GLOBAL EXISTENCE FOR A THIN FILM EQUATION WITH SUBCRITICAL MASS

. In this paper, we study existence of global entropy weak solutions to a critical-case unstable thin ﬁlm equation in one-dimensional case h t + ∂ x ( h n ∂ xxx h ) + ∂ x ( h n +2 ∂ x h ) = 0 , where n ≥ 1. There exists a critical mass M c = 2 √ 6 π 3 found by Witelski et al. (2004 Euro. J. of Appl. Math. 15, 223–256) for n = 1. We obtain global existence of a non-negative entropy weak solution if initial mass is less than M c . For n ≥ 4, entropy weak solutions are positive and unique. For n = 1, a ﬁnite time blow-up occurs for solutions with initial mass larger than M c . For the Cauchy problem with n = 1 and initial mass less than M c , we show that at least one of the following long-time behavior holds: the second moment goes to inﬁnity as the time goes to inﬁnity or h ( · ,t k ) (cid:42) 0 in L 1 ( R ) for some subsequence t k → ∞ .

For the Cauchy problem with n = 1 and initial mass less than Mc, we show that at least one of the following long-time behavior holds: the second moment goes to infinity as the time goes to infinity or h(·, t k ) 0 in L 1 (R) for some subsequence t k → ∞.

1.
Introduction. This paper deals with the following critical-case long-wave unstable thin film equation where h(x, t) denotes the height of the evolving free-surface and n ≥ 1 is the exponent of the mobility. We impose the following initial condition The model (1)- (2) can be used to describe pattern formation in physical systems that involve interfaces, c.f. [36]. Here we consider two classes of initial data: and 1462

JIAN-GUO LIU AND JINHUAN WANG
We list below some important properties of the Cauchy problem (1)- (3). First, non-negative solutions h(x, t) to (1)-(2) satisfy conservation of mass, i.e., Second, the equation (1) can be recast in a variational form where µ is the chemical potential. It is given by the variation of the free energy functional: In thin film equations, the negative chemical potential is referred to as the pressure, p = −µ = ∂ xx h + 1 3 h 3 . The variation equation (5) induces the following energydissipation relation for h ≥ 0: Third, following Bernis and Friedman [5], for any 0 ≤ h ≤ M we define a function which can be represented exactly by the following form Define another free energy functional Noticing that G M (h) = 1 h n , and from the equation (1), we deduce d dt According to (11), one has that G M (h) is bounded if it is initially so. This bound can be used to obtain non-negativity of solutions to the model (1)- (2). The details can be found in Subsection 2.2 below.
Notice that the second term in (1) involves the fourth order derivative and it is a stabilizing term. The third term is a destabilizing second derivative term. For short wave solutions, the stabilizing term dominates the destabilizing one so that the linearized equation of (1)-(2) is well-possed. However, for long wave solutions, the destabilizing term may dominate the stabilizing one such that the long wave instability may occur. The competition between stabilizing term and destabilizing term is represented by opposing signs for the corresponding terms in the free energy (6).
To the best of our knowledge, there are no results on existence of weak and strong solutions to (1) with unstable diffusion in multi-dimension before year 2014. In 2014, Taranets and King [35] proved local existence of nonnegative weak and strong solutions in a bounded domain Ω with smooth boundary in R d under a more restrictive threshold M 0 <M d . In one dimension (d = 1), their threshold is Below we review some results on the long-wave unstable thin film equation in one dimensional case.
A slightly more general version of the long-wave unstable thin film equation is given by and it was studied by Bertozzi, Pugh, and others in a series of papers [6]- [9], [13,26,34,36]. Here n > 0 denotes the exponent of the mobility, and m > 0 is the power of the destabilizing second order term. The classification for the critical (m = n + 2), super-critical (m > n + 2), and sub-critical (m < n + 2) cases can be obtained by the mass invariant scaling h λ = λh(λx, λ n+4 t). These three regimes were first introduced and studied by Bertozzi and Pugh [8] for (14). Details are discussed below.
In the subcritical case m < n+2, for relatively thick films (large λ) the stabilizing term (a pre-factor λ n+5 for this term in the re-scaling) dominates the destabilizing one (a pre-factor λ m+3 for this term in the re-scaling) and the blow-up is precluded. For relatively thin films (small λ), the destabilizing term dominates the stabilizing one and prevents spreading. Global existence and finite speed of propagation of the support of solutions were proved for any large initial data in [8] (also see [13]). On the contrary, for the supercritical case m > n + 2, the destabilizing term dominates the stabilizing one for relatively thick films so that solutions may blow up in finite time. For example, in [9] blow-up phenomenon is given for n = 1. In [13] blow-up phenomenon is obtained for n ∈ (0, 2). The reference [14] shows blow-up phenomenon and mass concentration for n ∈ (0, 3). For relatively thin films, the stabilizing term dominates the destabilizing one and solutions have infinite-time spreading.
For m = n + 2, the fourth order stabilizing term is balanced by the second order destabilizing term and this case is called the critical case. For the critical-case model, there is a critical mass M c that can be used to distinguish between global existence and finite time blow-up. As discussed below for the special case with n = 1, both global existence and blow-up may occur depending on whether the initial mass is less than or larger than the critical mass M c . Furthermore, let h(x, t) be a solution to (1)-(2), then the mass invariant re-scaling λh(λx, λ n+4 t) is also a solution to (1)-(2) (see [34]). Using this property, Slepčev and Pugh [34] proved that (14) cannot have self-similar blow-up solutions if n ≥ 3 2 . For 0 < n < 3 2 , there are compactly supported, symmetric, self-similar solutions that blow up in finite time. They also showed that any self-similar solution must have mass less that M c . In [36], for the n = 1 case of equation (1) Witelski, Bernoff, and Bertozzi studied infinite-time self-similar spreading behavior below the critical mass, and finite time blow-up self-similar solutions beyond the critical mass by numerical simulations and asymptotic analysis.
Another way to understand the critical mass M c is through the steady solution and the Sz. Nagy inequality as we discuss below. The critical mass sometimes is given by the mass of equilibrium solutions. For equilibrium solutions h eq , the dissipation term on the right side of (7) is zero, and the equilibrium chemical potential is given by x ∈ supp h eq , µ eq (x) ≥C, otherwise (15) for some constantC. In other words, equilibrium solutions are Nash equilibria [12]. Denote an equilibrium profile as h α with parameter α as the height at the center peak (assumed to be located at x = 0), i.e., h α (0) = α, ∂ x h α (0) = 0. From (15), one can solve for the equilibrium profile h α within its support Together with the decay property at infinity, one knows that a first integral is given by Evaluating the above integral at x = 0 and using h α (0) = α, ∂ x h α (0) = 0, one knows thatC = −α 3 /12. As shown in (i) below, h α has a compact support. Outside of its support we have µ eq ≡ 0 ≥C. Thus (15) holds. (17) can be recast as After some simple calculations, we have the following properties for h α : (i) The solution h α (x) has compact support, i.e., there exists x α > 0 such that h α (x) > 0 for |x| < x α and h α (x) = 0 as |x| ≥ x α , and x α satisfies where B(x, y) is the Beta function. From the above, one knows that the support of the solution to (16) can be arbitrarily narrow provided that α is large.
Noticing the second derivative ∂ xx h α is discontinuous at x α (hence µ eq is also discontinuous at x α ), i.e., =: M c , which is a universal constant independent of α and is the critical mass. (iv) F(h α (x)) = 0.
In fact, multiplying h α to (16) and integrating in (−x α , x α ), we get On the other hand, integrating (18) in (−x α , x α ), it holds Subtracting (19) from (20) gives that If the initial mass M 0 is larger than the critical mass M c , then there is an initial datum h 0 such that the solution to (1)-(2) blows up in finite time. In fact, taking where we have used (21) in the second equality. Let m 2 (t) := R |x| 2 h(x, t)dx be the second moment. Then for n = 1, a simple computation gives that the time derivative of the second moment satisfies d dt which implies that there exists a finite time t * such that m 2 (t * ) = 0 if the initial second moment is finite. With some computations we obtain (c.f. [12, formula Hence by (25) and the fact m 2 (t * ) = 0, we know that there is a T max ≤ t * such that lim sup t→Tmax h(·, t) L 4 = ∞.
Some blow-up results were also given in papers [9,36]. Consequently, M c can be used as the critical mass to distinguish between global existence and finite time blow-up for the thin film equation (1) with n = 1. For n > 1, (24) does not hold. The question of finite time blow-up is still open. In this paper we will prove that for n ≥ 1 and initial data satisfying M 0 < M c , there exists a global non-negative weak solution to the models (1)-(2) with the two classes of initial data (3) and (4). Throughout this paper, we will use C to denote positive constants, which may be different for each calculation.
Before defining weak solutions, we need to review literatures on possible singularities that may appear to the higher derivatives of weak solutions. For the thin film equation without the long-wave unstable term, Giacomelli, Knüpfer, and Otto [18], John [23], and Gnann [19] showed that for n = 1 solutions are smooth up to the free boundary. However, for the thin film equation with the long-wave unstable term, the control of solutions at the free boundary is subtle and we do not know if solutions have singularities or not. If there are singularities, however, they will occur in the set {h = 0}. Hence in the definition of an entropy weak solution, we need to define a set By Proposition 3 below, we know that P T is an open set and we can define distribution functions on P T . Now we give the following definition of an entropy weak solution to (1)-(2) with the periodic initial data (4).

Definition 1. (Entropy weak solution)
We say that a non-negative function and for any fixed T > 0, is an entropy weak solution to (1)-(2) provided that (i) For any 2L-periodic function φ ∈ C ∞ R × [0, T ] the following identity holds is a non-increasing function in t and satisfies the following energydissipation inequality The definition for an entropy weak solution to the Cauchy problem is similar. We omit details here. [8] introduce the notion of the "BF weak solution" to differentiate this solution from a weak solution in some sense of distributions in their paper. They referred to solutions of (12) with some regularity as BF weak solutions (see [8,Definition 3.1]). Later on for (12) with 0 < n < 3, it was proved in [3,4,6,7] that BF nonnegative weak solutions satisfy h(·, t) ∈ C 1 [−L, L] for almost all t > 0. We shall remark that this kind of solutions was referred to as strong solutions in literature such as [3,4]. To avoid confusion, we consistently use entropy weak solutions in this paper.

Remark 1. Bertozzi and Pugh
In Section 2 and Section 3, we will prove the following global existence theorem for the periodic problem (1)-(2).
Next, in Section 4, we will consider global existence of entropy weak solutions to the Cauchy problem (1)-(2) with initial data (3). Global existence of the Cauchy problem with compactly supported initial data can be proved by using (i) periodic extension, (ii) finite speed of propagation of the support of solutions, (iii) uniform estimates for H 1 -norm of solutions. The result is given as follows.
Theorem 3. Assume n = 1 and initial data h 0 satisfying (3), Then there is a global non-negative entropy weak solution to (1)- (2) and it satisfies the following uniform estimate Furthermore, we will prove the following long-time behavior in Section 4.
Theorem 4. Assume n = 1 and initial data h 0 satisfying (3), be a global non-negative entropy weak solution of (1)-(2) given by Theorem 3. Then at least one of the following results holds 2. Local existence and non-negativity. In this section, we will prove local existence and non-negativity of entropy weak solutions to the thin film equation (1)- (2) with initial data (4).

Local existence for a regularized problem. Define a standard mollifier
We consider local existence of solutions for the following regularized problem in a period domain (−L, L) Here is a H 1 -approximation sequence of initial data h 0 (x) and for ε sufficient small, it satisfies Here the first inequality used the embedding theorem and c depends only on h 0 H 1 (−L,L) . We refer to [1] for a different regularized method. Now we give a definition of entropy weak solutions to the regularized problem (36).
(Entropy weak solutions) For any fixed ε > 0 and T > 0, we say a 2L-periodic function is an entropy weak solution of the equation (36) provided that (i) For any 2L-periodic function φ ∈ C ∞ R × [0, T ] the following equality holds (ii) The energy-dissipation equality holds where t ∈ [0, T ].
the regularized problem (36) has an entropy weak solution h ε and it satisfies the following uniform in ε estimates and for any ( , the following Hölder continuity holds where C is independent of ε.
The proof of Proposition 2 is standard, for completeness we provide details in Appendix A.
Remark 2. As noticed by Bernis and Friedman [5], for n ≥ 4, regularized solutions h ε are positive, thus we can use directly the Sz. Nagy inequality and the energy functional for h ε to prove global existence of weak solutions. However, for 1 ≤ n < 4, there is no non-negativity of h ε . Instead, we will show that limit functions h of h ε are non-negative local weak solutions, and use the same strategy as above to obtain global existence of entropy weak solutions.

2.2.
Local existence of solutions for (1)- (2). In this subsection, we will prove local existence of solutions to the equation (1)-(2) with initial data (4). In order to prove non-negativity and uniform bound of ∂ xx h ε L 2 (0,T ;L 2 (−L,L)) , analogous to (8), we introduce the following entropy density functions where M is taken so that A simple computation gives the following lemma Lemma 1. The functions g M,ε (x) and G M,ε (x) in (46) satisfy the following properties 1. If x ≤ M , then for any ε > 0, Proof. The result (1) is obvious. We only need to prove (2).
Now we give local existence of entropy weak solutions to the periodic problem (1)-(2) and its non-negativity. The results on non-negativity of weak solutions were mostly proved by using Hölder continuity in previous papers. In the following proposition, we provide an alternative proof in Sobolev space and do not use Hölder continuity in hope to generalize the results to multi-dimensional thin film equations. Proof.
By Proposition 2 and Lemma 2, we know that solutions to (36) satisfy the following uniform estimates where the constant C is independent of ε. Consequently, as ε → 0, there exists a subsequence of h ε (still denoted by h ε ), and h satisfying (63)-(64) such that Using the Lions-Aubin lemma [27], there exists a subsequence of h ε (still denoted by h ε ) such that Hence h ε → h a.e. as ε → 0.
From (45), we have for any x 1 , x 2 ∈ [−L, L] and t 1 , t 2 ∈ [0, T ], the following estimate holds Step 2. Non-negativity of h almost everywhere. In this step, we use the contradiction method to prove that the limit function h is non-negative in (−L, L) × [0, T ] almost everywhere. If not, we have |{(x, t)|h(x, t) < We know that |D α0 | > 0, and t∈Dα 0 {t} × C α0 (t) ⊂ A. Due to (68), there exists a subsequence {ε k } ∞ k=1 of ε (ε k > 0, and ε k → 0 as k → +∞) such that for any (x, t) ∈ t∈Dα 0 {t} × C α0 (t), we have We define From the definition of A k and (71), we know that Noticing that for any k, B ∩ A k = ∅, from the definition of these two subsets, we On the other hand, for n = 1, by the formula (50) in Lemma 1, we know for any Thus by (73), we deduce lim sup which is a contradiction with the uniform estimate For n > 1, all the arguments above are exactly same except ln 1 ε in (75) and (74) is replaced by ε 1−n in view of using (51) in Lemma 1.
Step 3. h is a local solution of (1)- (2). Now we show that the non-negative limit function h in (65) is a local entropy weak solution to (1)-(2). Passing to the limit for h ε in (41) as ε → 0, we can obtain for any 2L-periodic function The details of the proof for (76) are given below. The convergence of the left side term and the second term on the right side in (41) can be directly obtained from the convergence of (66), (67) and (68), i.e., The limit for the first term on the right side in (41) is given by the following claim.
Claim. We have as ε → 0 where P T is defined in (26).
Hence we have Therefore for any η > 0, take δ j = η 2 j , we have the following relations which give Thus taking η → 0, we have We decompose I 1 as follows Similar to J 1 , we get To estimate L 2 , using the fact h ε k > δ 2 in D δ , and (64), we have Thus we have which implies that there exists a subsequence of h ε k (still denote h ε k ) such that Together with (68), we have for k → ∞, Hence from (81), (86), (93) and (94), we know that there exists K > 0 such that as k > K, Noticing (92), we get Therefore the following limit holds i.e., we obtain (79). This completes the proof of the claim. Therefore (77), (78) and (79) imply (76), i.e., h satisfies the weak form (30) in Definition 1.
Next we will prove that h is an entropy weak solution. The solution h ε of the regularized problem (36) satisfies the energy-dissipation equality (42). Here we recall it, By the convergent relations (65)- (68) and (79), we know that if ε → 0, Hence we get From (78) and (79), we have Together with F(h ε0 ) → F(h 0 ), we have that the non-negative function h satisfies the following energy-dissipation inequality Since F(h ε (·, t)) is decreasing in (0, T ) from (42), then Helly's selection theorem implies that F(h(·, t)) is also a decreasing function in (0, T ). Hence h is an entropy weak solution in [0,T] as that in Definition 1.
3. Global existence. In this section, we will prove global existence of entropy weak solutions to the problem (1)-(2) under the sharp initial condition M 0 < M c in the periodic domain. Namely, we prove that local weak solutions given by Proposition 3 are indeed global solutions. Firstly, we use the energy-dissipation inequality (96) and the Sz. Nagy inequality (22) to prove the following lemma. Proof. From (96) and non-negativity of h, we have To apply the Sz. Nagy inequality (22) in the periodic setting, we use a trick from [36] below. Suppose that h achieves its minimum h min at x * (t). Hence h min ≥ 0. Denoteh otherwise.
Using the Sz. Nagy inequality (22) forh(x), we have By the definition ofh(x), we obtain A simple computation gives Using Young's inequality, we get with some ν > 0. Hence from (100) and (101), we know that From conservation of mass, we have which means Thus we deduce (98) and (104), we have The formula (103) implies Hence (97)  At t = T 1 , h(x, t) still satisfies above (i), (ii) and (iii). Taking T 1 as a new initial time and h(·, T 1 ) as a new initial datum, we can obtain a non-negative entropy weak solution, which exists in t ∈ [T 1 , 2T 1 ]. We can continue this process and obtain a global solution in R + , and it satisfies h L ∞ (R+;H 1 (−L,L)) ≤ C(M 0 , F(h 0 )). Therefore again using the Sz. Nagy inequality (22), we have (32). Furthermore, estimates (27)-(31) hold for any T > 0. This completes the proof of Theorem 1.
Proof of Theorem 2. For n ≥ 2, we further assume L −L G M (h 0 ) dx < ∞. By Lemma 2, we have that (52) and (53) still hold. Hence we can prove that there is a global non-negative entropy weak solution, which satisfies Hölder continuity (70). Now we only need to prove positivity of weak solutions for n ≥ 4 in part (ii) of Theorem 2.
Step 1. Positivity of h ε for ε sufficiently small.
From Proposition 2 and Lemma 2, we know that solutions h ε of the regularized problem (36) satisfy the space Hölder continuity (45) and for any fixed By (8), we know that for n ≥ 4 and 0 ≤ h ≤ M , it holds that Hence from (105) and (106) we have A simple computation gives that and x * +δ where the constant C is independent of ε. Hence (107) and (108) imply which is a contradiction for sufficiently small ε.
Step 2. Positivity of limit functions h. From Step 1, for any fixed T and sufficient small ε, there exists a minimum point and for any x ∈ [−L, L], it holds In other words, the thin film equation is nondegenerate and its solutions have further regularity h ∈ L 2 (0, T ; H 3 (−L, L)) and P T = (−L, L) × (0, T ).
Let h 1 and h 2 be two solutions to (1) 4. Global existence and long-time behavior for the Cauchy problem. In this section, we first prove global existence for the Cauchy problem (1)-(2) with n = 1 as stated in Theorem 3.

Proof of Theorem 3.
Step 1. Local existence of weak solutions.
Since the initial data h 0 has compact support, h 0 ∈ H 1 (R), and supp h 0 ∈ [−a, a], we take a 2(a + A 0 )− periodic extension function h 0,period defined in R such that h 0,period satisfies the assumptions of Theorem 1, where A 0 is a constant and is given by (121) below. Hence from Theorem 1, we know that there exists a global non-negative weak solution h period to the periodic problem (1)-(4) with initial data h 0,period and h period satisfies (27)- (29). Now we apply results on finite speed of propagation for support of solutions in [4,8,13] to h period within the periodic domain (−(a + A 0 ), (a + A 0 )). Let ζ(t) denote the right boundary of support of solutions h period within the periodic domain (−(a + A 0 ), (a + A 0 )). Following the results in [8, Lemma 3.7 and Lemma 3.8] or [13] and using the method provided by Bernis in [4, Theorem 5.1], we have that there is a positive constant T * such that if 0 < t < T * where β = 1 4λ+5 is a decreasing function of λ, and k = λ+1 4λ+5 is a increasing function of λ. As explained in [8], the unstable term in (1) is attractive and it shrinks the support of solutions. We omit details here.
Using (110) and taking T 1 ∈ [0, T * ] such that one has that for 0 ≤ t ≤ T 1 , In other words, the support of solutions h period to the local equation (1) in every periodic domain stays within the interior of the periodic domain for any t ≤ T 1 . Thus we use h period within one period to construct h as below, for any t ∈ [0, Clearly, h(x, t) is a local non-negative entropy weak solution for the Cauchy problem (1)-(2) with initial data h 0 , and satisfies the following regularities and the energy-dissipation inequality (96).
Step 3. Global existence under the sharp condition M 0 < M c .
In the above two steps, we have shown that 1.
Taking T 1 as a new starting time and h(x, T 1 ) ∈ H 1 as a new initial data and repeating the arguments used in Step 1 and Step 2, we can obtain a non-negative entropy weak solution h to the Cauchy problem (1)-(2) and it satisfies supp h(·, t) ⊂ (−(a + 2A 0 ), a + 2A 0 ) in t ∈ [T 1 , 2T 1 ]. At t = 2T 1 , (i), (ii) and (iii) are also true. Hence we can continue this process and obtain a global solution for the Cauchy problem in R + and it satisfies for any fixed T > 0 This completes the proof of Theorem 3.
Proof of Theorem 4. Let h(x, t) be a global non-negative entropy weak solution of (1)-(2) given by Theorem 3 with initial data h 0 satisfying (3), M 0 < M c and F(h 0 ) < ∞. Since M 0 < M c , the Sz. Nagy inequality (22) implies that F(h(·, t)) > 0 for any t ≥ 0. Noticing that the free energy is decreasing in time t, we know that there is a F ∞ such that lim On the other hand, a simple computation gives d dt which says that the second moment is increasing in t. Now we prove that at least one of (34) and (35) holds. Suppose that By (124), we have that there exists a constantC > 0 such that m 2 (t) ≤C for any t ∈ (0, ∞). In this case, we claim that there is a sequence t k and h ∞ such that as In fact, since h ∈ L ∞ (R + , L 1 ∩ H 1 (R)) and the second moment is finite, we have 1. ∀ ε > 0, there exists a R ε > 0 such that Hence taking we obtain |x|>Rε h 4 dx < ε.
Then there is a subsequence t k (without relabel) and h 1,∞ such that 2. For fixed R ε satisfying (127), we know that h(x, t k ) ∈ L ∞ (R + ; H 1 (B(0, R ε ))). Thus by the Sobolev embedding theorem, one obtains that there is a strong convergent subsequence, still denoted by h(x, t k ), and h 2,∞ such that Let h ∞ be the combination of h 1,∞ and h 2,∞ defined in R. Hence, from (128), (129) and (130), we have that there is a K such that if k ≥ K, then which proves our claim (126). Thus we have By uniform in time estimate (33), we know that there is a subsequence of t k (still denoted by t k ) such that as t k → ∞ Therefore (131) and (132) give Finally, noticing that h(·, t k ) L 1 (R) = M 0 , h(·, t k ) L 4 (R) ≤ C and the second moment is finite, from the Dunford-Pettis theorem we can get Hence Fatou's lemma implies We have two cases: (i) h ∞ = 0, (ii) h ∞ = 0. In the case (i), the formula (134) implies that there exists a subsequence t k such that h(·, t k ) 0 as t k → ∞, i.e., (35) holds. In the case (ii), by the inequality (22), we know F(h ∞ ) > 0. Hence (133) gives F ∞ > 0. Note that which contradicts with (125). Hence (34) holds. This finishes the proof of Theorem 4.
Appendix A. The proof of Proposition 2. Denoteh δε := J δ * h δε + c √ ε. J δ is defined in Subsection 2.1. We introduce the following further regularized problem by using the modified method in [−L, L] × [0, ∞) with the 2L-periodic boundary condition. Here we notice Step 1.

So, we have
where C is a constant independent of T, δ, ε and h δε . Solving the above ordinary differential inequality, we obtain This implies that there exists a T = T ( h 0 H 1 (−L,L) ) independent of δ and ε such that the following estimates hold h δε L ∞ (0,T ;H 1 (−L,L)) ≤ C, where constants C are independent of δ and ε. A direct consequence of (138) and (140) is that where constants C are also independent of δ and ε.
Step 4. h ε is space-time Hölder continuous uniformly in ε. Indeed, for any x 1 , x 2 ∈ (−L, L) and t 1 , t 2 ∈ (0, T ), the estimate (45) holds. The property was proved in the paper [5]. For completeness, we provide the proof in Appendix B.
Appendix B. Hölder continuity. In this Appendix, we consider the uniform in ε Hölder continuity (70) of weak solutions h ε to the regularized problem (36). Before proving the main result, we first show the following two lemmas.
For any x 0 ∈ (−L, L), constructing an auxiliary function where 0 < α < 1 and K > 0 are two constants to be determined later, and a 0 (x) ∈ C ∞ 0 (R) is defined by satisfying |a 0 (x)| ≤ C. From (152), we know Taking K > T α L such that where C depends only on T , L and h L ∞ (0,T ;H 1 (−L,L)) .