomogeneous problem with f ̸ ≡ 0 and we provide an optimal regularity theory in the literature.


Introduction
In this paper, we are interested in finding a sharp higher integrability near the initial boundary to a weak solution to the parabolic system Here, 2n n+2 < p < ∞, Ω is a bounded domain in ℝ n with n ≥ 2, A(x, t, ζ) is modeled after the p-Laplacian operator, u 0 ∈ W 1,p (Ω, ℝ N ), F ∈ L p (Ω T ; ℝ Nn ) and f ∈ L q (Ω T , ℝ N ) for some N ≥ 1, where q = p(n+2) n is the parabolic Sobolev conjugate of p and q is Hölder conjugate of q.
In the case f ≡ 0, interior higher integrability results were proved by Kinnunen and Lewis in [13,14] by providing a suitable application of DiBenedetto's intrinsic geometry method from [9] to the setting of Gehring type estimates. Higher integrability results near the initial and lateral boundary were proved by Parviainen in [19,20]. These regularity results were extended to higher-order systems by Bögelein and Parviainen in [5].
On the other hand, in the case f ̸ ≡ 0, a nonlinear relation between the gradient of a weak solution and the non-divergence data f coming from parabolic embeddings naturally occurs. To be specific, there is an exponent α > 1 such that ‖∇u‖ p L p is related to ‖f ‖ q α L q . Indeed, there have been regularity estimates coming from such a nonlinear relation rather than Gehring type estimates. In particular, Kuusi and Mingione in [15] proved gradient L ∞ regularity and gradient continuity of a weak solution with an nonlinear exponent α based on the intrinsic geometry method. The recent paper [2] provides a new representation of the nonlinear relation. It replaces the exponent α by 1 from the intrinsic geometry method, obtaining interior higher integrability results for a system of p(x, t)-Laplacian type with scaling invariant estimates and extending gradient continuity results in [15] to the p(x, t)-Laplacian system. For the elliptic case, there is a divergence representation for the non-divergence data, and Gehring type estimates directly follow from [10,11,17].
As already mentioned, the main purpose of this paper is to establish higher integrability results near the initial boundary. In addition, we are considering a class of the nonlinearities whose structures are associated with the divergence data F. Regarding the non-divergence data, f ∈ L ( δp(n+2) n ) with δ ∈ (0, 1) is sharp in obtaining the reverse Hölder inequality by the parabolic Sobolev embedding theorem. A noteworthy feature of the present paper is to find the optimality and sharpness of the initial data u 0 from the intrinsic geometry method. Due to a suitable application of the Poincaré inequality in a type of the Caccioppoli inequality, ‖∇u 0 ‖ 2 L s (B ρ ) appears for some s > 1. The optimal exponent s can be found in the process of the use of the Fubini theorem. In fact, (−− ∬|∇u 0 | s dz) p s appears after using the reverse Hölder inequality, and the Minkowski integral inequality enforces the condition s = p. On the other hand, there is an alternative for the non-divergence data f to avoid applying the Minkowski integral inequality, thanks to the presence of radius ρ (see (4.3) below). Therefore, p is the optimal exponent for s, and we can find the sharp exponent β = min{2, p} in ‖∇u 0 ‖ β L p from the stopping time argument.
Our work can be applicable to different problems including obstacle problem and system of p(x, t)-Laplacian type problem as in [4,6] as well as Calderón-Zygmund type estimates as in [1,3,7,8,16] both when f ̸ ≡ 0 and when ∇u 0 ̸ ≡ 0. The paper is organized as follows. In Section 2, we introduce basic notation and definitions to state our main results. Section 3 is devoted to proving reverse Hölder inequality. Finally, in Section 4, we prove our main results.

Notations
We shall clarify all the notations that will be used in this paper. (i) We use ∇ to denote derivatives with respect the space variable x and ∂ t to denote the time derivative. (ii) In what follows, we always assume the bounds 2n n+2 < p < ∞. (iii) Let z 0 = (x 0 , t 0 ) ∈ ℝ n+1 be a point, ρ, s > 0 two given parameters, and let λ ∈ [1, ∞). We use the following notations: We use ∫ to denote the integral with respect to either space variable or time variable and use ∬ to denote the integral with respect to both space and time variables simultaneously. Analogously, we use − ∫ and −− ∬ to denote the integral averages as defined below: for any set A × B ⊂ ℝ n × ℝ, we define (v) We use the notation ≲ (a,b,...) to denote an inequality with a constant depending on a, b, . . . .

Definition 2.1.
Let Ω be a bounded domain in ℝ n with n ≥ 2 and u 0 ∈ L 2 (Ω, is a distributional energy solution in the sense

Structures of the operator
We now describe the assumptions on the nonlinear structures in (2.1). Assume A(x, t, ∇u) is a Carathéodory function, i.e., we have that (x, t) → A(x, t, ζ) is measurable for every ζ ∈ ℝ n and ζ → A(x, t, ζ) is continuous for almost every (x, t) ∈ Ω T . We further assume that, for a.e. (x, t) ∈ Ω T and for any ζ ∈ ℝ n , there exist two positive constants Λ 0 and Λ 1 such that the following bounds are satisfied by the nonlinear structures: where F ∈ L p (Ω T , ℝ Nn ).

Main results
Before stating our main theorem, we fix some constants which will be frequently used in this paper.
To apply the intrinsic geometry method developed in [13], let us define the following notations.
3) and f ∈ L a (Ω T , ℝ N ). Then there exists ε 0 (n, N, p, Λ 0 , Λ 1 , δ) such that, for any ε ∈ (0, ε 0 ) and for any ( Here, λ 0 is defined in Definition 2.4. Remark 2.6. As a consequence of Theorem 2.5, we can also obtain global estimates of the weak solution to where A, f and u 0 are assumed as in (2.1) and ϕ ∈ L p (0, T; Especially, ∇ϕ behaves like divergence data F does while ϕ t behaves exactly in the same way as non-divergence data f does. Before ending this section, we provide some important lemmas which will be used later in the proof of the main theorem. Let us state Gagliardo-Nirenberg's inequality (see [18]).

Estimates near the initial boundary
In this section, we assume that B 4ρ (x 0 ) ⊂ Ω, 0 < t 0 and λ ≥ 1. Also, note that, in the case 0 ∈ I λ 4ρ (t 0 ), for any ρ ≤ ρ 1 ≤ ρ 2 ≤ 4ρ, there holds We will show a reverse Hölder inequality in intrinsic cylinders under the following assumptions.
Note that, by a choice of δ ∈ (0, 1) in Definition 2.2, 0 < a (1 − p a ) < p holds, and this exponent a (1 − p a ) is chosen to preserve a nonlinear relation between the gradient of a weak solution and the non-divergence data. Indeed, using Young's inequality,

Caccioppoli inequality and Poincaré type inequality
Let us first state a Caccioppoli type inequality.

Lemma 3.2. Let u be a weak solution of
Proof. The proof basically follows from [5, Lemma 5.1]. For the sake of completeness, we present the details in our setting.
In the space direction, take a cut-off function η satisfying For the time direction, we divide two cases. In the case that 0 ∉ I λ 4ρ (t 0 ), consider a cut-off function ζ satisfying On the other hand, in the case that 0 ∈ I λ 4ρ (t 0 ), consider a cut-off function ζ satisfying Taking (u − (u 0 ) B ρ b (x 0 ) )η p ζ 2 as a test function in (2.1), we obtain Estimate of I: We use integration by parts and (2.2) to find that Estimate of I 1 : Using the triangle inequality along with the fact that ρ ≤ ρ a ≤ ρ b ≤ 4ρ b and (3.1), we obtain Estimate of I 2 : Applying Poincaré's inequality, we have Estimate of I 3 : We get Therefore, we obtain Estimate of II: Applying (2.3), we discover Estimate of III: Here, to obtain (a), we used Young's inequality with γ ∈ (0, 1). Estimate of IV: Clearly, there holds Combining all the above calculations and taking γ = γ(n, N, p, Λ 0 , Λ 1 ) small enough, the conclusion follows.
In our definition of a weak solution to (2.1), there is no differentiability assumption on u with respect to time. Therefore, we cannot apply Poincaré's inequality directly. Nevertheless, we shall use (2.1) to estimate continuity of u with respect to time.
The triangle inequality gives Estimate of I: Applying Poincaré's inequality in space direction, we obtain Estimate of II: Similarly, we have Estimate of III: Again, the triangle inequality implies To estimate the second term, we test η to Using (2.2), we obtain sup t∈I λ r (t 0 ) Therefore, we get This completes the proof.

Some crucial estimates
The purpose of this subsection is to refine the estimate u in C(I λ r (t 0 ) ∩ (0, T); L 2 (B r (x 0 ), ℝ N ). To this end, we first estimate the right-hand side of the inequality in Lemma 3.2 using Lemma 3.3 and assumption (3.2), and then return to the left-hand side of the inequality.
Proof. Applying Lemma 3.3 and Hölder's inequality, we find Therefore, making use of (3.2), this completes the proof.

Lemma 3.5. Under the assumptions and the conclusion in Lemma 3.4, we further have
Proof. Take 2ρ ≤ ρ a < ρ b ≤ 4ρ in Lemma 3.2 to get Estimate of I: Note that, under the restriction 2ρ ≤ ρ a < ρ b ≤ 4ρ, we see that Here, to obtain (a), we used Lemma 3.4 and Poincaré's inequality with (3.2) for initial data. Estimate of II: There holds that where, to obtain (a), we used Sobolev's inequality with respect to space direction and then Hölder's inequality with respect to the time integral, and to obtain (b), we used (3.5) and (3.2). Applying Young's inequality, for any γ ∈ (0, 1), there holds Estimate of III: After applying Hölder's inequality, we get Here, to obtain (a), we used Lemma 2.7 with σ = a, q = δp, r = 2 and ϑ = n n+2 . To obtain (b), we used Hölder's inequality with respect to the time integral.
We will not use following corollary in further estimates for our purpose, but it is worthwhile observing the estimate of the corollary.
Thus, applying Lemma 3.3 with θ = q to estimate the second term, we obtain Estimate of I: Applying Young's inequality along with the fact 2ϑ p < 2ϑ q ≤ 1, we get Estimate of II: Similarly, since 2ϑ(p−1) p < 1, Hölder's inequality and Young's inequality give Estimate of III: Since 2ϑ a < 1, Young's inequality gives Estimate of IV: Since 2ϑ p < 1, Hölder's inequality and Young's inequality give Therefore, combining all the estimates, the proof is completed.

Lemma 3.8. Under the assumptions and the conclusion in Lemma 3.7, we further have
Proof. We apply Lemma 2.7 with σ = p, q as defined in (3.7), r = 2 and ϑ = q p . Note that (3.7) implies np Therefore, we have Here, to obtain (a), we used pϑ q = 1, and to obtain (b), we used Lemma 3.5. Now, we apply Lemma 3.3 with θ = q to the first term on the right-hand side to get Estimate of I: Applying Young's inequality, there holds Estimate of II: Applying Hölder's inequality and Young's inequality, we get Estimate of III: We observe Therefore, Young's inequality gives Estimate of IV: Hölder's inequality and Young's inequality imply We combine all the estimates to complete the proof.

Lemma 3.9. Under the assumptions and the conclusion in Lemma 3.7, we further have
Proof. Apply Lemma 2.7 as in (3.6) and Lemma 3.5 to get Using Lemma 3.3 with θ = q to the first term on the right-hand side, we have Estimate of I: Applying Young's inequality, we obtain Estimate of II: Note that (p−1)n p(n+2) < 1. Apply Hölder's inequality and Young's inequality to get Estimate of III: Applying Hölder's inequality and Young's inequality, we have Estimate of IV: Again, Hölder's inequality and Young's inequality give The proof follows.
Proof. From Lemma 3.2 with ρ a = ρ and ρ b = 2ρ, there holds We apply Lemma 3.7, Lemma 3.8 and Lemma 3.9 to I, II and III. Use Young's inequality to estimate IV. Then there holds Taking γ = γ(n, N, p, Λ 0 , Λ 1 , δ) small enough, we finish the proof.

Proof of Theorem 2.5
In this section, we prove the main results. First of all, we shall find intrinsic cylinders such that (3.2) and (3.3) hold.