Regularity estimates for the p-Sobolev flow

We study doubly nonlinear parabolic equation arising from the gradient flow for p-Sobolev type inequality, referred as p-Sobolev flow from now on, which includes the classical Yamabe flow on a bounded domain in Euclidean space in the special case p=2. In this article we establish a priori estimates and regularity results for the $p$-Sobolev type flow, which are necessary for further analysis and classification of limits as time tends to infinity.

Our main result in this paper is the following theorem. Let Ω be a bounded domain with smooth boundary. Suppose that the initial data u 0 is positive in Ω, belongs to W 1,p 0 (Ω) ∩ L ∞ (Ω), and satisfies the volume constraint u 0 L q+1 (Ω) = 1. Let u be a weak solution of (1.1) in Ω ∞ ≡ Ω × (0, ∞) with the initial and boundary data u 0 . Then, u is positive and bounded in Ω ∞ and, together with its spatial gradient, locally Hölder continuous in Ω ∞ . Moreover, for t ∈ [0, ∞), λ(t) =ˆΩ |∇u(x, t)| p dx and λ(t) ≤ λ(0). (1. 2) The definition of a weak solution is given in Definition 3.2. The global existence of the p-Sobolev flow and its asymptotic behavior, that is the volume concentration at infinite time, will be treated in our forthcoming paper, based on the a-priori regularity estimates for the p-Sobolev flow, obtained in the main theorem above.
The ODE part of the p-Sobolev flow equation is of exponential type, since the order of solution in both the time derivative and lower-order terms are the same. Thus, the solution is bounded for all times by the maximum principle (Proposition 3.5). On the other hand, a priori the solution may vanish in a finite time, by the effect of fast diffusion. This undesirable behavior can be ruled out for the p-Sobolev flow (1.1) by the volume constraint, that is the preservation of volume at all time. In fact, we show the global positivity of solutions of (1.1) under the volume constraint (Proposition 5.4). The positivity of solutions is based on local energy estimates for truncated solution and De Giorgi's iteration method. For the porous medium and p-Laplace equations, see [28,29,6,7], and also [27]. Our task is to discover the intrinsic scaling to the doubly nonlinear operator in the p-Sobolev flow (Corollary 4.6). Then, the interior positivity is obtained from some covering argument, being reminiscent of the so-called Harnack chain (Corollary 4.8). This leads to the positivity and regularity on a non-convex domain and thus, may be of interest in geometry. Once the interior positivity is established, the positivity near the boundary of domain follows from the usual comparison function (Proposition 4.9). Finally, the Hölder continuity reduced to that of the evolutionary p-Laplacian equation, by use of the positivity and boundedness of solutions. The energy equality also holds true for a weak solution of the p-Sobolev flow, leading to the continuity on time of the p-energy and volume.
The doubly nonlinear equations have been considered by Vespri [30], Porizio and Vespri [21], and Ivanov [12,13]. See also [9,21,31,17,16]. The regularity proofs for doubly nonlinear equations are based on the intrinsic scaling method, originally introduced by DiBenedetto, and they have to be arranged in some way depending on the particular form of the equation. Here, the very fast diffusive doubly nonlinear equation such as the p-Sobolev flow (1.1) is treated, and the positivity, boundedness and regularity of weak solutions are studied and shown in some precise way. See [20] for existence of a weak solution.
Consider next the stationary equation for (1.1), which is described by the p-Laplacian type elliptic equation, obtained from simply removing the time derivative term from the first equation in (1.1). This stationary equation relates to the existence of the extremal function attaining the best constant of Sobolev's embedding inequality, W 1,p 0 (Ω) ֒→ L q+1 (Ω). If the domain Ω is star-shaped with respect to the origin, the trivial solution u ≡ 0 only exists, by Pohozaev's identity and Hopf's maximum principle. Thus, the p-Sobolev flow (1.1), if globally exists, may have finitely many volume concentration points at infinite time. This volume concentration phenomenon is one of our motives of studying the p-Sobolev flow (1.1). Moreover, if the domain Ω is replaced by a smooth compact manifold, we can study the generalization of Yamabe problem in the sense of p-Laplacian setting. This is another geometric motive for the p-Sobolev flow.
In fact, in the case that p = 2, our p-Sobolev flow (1.1) is exactly the classical Yamabe flow equation in the Euclidean space. The classical Yamabe flow was originally introduced by Hamilton in his study of the so-called Yamabe problem ( [32,2,3]), asking the existence of a conformal metric of constant curvature on n(≥ 3)-dimensional closed Riemannian manifolds ( [11]). Let (M, g 0 ) be a n(≥ 3)-dimensional smooth, closed Riemannian manifold with scalar curvature R 0 = R g 0 . The classical Yamabe flow is given by the heat flow equation Here we will notice that the condition for volume above naturally corresponds the volume constraint in (1.1). Hamilton ([11]) proved convergence of the Yamabe flow as t → ∞ under some geometric conditions. Under the assumption that (M, g 0 ) is of positive scalar curvature and locally conformal flat, Ye ([33]) showed the global existence of the Yamabe flow and its convergence as t → ∞ to a metric of constant scalar curvature. Schwetlick and Struwe ( [22]) succeeded in obtaining the asymptotic convergence of the Yamabe flow in the case 3 ≤ n ≤ 5, under an appropriate condition of Yamabe invariance Y (M, g 0 ), which is given by infimum of Yamabe energy E(u) =´M(c n |∇u| 2 g 0 + R 0 u 2 ) dvol g 0 among all positive smooth function u on M with Vol(M) = 1, for an initial positive scalar curvature. In Euclidean case, since R g 0 = 0 their curvature assumptions are not verified. In outstanding results concerning the Yamabe flow, the equation is equivalently transformed to the scalar curvature equation, and this is crucial for obtaining many properties for the Yamabe flow. In this paper, we are forced to take a direct approach dictated by the structure of the p-Laplacian leading to the degenerate/singular parabolic equation of the p-Sobolev flow. Let us remark that our results cover those of the classical Yamabe flow in the Euclidian setting.
The structure of this paper is as follows. In Section 2 we prepare some notations and technical analysis tools, which are used later. Section 3 provides basic definitions of weak solutions, and also some basic study the doubly nonlinear equations of p-Sobolev flow type, including (1.1), and derivation of the minimum and maximum principles. Moreover, we establish the comparison theorem and make the Caccioppoli type estimates, which have a crucial role in Section 4. In the next section, Section 4, we prove the expansion of positivity for the doubly nonlinear equations of p-Sobolev flow type. In Section 5 we show the positivity, the energy estimates and, consequently, the Hölder regularity for the p-Sobolev flow (1.1). Finally, in Appendix A, B and C, for the p-Sobolev flow, we give detailed proofs of facts used in the previous sections.

Preliminaries
We prepare some notations and technical analysis tools, which are used later.

Notation
Let Ω be a bounded domain in R n (n ≥ 3) with smooth boundary ∂Ω and for a positive T ≤ ∞ let Ω T := Ω × (0, T ) be the cylindrical domain. Let us define the parabolic boundary of Ω T by We prepare some function spaces, defined on space-time region. For 1 ≤ p, q ≤ ∞, L q (t 1 , t 2 ; L p (Ω)) is a function space of measurable real-valued functions on a space-time region Ω × (t 1 , t 2 ) with a finite norm v L q (t 1 ,t 2 ; L p (Ω)) := When p = q, we write L p (Ω×(t 1 , t 2 )) = L p (t 1 , t 2 ; L p (Ω)) for brevity. For 1 ≤ p < ∞ the Sobolev space W 1,p (Ω) is consists of measurable real-valued functions that are weakly differentiable and their weak derivatives are p-th integrable on Ω, with the norm . . , v xn ) denotes the gradient of v in a distribution sense, and let W 1,p 0 (Ω) be the closure of C ∞ 0 (Ω) with resptect to the norm · W 1,p . Also let L q (t 1 , t 2 ; W 1,p 0 (Ω)) denote a function space of measurable real-valued functions on space-time region with a finite norm v L q (t 1 ,t 2 ; W 1,p 0 (Ω)) := denote an open ball with radius ρ > 0 centered at some x 0 ∈ R n . Let E ⊂ R n be a bounded domain. For a real number k and for a function v in L 1 (E) we define the truncation of v by For a measurable function v in L 1 (E) and a pair of real numbers k < l, we set Let z = (x, t) ∈ R n × R be a space-time variable and dz = dxdt be the space-time volume element.

Technical tools
We first recall the following De Giorgi's inequality (see [6]).

Proposition 2.1 (De Giorgi's inequality).
Let v ∈ W 1,1 (B) and k, l ∈ R satisfying k < l. Then there exists a positive constant C depending only on p, n such that Let q = np/(n − p) − 1 as before. Following [6], we define the auxiliary function for u ≥ 0 and k ≥ 0. Changing a variable η = ξ 1/q , we have Then we formally get If k = 0, we abbreviate as Let 0 < t 1 < t 2 ≤ T and let K be any domain in Ω. We denote a parabolic cylinder by K t 1 ,t 2 := K × (t 1 , t 2 ). We recall the Sobolev embedding of parabolic type.
). There exists a constant C depending only on n, p, r such that for every v ∈ L ∞ (t 1 , t 2 ; L r (K)) ∩ L p (t 1 , t 2 ; W 1,p 0 (K)) We next use so-called fast geometric convergence which will be employed later on many times. See [6] for details. Lemma 2.3 (Fast geometric convergence, [6]). Let {Y m } ∞ m=0 be a sequence of positive numbers, satisfying the recursive inequlities where C, b > 1 and α > 0 are given constants independent of m. If the initial value Y 0 satisfies We also need a fundamental algebraic inequality, associated with the p -Laplace operator (see [5]). Lemma 2.4. For all p ∈ (1, ∞) there exist positive constants C 1 (p, n) and C 2 (p, n) such that for all ξ, η ∈ R n

10)
where dot · denotes the inner product in R n . In particular, if p ≥ 2, then 3 Doubly nonlinear equations of p-Sobolev flow type Let T ≤ ∞. We study the following a doubly nonlinear equation of p-Sobolev flow type: where u = u(x, t) : Ω T → R be unknown real valued function, and c and M are nonnegative constant and positive one, respectively. This section is devoted to some a priori estimates of a weak solution to (3.1). Firstly, we recall the definition of weak solution of (3.1).
A measurable function u defined on Ω × [0, T ] is called a weak solution to (3.1) if it is simultaneously a weak sub and supersolution; that is, Similarly, Definition 3.2. A measurable function u defined on Ω T is called a weak solution of (1.1) if the following (D1)-(D4) are satisfied.

Nonnegativity and boundedness
We next claim that a weak supersolutions to (3.1) are nonnegative, i.e., they satisfy the weak minimum principle.

3) is bounded from below aŝ
Taking the limit in (3.3) as δ ց 0 and t 1 ց 0, and combining (3 and, by Gronwall's lemma,ˆΩ ∈ Ω T and the claim is verified.
We next show the boundedness of the solution.
. Let 0 < t 1 < t ≤ T and σ t 1 ,t be the same time cut-off function as in the proof of Proposition 3.4. The function e −ct σ t 1 ,t φ δ (u) is an admissible test function in (D2) in Definition 3.1. Choose a test function as e −ct σ t 1 ,t φ δ (u) in (D2) in Definition 3.1 to havê The first term on the left of (3.9) is computed aŝ Since, on the support of φ δ , {u > e ct/q M}, the second term is estimated as Gathering (3.9), (3.10) and (3.11), we obtain Taking the limit as δ ց 0 in (3.12), by the Lebesgue dominated convergence theorem, we have that as t 1 ց 0. Hence, we pass to the limit as t 1 ց 0 in (3.13) to havê T ], and we arrive at the assertion.

Comparison theorem
We We Theorem 3.6 (Comparison theorem, [1]). Let u and v be a weak supersolution and subsolution to (3.1) in Proof. As before, for a small δ > 0, let us define the Lipschitz function φ δ by Notice that and thus, Subtract (3.14) from (3.15) in Lemma 2.4 to obtain We find that the first term on the right hand side of (3.16) is bounded above as for a positive constant c ′ . Thus (3.16) and (3.17) lead tô Since ∂ t (|u| q−1 u) and ∂ t (|v| q−1 v) belong to L 2 (Ω T ), by the Lebesgue's dominated convergence theorem, we can take the limit as δ ց 0 in (3.18) and then, as t 1 ց 0 to obtainˆΩ where we used that φ δ (v − u) → χ {v>u} as δ ց 0 and that, from (2.10), u ≥ v is equivalent to |u| q−1 u ≥ |v| q−1 v and, by the initial trace condition, Thus Gronwall's lemma yields that in Ω, 0 ≤ t ≤ T . Hence the proof is complete.

Caccioppoli type estimates
We present the Caccioppoli type estimates, which have a crucial role in De Giorgi's method (see Section 4). From Proposition 3.4 we find that if u 0 ≥ 0 in Ω, a weak solution u of (3.1) is nonnegative in Ω T . Thus we can consider(3.1) as In what follows, we always assume that u 0 ≥ 0 in Ω and address (3.1') in place of (3.1). Let K be a subset compactly contained in Ω, and 0 < t 1 < t 2 ≤ T . Here we use the notation K t 1 ,t 2 = K × (t 1 , t 2 ). Let ζ be a smooth function such that 0 ≤ ζ ≤ 1 and ζ = 0 outside K t 1 ,t 2 . By use of A + (k, u) and A − (k, u), the local energy inequality can be derived.
Using the formula (2.6), the first term on the left hand side of (3.21) is computed as By use of Young's inequality, the second term on the left hand side of (3.21) is estimated from below by We gather (3.21), (3.22) and (3.23) to obtain, for any t ∈ (t 1 , t 2 ), Thus, we arrive at the conclusion.
The following so-called Caccioppoli type estimate follows from Lemma 3.7.
Proposition 3.8 (Caccioppoli type estimate). Let k ≥ 0. Let u be a nonnegative weak supersolution of (3.1). Then, there exists a positive constant C depending only on p, n such that ess sup The lower boundedness is obtained as follows: where, in the last line, we use k > k − u ≥ 0 since 0 ≤ u ≤ k/2. Also, the upper boundedness follows from Gathering Lemma 3.7, (3.26), (3.27) and (3.28), we arrive at the conclusion.

Expansion of positivity
In this section, we will establish the expansion of positivity of a nonnegative solution to the doubly nonlinear equations of p-Sobolev flow type (3.1').
We make local estimates to show the expansion on space-time of positivity of a nonnegative weak (super)solution of (3.1'). For any positive numbers ρ, τ and any point z 0 = (x 0 , t 0 ) ∈ Ω T , we denote a local parabolic cylinder of radius ρ and height τ with vertex at z 0 by For brevity, we write Q(τ, ρ) as Q(τ, ρ)(0).

Expansion of interior positivity I
In this subsection we will study expansion of local positivity of a weak solution of (3.1').
Proof. We may assume z 0 = 0 as before. By Proposition 4.1, there exist positive numbers δ, ε ∈ (0, 1) such that for all t ∈ [0, δL q+1−p ρ p ]. Set θ = δL q+1−p and let ζ = ζ(x) be a piecewise smooth cutoff function satisfying From (4.13) and the Caccioppoli type inequality (3.25), applied for the truncated solution (k j − u) + over Q 4ρ with the level k j = 1 2 j εL (j = 0, 1, . . .), we obtain (4.14) Here we note that the constant C depends only on n, p and, in particular, is independent of ρ, L. On the other hand, applying De Giorgi's inequality (2.3) in Proposition 2.1 to k = k j+1 and l = k j , we have, for all t, 0 ≤ t ≤ δL q+1−p ρ p = θρ p , Let A j (t) := B 4ρ ∩ {u(t) < k j } and then, by (4.12), it holds that Combine (4.16) with (4.15) to have that Integrating above inequality (4.17) in t ∈ (0, θρ p ) yields where we put |A j | :=ˆθ By use of Hölder's inequality, (4.14) and (4.18), we have and thus, (4.20) Let J ∈ N be determined later. Summing (4.20) over j = 0, 1, . . . , J − 1, we obtain (4.21) Indeed, by use of |A 0 | ≥ |A j | ≥ |A J | for j ∈ {0, 1, . . . , J}, we find that Therefore, from (4.21), it follows that Thus, for any ν ∈ (0, 1), we choose sufficiently large J ∈ N satisfying (4.23) Here we note that J depends only on p, n, α, δ and ν. We finally take ε ν = ε 2 J and then (4.22) yields that which is the very assertion.  Under such choice as above we note that k J = δ 2 I+J 1 q+1−p L and obtain that which is a positive integer. Following a similar argument to [7, p.76], we next divide Q 4ρ (z 0 ) into finitely many subcylinders. For any ν ∈ (0, 1), let J be determined in (4.23). We divide Q 4ρ (z 0 ) along time direction into parabolic cylinders of number s 0 := 2 I+J with each timelength k q+1−p J ρ p , and set Then there exists a Q (ℓ) which holds that The following theorem enables us to have the positivity of a solution of (3.1') in a small interval. It plainly holds true that The cutoff function ζ is taken of the form ζ(x, t) = ζ 1 (x)ζ 2 (t), where ζ i (i = 1, 2) are Lipschitz functions satisfying Thus, applying the local energy inequality (3.25) over B m and Q m to the truncated solution (κ m − u) + and above ζ, we obtain where we used that where we note that q + 1 = p(n+q+1) n in the second line. The left hand side of (4.29) is estimated from below aŝ (4.30) Hence, by (4.29) and (4.30), we havê Dividing the both side of (4.31) by |Q m+1 | > 0, we have where the above inequality (4.32) is rewritten as then Y m → 0 as m → ∞.
Proof. Suppose that (x 0 , t 0 ) be the origin, as before. Let L := inf B 4ρ u(0) > 0. Since |B ρ ∩{u(0) ≥ L}| = |B ρ |, by Proposition 4.1, there exist positive numbers δ, ε depending only on n, p and independent of L such that for all t ∈ [0, δL q+1−p ρ p ]. Let Q θ 4ρ (z 0 ) = B 4ρ (x 0 ) × (0, θρ p ) ∈ Ω T , where 0 < θ < δL q+1−p ρ p is a parameter determined later. By Lemma 4.2 with some minor change, for any ν ∈ (0, 1) there exists a positive number ε ν depending only on p, n, δ and ν such that Here we notice that in the proof of Proposition 4.1 and Lemma 4.2, we do not need to use the cutoff on time. In the following we modify the proof of Theorem 4.4 to that without any cutoff on time.
By translation, we may assume x 0 = 0 as before. For m = 0, 1, . . . , we put and J is to be determined in (4.23), and also set Clearly it holds true that From the Caccioppoli type inequality (3.25), applied for the truncated solution Q m to the truncated (κ m − u) + again, we obtain that ess sup where, in the second line, we used (κ m − u(0)) + = 0 in B m . By the very same argument as in the proof of Theorem 4.4, we have that

Expansion of interior positivity II
We continue to study the expansion of positivity of a nonnegative solution. Let Ω ′ be a subdomain contained compactly in Ω. Using Theorem 4.4 and a method of chain of finitely many balls as used in Harnack's inequality for harmonic functions, which is so-called Harnack chain (see [8,Theorem 11, and [4,15] in the p-parabolic setting), we have the following theorem. Here we use the special choice of parameters, as explained before Theorem 4.4.
Proof. We will prove the assertion in five steps.

Positivity near the boundary
We next study the positivity of the solution to the doubly nonlinear equations of p-Sobolev flow type (3.1') near the boundary. In what follows, assume that the bounded domain Ω satisfies the interior ball condition, that is, for every boundary point ξ ∈ ∂Ω, there exist a point x 0 ∈ Ω and some ρ > 0 such that where B ρ (x 0 ) denotes the closure of B ρ (x 0 ). Proof. We will follow the similar idea as [24]. Since, Ω satisfies the interior ball condition, we have, for every boundary point ξ ∈ ∂Ω, Take ρ ′ ∈ (0, ρ) and define the annulus where r := |x − x 0 | < ρ and α > 0 is to be determined later. Since we have ∆ p v = (p − 2)|∇v| p−4 e −3αr 2 8α 3 r 2 (−1 + 2αr) + |∇v| p−2 e −αr 2 2α(−n + 2αr 2 ) (4.55) and thus, we can choose a sufficiently large α so that where α is chosen depending on ρ and ρ ′ . Therefore, by ∂ t v q = 0 Now, let m := min min u . We will show that mv(x, t) is a lower comparison function for the solution. We note that the solution u is uniformly (Hölder) continuous in Ω T = Ω × (0, T ) (see Section 5.2 below), again, we can choose α > 0 to be so large that, on the initial boundary A × {t = 0}, u(x, 0) = u 0 (x) ≥ mv(x, 0) = m(e −αr 2 − e −αρ 2 ).
From (4.56), (4.57) and (4.58), we find that where A T := A × (0, T ) and ∂ p A T is the parabolic boundary of A T and thus, we have that mv(x, t) is lower comparison function for u in A T = A × (0, T ). By Theorem 3.6, we arrive at u ≥ mv > 0 in A T . In what follows, we consider the p-Sobolev flow (1.1). We first notice the nonnegativity of a solution of the p-Sobolev flow and its proof. Proof. Let 0 < t 1 < t ≤ T be arbitrarily taken and fixed. Let σ t 1 ,t be the same Lipschitz cut-off function on time as in the proof of Proposition 3.4. The function −(−u) + σ t 1 ,t is an admissible test function in (D2) of Definition 3.2, since ∂ t (|u| q−1 u) ∈ L 2 (Ω T ) by (D1) of Defintion 3.2 and, −(−u) + σ t 1 ,t is in L q+1 (Ω × (t 1 , t)). Thus, we havê Applying the very same argument as in the proof of Proposition 3.4 to (5.2), we obtain that q q + 1ˆΩ (−u(t)) q+1 + dx ≤ˆt 0 (λ(τ )) +ˆΩ (−u(τ )) q+1 + dxdτ.
We now state the fundamental energy estimate.
Proposition 5.2 (Energy equality). Let u be a nonnegative solution to (1.1). Then the following identities are valid: In particular, The proof of this proposition is postponed, and will be given in Appendix B. Proof. By (5.3) we have that λ(t) ≤ λ(0). Therefore, u is a weak subsolution of (3.1) with M = u 0 L ∞ (Ω) and c = λ(0). The result then follows by Proposition 3.5.
In general, the solution to (3.1') may vanish at a finite time, however, under the volume constraint as in (1.1), the solution may positively expand in all of times (see Corollary 4.8). This is actually the assertion of the following proposition.

Hölder and gradient Hölder continuity
In this section, we will prove the Hölder and gradient Hölder continuity of the solution to p-Sobolev flow (1.1) with respect to space-time variable.
Suppose u 0 > 0 in Ω. Then by Propositions 5.4 and 5.3, for any Ω ′ compactly contained in Ω and T ∈ (0, ∞), we can choose a positive constantc such that Under such positivity of a solution in the domain as in (5.5), we can rewrite the first equation of (1.1) as follows : Set v := u q , which is equivalent to u = v 1 q and put g := 1 q v 1/q−1 and then, we find that the first equation of (1.1) is equivalent to and thus, v is a positive and bounded weak solution of the evolutionary p-Laplacian equation (5.6). By (5.5) g is uniformly elliptic and bounded in Ω ′ ∞ . Then we have a local energy inequality for a local weak solution v to (5.6) in Appendix C.1 (see [6]).
The following Hölder continuity is proved via using the local energy inequality, Lemma C.1 in Appendix C.1 and standard iterative real analysis methods. See [6, Chapter III] or [27, Section 4.4, pp.44-47] for more details.
By a positivity and boundedness as in (5.5) and a Hölder continuity in Theorem 5.6, we see that the coefficient g p−1 is Hölder continuous and thus, obtain a Hölder continuity of its spacial gradient. The outline of proof of Theorem 5.7 is presented in Appendix C.
By an elementary algebraic estimate and a interior positivity, boundedness and a Hölder regularity of v and its gradient ∇v in Theorems 5.6 and 5.7, we also have a Hölder regularity of the solution u and its gradient ∇u. A Some fundamental facts A.1 L 2 estimate of the time derivative We will show the existence in L 2 (Ω T ) of time-derivative for a weak solution to (1.1).
Lemma A.1. Let u be a nonnegative solution to (1.1).Then there exists ∂ t u in a weak sense, such that ∂ t u ∈ L 2 (Ω T ).
In the proof above, we used the following lemma as for the convergence of Dirac measure. which is our claim.
Dividing above formula by t − t 1 , we have According to the volume preserving condition again, passing the limit as t ց t 1 in the formula above, we obtain that λ(t 1 ) =ˆΩ |∇u(x, t 1 )| p dx, which is our first assertion.
(ii) We notice the boundedness of the solution u of the p-Sobolev flow. This is shown as follows: By Proposition 5.2 (i) above, λ(t) = ∇u(t) p L p (Ω) and thus, λ(t) ∈ L ∞ (0, T ) by (D1) in Definition 3.2. We also have Proposition 5.2 (i) that (u) + is bounded in Ω T as in Proposition 5.3, and thus, u itself bounded by Proposition 5.1. Consequently, the function σ t 1 ,t ∂ t u is an admissible test function in (D2) of Definition 3.2 by Lemmata A.1 and A.3. We now take a test function as σ t 1 ,t ∂ t u in (D2) of Definition 3.2 and then Ωt 1 ,t ∂ t (u q )σ t 1 ,t ∂ t u dz +ˆΩ where the manipulation in the second and third lines are justified by Lemma A.3 in Appendix A. By the volume conservation´Ω u(x, t) q+1 = 1, t ≥ 0, the right hand side of (B.4) is calculated aŝ From (B.5), (B.6) and (B.7), it follows that qˆΩ t 1 ,t u q−1 (∂ t u) 2 dz + 1 p λ(t) − 1 p λ(t 1 ) = 0.
Letting t 1 = 0, we have the desired result.
C.2 Outline of proof of Theorem 5.7 We recall the outline of proof of Theorem 5.7 here.
Proof. By the Hölder continuity in Theorem 5.6, the equation (5.6) is an evolutionary p-Laplacian system with Hölder continuous elliptic and boudedness coefficients g and lower order terms v. We apply the gradient Hölder regularity for the evolutionary p-Laplacian systems with lower order terms in [19,Theorem 1,p.390] (also see [14]).
Here the so-called Campanatto's perturbation method is applied to the gradient Hölder regularity for the evolutionary p-Laplacian systems with Hölder coefficients and lower order terms. We also refer to the book in [6, Theorem 1.1, p.245].