Local Hölder Regularity of Weak Solutions for Singular Parabolic Systems of p-Laplacian Type CAUCHY –Jurnal Matematika Murni dan Aplikasi Volume 7(1) (2021), Pages 136-141 p-ISSN: 2086-0382; e-ISSN: 2477-3344 Submitted: August 07, 2021 Reviewed: October 11, 2021 Accepted: October 21, 2021 DOI: https://doi.org/10.18860/ca.v7i1.13105 Local Hölder Regularity of Weak Solutions for Singular Parabolic Systems of p-Laplacian Type Khoirunisa Khoirunisa1, Corina Karim2, M. Muslikh3 1,2,3Department of Mathematics, Universitas Brawijaya Email: khoir.n97@gmail.com, co_mathub@ub.ac.id, mslk@ub.ac.id ABSTRACT Local Hölder regularity of weak solutions for degenerate parabolic systems of p-Laplacian type was proved by Corina, K. In this paper, we aim to show the local Hölder regularity of weak solutions in the singular case with 2𝑚 𝑚+2 < 𝑝 < 2, 𝑚 ≥ 2, so that the Local Hölder regularity of weak solutions for parabolic systems of p-Laplacian type can hold for both cases. By applying Poincaré inequality, we show that its weak solutions within Hölder space, or we can tell that the local Hölder regularity for the singular case is valid. Keywords: Hölder regularity; singular case; weak solutions INTRODUCTION Hölder regularity of weak solutions for parabolic equations in singular and degenerate case was introduced in [1] dan [2], where the coefficients are measurable and satisfy elliptic condition. However, the result in [2] is not showing the boundary estimates. After that, Bögelein was interested to prove for the boundary regularity from the previous result in [2], see [3]. Then, [4] investigated the same problem for nonlinear parabolic equations in the degenerate case. In 2002, DiBenedetto et al discussed about the regularity of weak solutions for quasilinear parabolic equations in singular and degenerate case to proving their Hölder character in [5]. The result in [2] was extended by Misawa to a larger class of right-hand side terms, but the singular case was ecluded here, see [6]. Meanwhile, for singular case, we can see the Hölder regularity in [7]. In addition, the Holder regularity of gradient solutions for all cases was proved in [8]. The study about Hölder regularity of gradients solution was continued in [9] for evolutionary p-Laplacian systems where the coefficients is Hölder continuous. Based on [10], the global weak solutions for similar case is exist. They used variational method to prove the existence of weak solutions globally. In 2018, Karim started to investigated about simpler p-Laplacian type in singular parabolic systems and showed that the weak solutions is bounded [11]. To prove the local boundedness in [11], we can adopt the method to prove energy estimates of singular parabolic equations in [12]. On the other hand, the existence of weak solutions in singular case was proved in [13] by using the Galerkin method. Furthermore, the intrinsic scaling method was used to treat the weak solutions to prove the Hölder regularity for degenerate case [14]. The intrinsic scaling https://doi.org/10.18860/ca.v7i1.13105 mailto:khoir.n97@gmail.com mailto:co_mathub@ub.ac.id Local Hölder Regularity of Weak Solutions for Singular Parabolic Systems of p-Laplacian Type Khoirunisa Khoirunisa 137 method is based on [15], which used the intrinsic scaling method to approach the regularity in degenerate and singular partial differential equations. Motivated by the last result in[14], we would investigated the Hölder regularity of weak solutions in singular case. Let Ω ⊂ ℝ𝑛 be a bounded domain, 𝑛 ≥ 2, and 𝜕Ω is smooth boundary. The unknown function 𝑢: (0, 𝑇) × Ω → ℝ𝑚is vector valued function, 𝑢 = (𝑢1(𝑧), 𝑢2(𝑧), … , 𝑢𝑚(𝑧)), where 𝑧 = (𝑡, 𝑥1, 𝑥2, … , 𝑥𝑛 ). Let 2𝑚 𝑚+2 < 𝑝 < 2, 𝑚 ≥ 2, consider the parabolic systems { 𝜕𝑡 𝑢 − div (|𝐷𝑢| 𝑝−2𝐷𝑢) = 0 in(0, 𝑇) × Ω, 𝑢(0, 𝑥) = 𝑢0(𝑥) on 𝜕𝑝(0, 𝑇) × Ω, (1) where 𝑢0(𝑥) ∈ 𝑊 1,𝑝(Ω, ℝ𝑚). The result in [13] shows that for any initial condition, there exists weak solutions of (1) from Ω into ℝ𝑚. On the other hand, the local boundedness of the weak solution of (1) was proved by Karim in [11]. They modified the intrinsic scaling from the original work by DiBenedetto [12]. They used the intrinsic scaling for singular case. Their main theorem established that intrinsic scaling well-worked to prove the local boundedness of weak solution of (1). We now turn to notion of Hölder continuous functions. For any 𝑥, 𝑦 ∈ Ω̅, if 𝑢 ∈ 𝐶0(Ω̅) satisfy sup 𝑥,𝑦∈ Ω̅,𝑥≠𝑦 |𝑢(𝑥) − 𝑢(𝑦)| |𝑥 − 𝑦|𝛼 < ∞, then 𝑢 ∈ 𝐶0,𝛼 (Ω̅). Karim in [14] shows that the weak solutions of (1) in degenerate case satisfy the following theorem. Theorem 1.[14] Let 𝑝 ≥ 2 and u is weak solutions of (1) in 𝑄(𝜆2−𝑝𝜌2, 𝜌)(𝑧0). Then, 𝑢 is locally Hölder continuous with some 0 < 𝛼 < 1. Furthermore, for any 𝑄3𝜌0 (𝑧0 ′ ) ⊂ 𝑄 with 𝑧0 ′ ∈ 𝑄 and 0 < 𝜌0 < 1, there exist 𝐶 > 0 such that |𝑢(𝑧) − 𝑢(𝑧′)| ≤ 𝐶 {|𝑡 − 𝑡′ | 𝛼 𝑝 + |𝑥 − 𝑥′|𝛼 }, holds for any 𝑧, 𝑧′ ∈ 𝑄𝜌0 where 𝑄𝑟 (𝑧0) ⊂ 𝑄(𝜆 2−𝑝𝜌2, 𝜌)(𝑧0) ⊂ 𝑄𝜌0 (𝑧0 ′ ) ⊂ 𝑄3𝜌0 (𝑧0 ′ ). Theorem 1 implies that the local Hölder regularity of weak solutions (1) in degenerate case is proved. Here, we aim to prove for the singular case, so that the local Hölder regularity of weak solutions (1) can be proven for both cases or 2𝑚 𝑚+2 < 𝑝 < ∞. METHODS We can prove the local Hölder regularity of weak solutions for parabolic systems of p- Laplacian type in singular case by showing that the weak solutions are elements of Hölder space. Our method is using Poincaré inequality to show that the weak solutions within Campanato space, then by isomorphism between Campanato and Hölder space, it is easy to see that its weak solutions are elements of Hölder space. The Poincaré inequality that we use is in the following theorem. Theorem 2. (Poincaré Inequality).[16] Let Ω ⊂ ℝ𝑛 be a bounded open set and 1 ≤ 𝑝 ≤ ∞. Then, there exists positive constant 𝐶 = 𝐶(Ω, 𝑝) so that ||𝑢 − 𝑢Ω||𝐿𝑝 ≤ 𝐶||𝐷𝑢|| 𝐿𝑝 , (2) Local Hölder Regularity of Weak Solutions for Singular Parabolic Systems of p-Laplacian Type Khoirunisa Khoirunisa 138 where 𝑢Ω = 1 |Ω| ∫ 𝑢 Ω 𝑑𝑥. RESULTS AND DISCUSSION For singular equation (1) with 1 < 𝑝 < 2, we have the cylinder of the type 𝑄(𝜌)(𝑧0)(𝑡0 − 𝜌 2, 𝑡0 + 𝜌 2) × 𝐵 (𝜆 𝑝−2 2 𝜌, 𝑥0). By switching the cylinder 𝑄(𝜌)(𝑧0) to 𝑄(1), we have 𝑣(𝑠, 𝑦) = 𝑢(𝑡0 + 𝜆 2−𝑝𝜌𝑠, 𝑥0 + 𝜌𝑦) 𝜆𝜌 ; (𝑠, 𝑦) ∈ 𝐵(1) × (−1,1) ≡ 𝑄(1). Suppose that on such a certain cylinder the relations 1 |𝑄(𝜌)(𝑧0)| ∫ |𝐷𝑢|𝑝𝑑𝑧 ≈ 𝜆𝑝. (3) 𝑄(𝜌)(𝑧0) Hence, we have 𝜕𝑡 𝑢 − 𝜆 𝑝−2div(𝐷𝑢) = 0, in 𝑄(𝜌)(𝑧0). Let 𝑢 ∈ 𝐿∞(0 , 𝑇; 𝑊1,𝑝(Ω, ℝ𝑚)) be a weak solutions of (1) in cylinder 𝑄 = (0, 𝑇) × Ω where 𝑧0 = (𝑡0, 𝑥0) ∈ 𝑄, 𝜆 ≥ 1and 0 < 𝜌 < 1 such that 𝑄 (𝜌 2, 𝜆 𝑝−2 2 𝜌) ⊂ 𝑄. While, our main theorem is the following. Theorem 3.Let 2𝑚 𝑚+2 < 𝑝 < 2, 𝑚 ≥ 2 and u is a weak solution of (1) in 𝑄(𝜌2, 𝜆 𝑝−2 2 𝜌)(𝑧0). Then 𝑢 is locally 𝛼-Hölder continuous with some 0 < 𝛼 < 1. Furthermore, for any 𝑄3𝜌0 (𝑧0 ′ ) ⊂ 𝑄 with 𝑧0 ′ ∈ 𝑄 and 0 < 𝜌0 < 1, there 𝐶 > 0 such that |𝑢(𝑧) − 𝑢(𝑧′)| ≤ 𝐶 {|𝑡 − 𝑡′ | 𝛼 𝑝 + |𝑥 − 𝑥′|𝛼 }, holds for any 𝑧, 𝑧′ ∈ 𝑄𝜌0 (𝑧0 ′ ) where 𝑄𝑟 (𝑧0) ⊂ 𝑄(𝜌 2, 𝜆 𝑝−2 2 𝜌)(𝑧0) ⊂ 𝑄𝜌0 (𝑧0 ′ ) ⊂ 𝑄3𝜌0 (𝑧0 ′ ) Proof. Figure 1. Any cylinder in 𝑄 Local Hölder Regularity of Weak Solutions for Singular Parabolic Systems of p-Laplacian Type Khoirunisa Khoirunisa 139 Let 𝑄3𝜌0 (𝑧0 ′ ) ≔ 𝑄 (3(𝜌0) 2, 3(𝜌0) 2 𝑝) ⊂ 𝑄 is any cylinder centered in 𝑧0 ′ = (𝑡0 ′ , 𝑥0 ′ ) ∈ 𝑄, and 0 < 𝜌0 < 1. Let 𝑧0 = (𝑡0, 𝑥0) ∈ 𝑄𝜌0 (𝑧0 ′ ) ≔ 𝑄 ((𝜌0) 2, (𝜌0) 2 𝑝) (𝑧0 ′ ), where 𝜆 ≥ 1. It is easy to see that ∫ |𝐷𝑢|𝑝 𝑑𝑧 ≤ |𝑄𝑟 (𝑧0)| 1 |𝑄(𝜌2, 𝜆 𝑝−2 2 𝜌)(𝑧0)| ∫ |𝐷𝑢|𝑝 𝑑𝑧 𝑄(𝜌2,𝜆 𝑝−2 2 𝜌)(𝑧0)𝑄𝑟(𝑧0) By using relation (3) we have 1 |𝑄𝑟(𝑧0)| ∫ |𝐷𝑢|𝑝 𝑑𝑧 𝑄𝑟(𝑧0)) ≤ 𝐶𝑟𝑚+𝑝𝛼 (4) where0 < 𝑟 < 1, 0 < 𝛼 < 1, and 𝐶 = 𝜋𝑟2−𝑚. Next, recall the Poincaré inequality. Then, substitute (4) to (2) we have ∫ |𝑢 − 𝑢𝑟 | 𝑝 𝑑𝑧 ≤ 𝐶𝑟𝑝 ∫ |𝐷𝑢|𝑝 𝑑𝑧, 𝑄𝑟(𝑧0)𝑄𝑟(𝑧0) ≤ 𝐶𝑟𝑚+𝑝+𝑝𝛼 , holds for any 𝑧0 ∈ 𝑄𝜌0 (𝑧0) and 0 < 𝑟 < 2(𝜌0) 2 𝑝. We apply the characterization between Hölder continuous functions and Campanato space, which implies that for 𝑢 ∈ 𝐿𝑝 (𝑄𝜌0 (𝑧0 ′ )) and sup 𝑄𝜌0 (𝑧0 ′ ) 1 𝑟𝑚+𝑝+𝑝𝛼 ∫ |𝑢 − 𝑢𝑟 | 𝑝 𝑑𝑧 ≤ 𝐶, (5) 𝑄𝑟(𝑧0) then 𝑢 ∈ 𝐶0,𝛼 (𝑄𝜌0 (𝑧0 ′ )). Thus, we have for any two points (𝑡1, 𝑥1), (𝑡2, 𝑥2) ∈ 𝑄𝜌0 (𝑧0 ′ )with |𝑡1 − 𝑡2| = 𝑟 𝑝, |𝑢(𝑡1, 𝑥1) − 𝑢(𝑡2, 𝑥1)| ≤ 𝐶|𝑡1 − 𝑡2| 𝛼 𝑝 , (6) and let |𝑥1 − 𝑥2| = 𝑟, then |𝑢(𝑡2, 𝑥2) − 𝑢(𝑡2, 𝑥1)| ≤ 𝐶|𝑥1 − 𝑥2| 𝛼 . (7) Moreover, we can conclude from (6),(7) and triangle inequality that |𝑢(𝑧) − 𝑢(𝑧′)| ≤ 𝐶 {|𝑡 − 𝑡′ | 𝛼 𝑝 + |𝑥 − 𝑥′|𝛼 }, in 𝑄𝜌0 (𝑧0 ′ ) or we have Local Hölder Regularity of Weak Solutions for Singular Parabolic Systems of p-Laplacian Type Khoirunisa Khoirunisa 140 𝑢 ∈ 𝐶 , 𝛼 𝑝 ,𝛼 (𝑄𝜌0 (𝑧0 ′ )). Since, 𝑄3𝜌0 (𝑧0 ′ ) ⊂ 𝑄 is arbitrary, then 𝑢 ∈ 𝐶 𝑙𝑜𝑐 , 𝛼 𝑝 ,𝛼 (𝑄, ℝ𝑛 ). By this result, we can tell that the local Hölder regularity of weak solutions for parabolic systems of p-Laplacian type in singular case is proved. CONCLUSIONS Based on the previous results and discussion, it can be concluded that in singular case the weak solutions for parabolic systems of p-Laplacian type are elements of Hölder space, 𝑢 ∈ 𝐶 , 𝛼 𝑝 ,𝛼 (𝑄𝜌0 (𝑧0 ′ )). Since 𝑄3𝜌0 (𝑧0 ′ ) ⊂ 𝑄 is arbitrary, then 𝑢 ∈ 𝐶 𝑙𝑜𝑐 , 𝛼 𝑝 ,𝛼 (𝑄, ℝ𝑛 ) or we can say that the local Hölder regularity of weak solutions is proved. 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