Electronic Journal of Differential Equations, Vol. 2024 (2024), No. 58, pp. 1–14. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu DOI: 10.58997/ejde.2024.58 MILD SOLUTIONS TO FOURTH-ORDER PARABOLIC EQUATIONS MODELING THIN FILM GROWTH WITH TIME FRACTIONAL DERIVATIVE QIANG LIU, WANYU ZHU, HAILONG YE Abstract. In this article, we study initial-boundary problems for fourth-order nonlinear parabolic equations modeling thin film growth with Caputo-type time fractional derivative. By means of the theory of abstract fractional calcu- lus and Lp − Lq estimates, we establish the existence and uniqueness of local mild solutions in the spaces C([0, T ];L βN 2−β (Ω)) with 1 < β < 2. Moreover, the local solutions can be extended globally if the initial data is sufficiently small. 1. Introduction and main result Thin film growth processes play a crucial role in various scientific and technolog- ical applications, ranging from semiconductor manufacturing to material sciences [14, 17, 23]. Understanding the dynamics of thin film growth is essential for opti- mizing the quality, stability, and functionality of thin films in these applications. Mathematical modeling provides a powerful tool to capture the intricate dynamics involved in such processes and to develop predictive models that guide experimental design and optimization. In recent years, there has been a growing interest in utilizing fractional calculus to describe and analyze complex phenomena that exhibit non-local and memory ef- fects. In this article, we focus on boundary value problems of fourth-order parabolic equation modeling thin film growth with time fractional derivative, cD α t u(t) + ∆2u = ∇ · f(∇u), x ∈ Ω, t > 0, ∂νu|∂Ω = ∂ν∆u|∂Ω = 0, t > 0, u(x, 0) = φ(x), x ∈ Ω. (1.1) Here Ω ∈ RN with N ≥ 2 is a bounded smooth domain, ∂Ω denotes the boundary of Ω and ν is the unit outer vector normal to Ω, cDα t is the Caputo derivative with order α ∈ (0, 1), which is defined by cD α t u(t) = 1 Γ(1− α) d dt ∫ t 0 (t− s)−α(u(s)− u(0)) ds, 2020 Mathematics Subject Classification. 35G25, 35K90. Key words and phrases. Thin-film equation; Caputo fractional derivative; mild solution; Mittag-Leffler functions. ©2024. This work is licensed under a CC BY 4.0 license. Submitted March 30, 2024. Published October 4, 2024. 1 2 Q. LIU, W. ZHU, H. YE EJDE-2024/58 where the Gamma function defined by Γ(λ) := ∫∞ 0 tλ−1e−tdt. Here the term u represents the scaled film height, and ∆2u denotes the capillarity-driven surface diffusion whereas ∇ · f(∇u) denotes the upward hopping of atoms. Throughout this paper, we will assume that f ∈ C1(RN ,RN ) with f(0) = Df(0) = 0 and for some β > 1, and f satisfies the following growth condition |f ′(ξ1)− f ′(ξ2)| ≤ C(|ξ1|β−1 + |ξ2|β−1)|ξ1 − ξ2| (1.2) for any ξ1, ξ2 ∈ RN . As a simple example of (1.2), we can take f(ξ) = |ξ|βξ. When α = 1, the fractional time Caputo derivative is replaced with the classical integer derivative ut. The problem (1.1) becomes the usual thin film growth model, which is well studied by many researchers. King et al. [9] proved the existence, uniqueness, regularity and large time behavior of solutions in Sobolev function space. Using Kato’s Method, Sandjo et al. [16] established existence, uniqueness and regularity of the solution in spaces of C0([0, T ];Lp(Ω)) with p = nβ 2−β , 1 < β < 2. Furthermore, they illustrated the qualitative behavior of the approximate solution through some numerical simulations. Ishige et al. [6] give sufficient conditions on the existence of global solutions, the maximal existence time and blow up rate for 0 < β ≤ 2. Y. Feng et al. [5] studied the existence of local mild solutions for any initial data lies in L2 on the two-dimensional torus with and without advection. We refer the interested reader to the references therein. Fractional models extend the classical model with non-integer orders differentia- tion and integration, allowing the incorporation of memory and long-range depen- dencies into models. There are many numerical simulations for the time fractional thin film growth equations. T. Tang et al. [18] proved the time-fractional molecular beam epitaxy models admit an energy dissipation law as integral cases and pro- posed a class of finite difference schemes inherited the theoretical energy stability. Chen et al. [4] developed an efficient and accurate, full discrete, linear numerical approximation for the time-fractional thin film model with the classical Caputo fractional derivative of order α and shown the models possessed an energy dissipa- tion law. Wang et al. [20] proposed a variable-step L1 scheme for the time-fractional molecular beam epitaxy model and also investigated the stability and convergence of the stabilized convex splitting scheme. However, to the best of our knowledge, the well-posedness for the solutions of the time fractional thin film growth equation is not clear, which is the main motivation of the present work. We refer [19] for well posedness of the linear and semilinear time fractional order Cauchy problem with almost sectorial operators, and [22] for well posedness of the time fractional order Cahn-Hilliard equation in R3. In this article, we focus on the existence and uniqueness of mild solutions on problem (1.1). Because of the observation that the biharmonic operator can be regarded as a sectorial operator on some spaces, we follow some ideas in [3], properly adapted to our problem. We denote A = ∆2 defined on Lp(Ω), 1 < p < ∞ with its domain D(A) = {u ∈ W 4,p(Ω) : ∂νu|∂Ω = ∂ν∆u|∂Ω = 0}. It is clear [15] that the following homogeneous boundary value problem is normally elliptic in Ω, ∂tu = −Au, x ∈ Ω, t > 0, u(x, 0) = φ(x), x ∈ Ω. (1.3) EJDE-2024/58 EQUATIONS MODELING THIN FILM GROWTH 3 Hence, the fourth-order operator −∆2 with corresponding Neumann boundary ( ∂ ∂ν , ∂∆ ∂ν ) is the infinitesimal generator of an analytic semigroup T (t) = e−t∆2 in Lp(Ω). For a detailed discussion on these results, we refer the reader to [1, 2]. The time fractional version of (1.3) with lower order term can be written as cD α t u = −Au+ F (u), x ∈ Ω, t > 0 u(x, 0) = φ(x), x ∈ Ω. (1.4) We formally give a mild solution for (1.4), where the rigorous deduction could be found in [3, 8, 10, 19]. We denote by L Laplace transform operator. By the convolution property of Laplace transform, we have L( cDα t u) = λαL(u)− λα−1φ, as in [10]. So taking the Laplace transform to (1.4) gives L(u) = λα−1(λα +A)−1φ+ (λα +A)−1L(F (u)). Application of Laplace inversion [3] implies u(t) = Eα (−tαA)φ+ ∫ t 0 (t− s)α−1Eα,α (−(t− s)αA)F (u)(s)ds. where Eα(−tαA) and Eα,α(−tαA) are the Mittag-Leffler operators (see Section 2). This formal computation then motivates the definition of the mild solution of (1.1) as follows: Definition 1.1. Let 0 < α < 1 and T > 0 . (i) A function u such that u ∈ C([0, T ];L βN 2−β (Ω)) defined by u(x, t) = Eα(−tαA)φ+ ∫ t 0 (t− s)α−1Eα,α(−(t− s)αA)∇ · f(∇u) ds, (1.5) is called a local mild solution of (1.1). (ii) If T = ∞, we say that u is a global mild solution of (1.1). We are now in a position to state the main result of this article. Theorem 1.2. Suppose Ω ⊆ RN is a bounded domain with C4 boundary, 0 < α < 1 < β < 2 and φ ∈ L βN 2−β (Ω). Then there exists T > 0 such that (1.1) admits a unique mild solution u ∈ C([0, T ];L βN 2−β (Ω)) satisfying max { sup 0≤t≤T t α 2β− αγ 4β2 ∥∇u(t)∥ β2N γ , sup 0≤t≤T t α 2 ∥∇2u(t)∥ βN 2−β } < ∞, where 0 < γ < min{2β, β(N+1)−2}. Furthermore, if ∥φ∥ βN 2−β is sufficiently small, the solution u can be extended to be global, that is, T = ∞. To explain the meaning of the result, we take f(ξ) = |ξ|βξ in (1.1)1 as an example. Notice that a smooth function u(x, t) solves the equation in (1.1) for t > 0 if and only if uλ(x, t) = λ 2α β −αu(λαx, λ4t) does so too with each given constant λ. In addition, for the initial data φ, under the transformation φ 7→ φλ, the L βN 2−β (Ω) norm is invariant. Therefore, we expect the global existence and uniqueness of solutions when the initial data is sufficiently small in the critical space L βN 2−β (Ω). 4 Q. LIU, W. ZHU, H. YE EJDE-2024/58 It is worth mentioning that the singularity together with strong nonlinearity arising from the time fractional derivative cD α t u and the nonlinear term ∇· f(∇u) make its mathematical analysis more difficult in comparison with the fourth-order parabolic equation (1.3)1. For example, the Mittag-Leffler operators Eα(−tαA) and Eα,α(−tαA) do not satisfy the semigroup properties. So we need to overcome these essential difficulties to get some a priori estimates and extend the local solution to a global one. For more details, one can refer to Section 3. This article is organized as follows. In the next section, we introduce some elementary properties of Mittag-Leffler operators, which are essential throughout the whole paper and give the main results of this paper. In Section 3, we establish the existence and uniqueness of mild solutions using appropriate functional spaces and Banach fixed pointed theorem. 2. A priori estimates Throughout this paper, C stands for a generic positive constant which may vary from line to line. For the analytic semigroup T (t), we have the Lp − Lq estimates, which are state as follows. Proposition 2.1 ([15, 21]). Let 1 < q ≤ p < ∞ and j = 0, 1, 2, 3. For u ∈ Lq(Ω), we have ∥∇jT (t)u∥Lp(Ω) ≤ C t− N 4 ( 1 q− 1 p )− j 4 ∥u∥Lq(Ω), t > 0. Let us recall some properties of Mittag-Leffler operators. For α ∈ (0, 1), we denote the entire function Mα : C → C the Mainardi function by Mα(z) := ∞∑ n=0 (−z)n n!Γ(1− α(1 + n)) , which is a particular case of the Wright type function introduced by Mainardi in [12] to characterize the fundamental solutions for some standard boundary value problems in physics. The following classical result gives some essential relations used in this article to obtain the main estimates. Proposition 2.2 ([19]). Let 0 < α < 1 and −1 < γ < ∞. If we restrict Mα to the positive real line, then it holds that Mα ∈ S([0,∞)), Mα(t) ≥ 0 for all t ≥ 0 and ∫ ∞ 0 tγMα(t) dt = Γ(γ + 1) Γ(αγ + 1) , where S([0,∞)) is the Schwartz space on [0,∞). Now, for each α ∈ (0, 1), we define the Mittag-Leffler families Eα(−tαA) = ∫ ∞ 0 Mα(s)T (st α) ds, Eα,α(−tαA) = ∫ ∞ 0 αsMα(s)T (st α) ds. It is interesting to notice that the Mainardi functions act as a bridge between the fractional and the classical abstract theories, for more details see [19, 11, 3]. The next result comprises the main assertions about the theory of abstract frac- tional calculus. EJDE-2024/58 EQUATIONS MODELING THIN FILM GROWTH 5 Proposition 2.3 ([19]). Eα(−tαA) and Eα,α(−tαA) are well defined from Lp(Ω) to Lp(Ω), p ∈ (1,∞). Moreover, for t ≥ 0, Eα(−tαA) and Eα,α(−tαA) are uniformly continuous in the uniform operator topology on Lp(Ω). Our proof relies on the well-known Weierstrass M-test in Banach space and Lp − Lq estimates for Mittag-Leffler operators. Lemma 2.4 (Weierstrass M-test [7]). Let X denote a Banach space equipped with the norm ∥ · ∥X . Suppose {ωj}j≥0 is a sequence of continuous functions from [0, T ] to X such that sup 0≤t≤T ∥ωj(t)∥X ≤ Mj , j = 0, 1, 2, . . . where 0 < T ≤ ∞ and the sequence {Mj}j≥0 satisfies ∑∞ j=0 Mj < ∞, then the sequence {ωj}j≥0 converges uniformly on [0, T ], that is, ∞∑ j=0 ωj ∈ C([0, T ];X). Using Proposition 2.1, we can obtain similar Lp −Lq estimates for both families of Mittag-Leffler operators, which also depend on the exponent of differentiation α. Proposition 2.5. Let 1 < q ≤ p < ∞ and 1 q − 1 p < 4−j N with any nonnegative integer j < 4. For u ∈ Lq(Ω), we have ∥∇jEα(−tαA)u∥Lp(Ω) ≤ Ct− Nα 4 ( 1 q− 1 p )− jα 4 ∥u∥Lq(Ω), t > 0. (2.1) Proof. Noting that if 0 ≤ 1 q − 1 p < 4−j N with j < 4, we have −N 4 ( 1 q − 1 p )− j 4 > −1. By Proposition 2.1 and 2.2, we obtain ∥∇jEα(−tαA)u∥Lp(Ω) ≤ ∫ ∞ 0 Mα(s)∥∇jT (stα)u∥Lp(Ω) ds ≤ C (∫ ∞ 0 Mα(s)s −N 4 ( 1 q− 1 p )− j 4 ds )( t− Nα 4 ( 1 q− 1 p )− jα 4 ∥u∥Lq(Ω) ) ≤ Ct− Nα 4 ( 1 q− 1 p )− jα 4 ∥u∥Lq(Ω). The proof is complete. □ Proposition 2.6. Let 1 < q ≤ p < ∞ and 1 q − 1 p < 8−j N with a nonnegative integer j < 4. Then we have ∥∇jEα,α(−tαA)u∥Lp(Ω) ≤ Ct− Nα 4 ( 1 q− 1 p )− jα 4 ∥u∥Lq(Ω), t > 0. The proof of the above propostion is similar to that of Proposition 2.5, so we omit it. 6 Q. LIU, W. ZHU, H. YE EJDE-2024/58 3. Proof of Theorem 1.2 In this section, based on the Lp−Lq estimates in section 2, we will prove Theorem 1.2 by using the method of successive approximation. To simplify the notation, we denote the norm ∥ · ∥Lp(Ω) by ∥ · ∥p. Recalling the integral equation (1.5), we define a sequence {uj}j≥0 as follows, u0(x, t) = Eα(−tαA)φ, uj(x, t) = Eα(−tαA)φ+ ∫ t 0 (t− s)α−1Eα,α(−(t− s)αA)∇ · f(∇uj−1) ds (3.1) for positive integers j and t > 0. In additional, we define R(t)j = max { sup 0 0 and j ≥ 0, where 0 < γ < min{2β, β(N + 1)− 2}. Next, we show uj(t) belongs L βN 2−β (Ω) and is continuous under some smallness conditions. The proofs require the following iteration lemma. Lemma 3.1 ([13]). Let λ, β > 0 and bj be a nonnegative sequence such that bj ≤ b0 + λb1+β j−1 for all positive integer j. If 2λ(2b0) β < 1, then for each nonnegative integer j, we have bj ≤ b0 1− λ(2b0)β . Lemma 3.2. For any T > 0, there exists a constant ε0 > 0 independent of T such that if R(T )0 ≤ ε0, each uj(t) is well defined as an element of L βN 2−β (Ω) for any t > 0, and R(T )j ≤ 2R(T )0, j ≥ 0. (3.3) Proof. Because f ′(ξ) behaves like |ξ|β , we have ∥f ′(∇u)∥s ≤ C∥∇u∥βs . By Hölder’s inequality, it is easy to verify that ∥∇ · f(∇uj(s))∥ βN 2−β+γ = ∥∇2uj(s) · f ′(∇uj(s))∥ βN 2−β+γ ≤ ∥∇2uj(s)∥ βN 2−β ∥f ′(∇uj(s))∥ βN γ = s−α+αγ 4β ( s α 2 ∥∇2uj(s)∥ βN 2−β )( s α 2 −αγ 4β ∥∇uj(s)∥ββ2N γ ) ≤ s−α+αγ 4β ( sup 0 0, we have R(T )j+1 ≤ R(T )0 + CR(T )1+β j , where C > 0 is independent of T . According to Lemma 3.1, there exists a constant ε0 > 0 such that R(T )0 ≤ ε0, and 2C(2R(T )0) β < 1, which yields R(T )j ≤ 2R(T )0, j ≥ 0. At last, we obtain ∥uj+1∥ βN 2−β ≤ ∥Eα(−tαA)φ∥ βN 2−β + ∫ t 0 (t− s)α−1∥Eα,α(−(t− s)αA)∇ · f(∇uj)∥ βN 2−β ds ≤ R(T )0 + C ∫ T 0 (t− s)α−1−αγ 4β ∥∇ · f(∇uj)∥ βN 2−β+γ ds ≤ R(T )0 + CR(T )1+β j ∫ T 0 (t− s)α−1−αγ 4β s−α+αγ 4β ds ≤ R(T )0 + CR(T )1+β j ≤ CR(T )0, which means uj(t) ∈ Lp(Ω) for any 0 < t ≤ T and j ≥ 0. □ Next, we shall show the strong continuity of uj(t) for t ∈ [0, T ]. Lemma 3.3. Under the assumption R(T )0 ≤ ε0 in Lemma 3.2, for any j ≥ 0, we have uj ∈ C([0, T ];L βN 2−β (Ω)), T > 0, j = 0, 1, 2, · · · . 8 Q. LIU, W. ZHU, H. YE EJDE-2024/58 Proof. Fixing t0 ∈ (0, T ), we have the estimate ∥uj+1(t)− uj+1(t0)∥ βN 2−β ≤ ∥(Eα(−tαA)− Eα(−tα0A))φ∥ βN 2−β + ∫ t t0 (t− s)α−1 ∥Eα,α(−(t− s)αA)∇ · f(∇uj)∥ βN 2−β ds + ∫ t0 0 ∥∥(t− s)α−1Eα,α(−(t− s)αA)∇ · f(∇uj) − (t0 − s)α−1Eα,α(−(t0 − s)αA)∇ · f(∇uj) ∥∥ βN 2−β ds = I1 + I2 + I3, (3.7) where t0 < t ≤ T . Using the strong continuity of Eα(−tαA) on Lp(Ω) for t ∈ [0,∞), we deduce easily that the first term I1 goes to zero as t → t+0 . By Proposition 2.6, the estimate of the second term yields I2 = ∫ t t0 (t− s)α−1 ∥Eα,α(−(t− s)αA)∇ · f(∇uj)∥ βN 2−β ds ≤ C ∫ t t0 (t− s)α−1−αγ 4β ∥∇ · f(∇uj)∥ βN 2−β+γ ds ≤ CR(T )1+β 0 ∫ t t0 (t− s)α−1−αγ 4β s−α+αγ 4β ds ≤ CR(T )1+β 0 ( t− t0 t0 )1−α+αγ 4β . Obviously, this term vanishes as t → t+0 . For the estimate of the third term, we first denote g(t, s) = ∥∥(t− s)α−1Eα,α(−(t− s)αA)∇ · f(∇uj) − (t0 − s)α−1Eα,α(−(t0 − s)αA)∇ · f(∇uj) ∥∥ βN 2−β , where 0 < s < t0 < t. Combining this with (3.4), it is easy to see that g(t, s) ≤ |(t− s)α−1 − (t0 − s)α−1| · ∥∥Eα,α(−(t− s)αA)∇ · f(∇uj(s)) ∥∥ βN 2−β + (t0 − s)α−1 ∥∥Eα,α(−(t− s)αA)∇ · f(∇uj(s)) − Eα,α(−(t0 − s)αA)∇ · f(∇uj(s)) ∥∥ βN 2−β ≤ C|(t− s)α−1 − (t0 − s)α−1|(t− s)− αγ 4β · ∥∇ · f(∇uj(s))∥ βN 2−β+γ + (t0 − s)α−1 ∥∥Eα,α(−(t− s)αA)∇ · f(∇uj(s)) − Eα,α(−(t0 − s)αA)∇ · f(∇uj(s)) ∥∥ βN 2−β ≤ C|(t− s)α−1 − (t0 − s)α−1|(t− s)− αγ 4β s−α+αγ 4β R(T )0 + (t0 − s)α−1 ∥∥Eα,α(−(t− s)αA)∇ · f(∇uj(s)) − Eα,α(−(t0 − s)αA)∇ · f(∇uj(s)) ∥∥ βN 2−β . (3.8) EJDE-2024/58 EQUATIONS MODELING THIN FILM GROWTH 9 Then combining with the strong continuity of Eα,α(−tαA) on L βN 2−β (Ω) for t ∈ (0,∞) , we have lim t→t+0 g(t, s) = 0 for each fixed s ∈ (0, t0). In addition, the last term of (3.8) is integrable. Applying dominated convergence theorem yields the third term I3 also tends to zero as t → t+0 . Indeed, for 0 < s < t0 < t, we have I3 = ∫ t0 0 g(t, s) ds ≤ C ∫ t0 0 (t0 − s)α−1−αγ 4β ∥∇ · f(∇uj)∥ βN 2−β+γ ds, ≤ CR(T )1+β j ∫ t0 0 (t0 − s)α−1−αγ 4β s−α+αγ 4β ds ≤ CR(T )1+β 0 . Similarly, we can also prove the same limit as t → t−0 with t0 ∈ (0, T ]. Thus, lim t→t0 ∥uj+1(t)− uj+1(t0)∥ βN 2−β = 0, t0 ∈ (0, T ]. As for the continuity up to t = 0 of uj+1, Observe that ∥uj+1(t)− uj+1(0)∥ βN 2−β ≤ ∥Eα(−tαA)φ− φ∥ βN 2−β + ∫ t 0 (t− s)α−1 ∥Eα,α(−(t− s)αA)∇ · f(∇uj(s))∥ βN 2−β ds ≤ ∥Eα(−tαA)φ− φ∥ βN 2−β + C ∫ t 0 (t− s)α−1−αγ 4β ∥∇ · f(∇uj)∥ βN 2−β+γ dsr ≤ ∥Eα(−tαA)φ− φ∥ βN 2−β + CR(t)1+β j ∫ t 0 (t− s)α−1−αγ 4β s−α+αγ 4β ds ≤ ∥Eα(−tαA)φ− φ∥ βN 2−β + CR(t)1+β 0 . Obviously, if lim t→0+ R(t)0 = 0, (3.9) then combining with the strong continuity of Eα(−tαA), we obtain that lim t→0+ ∥uj+1(t)− uj+1(0)∥ βN 2−β = 0. To prove (3.9), we notice that since φ ∈ L βN 2−β (Ω), for any ε > 0, there exists φ̃ ∈ C∞ 0 (Ω) such that ∥φ− φ̃∥ βN 2−β < ε. Then we have t α 2β− αγ 4β2 ∥∇u0(t)∥ β2N γ = t α 2β− αγ 4β2 ∥∇Eα(−tαA)φ∥ β2N γ ≤ t α 2β− αγ 4β2 ∥∇Eα(−tαA)(φ− φ̃)∥ β2N γ + t α 2β− αγ 4β2 ∥∇Eα(−tαA)φ̃∥ β2N γ ≤ ∥φ− φ̃∥ βN 2−β + t α 4 ∥∇φ̃∥ βN 2−β ≤ ε 10 Q. LIU, W. ZHU, H. YE EJDE-2024/58 for small t > 0, which implies lim t→0+ t α 2β− αγ 4β2 ∥∇u0(t)∥ β2N γ = 0. (3.10) Similarly, we can obtain lim t→0+ t α 2 ∥∇2u0(t)∥ βN 2−β = 0. (3.11) Combining (3.10) with (3.11), we derive (3.9). Summing up, we see that uj ∈ C([0, T ];L βN 2−β (Ω)) for any T > 0 and j ≥ 0. □ Proof of Theorem 1.2. To prove the main result, we only need to show the uniform convergence of the sequence {uj}j≥0 under the assumption that R(T )0 ≤ ε0. By Lemma 2.4, we define the sequence ω0(x, t) = u0(x, t), ωj(x, t) = uj(x, t)− uj−1(x, t), j ≥ 1. (3.12) Obviously, ωj ∈ C([0, T ];L βN 2−β (Ω)) for any T > 0 and j ≥ 0. In addition, we have the identity ωj+1(x, t) = uj+1(x, t)− uj(x, t) = ∫ t 0 (t− s)α−1Eα,α(−(t− s)αA) ( ∇ · f(∇uj)−∇ · f(∇uj−1) ) ds = ∫ t 0 (t− s)α−1Eα,α(−(t− s)αA) ( ∇2uj · f ′(∇uj)−∇2uj−1 · f ′(∇uj−1) ) ds = ∫ t 0 (t− s)α−1Eα,α(−(t− s)αA)∇2ωj · f ′(∇uj) ds (3.13) + ∫ t 0 (t− s)α−1Eα,α(−(t− s)αA)∇2uj−1 · ( f ′(∇uj)− f ′(∇uj−1) ) ds. As in Lemma 3.2, to estimate ωj , we to derive a priori estimates for ∇ωj and ∇2ωj . So we define R̃(t)j = max { sup 0 0 and j ≥ 0, where 0 < γ < min{2β, β(N+1)−2}. Obviously, R̃(t)0 = R(t)0. Let R(T )0 ≤ ε0. By Hölder’s inequality, we obtain that for 0 < s ≤ t ≤ T , ∥∇2ωj(s) · f ′(∇uj(s))∥ βN 2−β+γ ≤ ∥∇2ωj(s)∥ βN 2−β ∥f ′(∇uj(s))∥ βN γ = s−α+αγ 4β ( s α 2 ∥∇2ωj(s)∥ βN 2−β )( s α 2 −αγ 4β ∥∇uj(s)∥ββ2N γ ) ≤ s−α+αγ 4β R̃(T )jR(T )βj ≤ Cs−α+αγ 4β R̃(T )jR(T )β0 . (3.15) EJDE-2024/58 EQUATIONS MODELING THIN FILM GROWTH 11 In addition, the growth condition on f ′ and Hölder’s inequality yield, for 0 < s ≤ t ≤ T , ∥f ′(∇uj(s))− f ′(∇uj−1(s))∥ βN γ ≤ ∥∥∥∇ωj(s) ( |∇uj(s)|β−1 + |∇uj−1(s)|β−1 )∥∥∥ βN γ ≤ ∥∇ωj(s)∥ β2N γ ∥∥∥|∇uj(s)|β−1 + |∇uj−1(s)|β−1 ∥∥∥ β2N γ(β−1) ≤ ∥∇ωj(s)∥ β2N γ ( ∥∇uj(s)∥β−1 β2N γ + ∥∇uj−1(s)∥β−1 β2N γ ) ≤ s− α 2 +αγ 4β R̃(T )j ( R(T )β−1 j +R(T )β−1 j−1 ) . Then for R(T )0 ≤ ε0, we have ∥∇2uj−1(s) · ( f ′(∇uj(s))− f ′(∇uj−1(s)) ) ∥ βN 2−β+γ ≤ ∥∇2uj−1(s)∥ βN 2−β ∥f ′(∇uj(s))− f ′(∇uj−1(s))∥ βN γ ≤ s−α+αγ 4β R̃(T )jR(T )j−1 ( R(T )β−1 j +R(T )β−1 j−1 ) ≤ Cs−α+αγ 4β R̃(T )jR(T )β0 . (3.16) Applying the differential operator ∇ to (3.13), and combining this with (3.15)– (3.16), we have ∥∇ωj+1∥ β2N γ ≤ ∫ t 0 (t− s)α−1∥∇Eα,α(−(t− s)αA)∇2ωj · f ′(∇uj)∥ β2N γ ds + ∫ t 0 (t− s)α−1∥∇Eα,α(−(t− s)αA)∇2uj−1 · ( f ′(∇uj)− f ′(∇uj−1) ) ∥ β2N γ ds ≤ C ∫ t 0 (t− s) α−1+ αγ 4β2 − α 2β−αγ 4β ∥∇2ωj · f ′(∇uj)∥ βN 2−β+γ ds + C ∫ t 0 (t− s) α−1+ αγ 4β2 − α 2β−αγ 4β ∥∇2uj−1 · ( f ′(∇uj)− f ′(∇uj−1) ) ∥ βN 2−β+γ ds ≤ CR̃(T )jR(T )β0 ∫ t 0 (t− s) α−1+ αγ 4β2 − α 2β−αγ 4β s−α+αγ 4β ds ≤ Ct αγ 4β2 − α 2β R̃(T )jR(T )β0 , j ≥ 1. (3.17) Similarly, we have ∥∇2ωj+1∥ βN 2−β ≤ ∫ t 0 (t− s)α−1∥∇2Eα,α(−(t− s)αA)∇2ωj · f ′(∇uj)∥ βN 2−β ds + ∫ t 0 (t− s)α−1∥∇2Eα,α(−(t− s)αA)∇2uj−1 · ( f ′(∇uj)− f ′(∇uj−1) ) ∥ βN 2−β ds ≤ C ∫ t 0 (t− s) α 2 −1−αγ 4β ∥∇2ωj · f ′(∇uj)∥ βN 2−β+γ ds 12 Q. LIU, W. ZHU, H. YE EJDE-2024/58 + C ∫ t 0 (t− s) α 2 −1−αγ 4β ∥∇2uj−1 · ( f ′(∇uj)− f ′(∇uj−1) ) ∥ βN 2−β+γ ds ≤ CR̃(T )jR(T )β0 ∫ t 0 (t− s) α 2 −1−αγ 4β s−α+αγ 4β ds ≤ Ct− α 2 R̃(T )jR(T )β0 , j ≥ 1. (3.18) Combining (3.14), (3.17) with (3.18), for any fixed T > 0, we have R̃(T )j+1 ≤ CR̃(T )jR(T )β0 , j = 1, 2, . . . (3.19) where C > 0 is independent of T . Noting that ω1 = ∫ t 0 (t− s)α−1Eα,α(−(t− s)αA)∇ · f(∇u0) ds, following the same idea in the proof (3.17) and (3.18), we obtain R̃(T )1 ≤ CR̃(T )0R(T )β0 = CR(T )1+β 0 . (3.20) Combining (3.19) with (3.20), we finally derive R̃(T )j ≤ R(T )0 ( CR(T )β0 )j , j = 0, 1, 2, . . . provided that R(T )0 ≤ ε0. With the help of a priori estimates above, now we can estimate ωj , ∥ωj+1∥ βN 2−β ≤ ∫ t 0 (t− s)α−1∥Eα,α(−(t− s)αA)∇2ωj · f ′(∇uj)∥ βN 2−β ds + ∫ t 0 (t− s)α−1∥Eα,α(−(t− s)αA)∇2uj−1 · ( f ′(∇uj)− f ′(∇uj−1) ) ∥ βN 2−β ds ≤ C ∫ t 0 (t− s)α−1−αγ 4β ∥∇2ωj · f ′(∇uj)∥ βN 2−β+γ ds + C ∫ t 0 (t− s)α−1−αγ 4β ∥∇2uj−1 · ( f ′(∇uj)− f ′(∇uj−1) ) ∥ βN 2−β+γ ds ≤ CR̃(T )jR(T )β0 ∫ t 0 (t− s)α−1−αγ 4β s−α+αγ 4β ds ≤ R(T )0 ( CR(T )β0 )j+1 , j ≥ 1. (3.21) Similarly, we obtain that ∥ω1∥ βN 2−β ≤ CR(T )1+β 0 . Therefore, Mj := sup 0≤t≤T ∥ωj(t)∥ βN 2−β ≤ R(T )0 ( CR(T )β0 )j , j ≥ 0 and the sequence {Mj}j≥0 is summable provided that R(T )0 < min{ε0, C− 1 β }. (3.22) EJDE-2024/58 EQUATIONS MODELING THIN FILM GROWTH 13 Since limT→0+ R(T )0 = 0, we can choose T > 0 small enough such that (3.22) holds. Then Lemma 2.4 implies that {uj}j≥0 is a Cauchy sequence in C([0, T ];L βN 2−β (Ω)), and converges to a unique solution u ∈ C([0, T ];L βN 2−β (Ω)) of the integral equation (1.5). Moreover, recalling that u0(x, t) = Eα(−tαA)φ, and combining this with Proposition 2.5, we obtain R(T )0 ≤ max { sup 0≤t≤T t α 2β− αγ 4β2 ∥∇u0(t)∥ β2N γ , sup 0≤t≤T t α 2 ∥∇2u0(t)∥ βN 2−β } ≤ C∥φ∥ βN 2−β . So if ∥φ∥ βN 2−β is sufficiently small, the solution u can be extended to be global. The proof of Theorem 1.2 is complete. □ Acknowledgements. 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Zangwill; Some causes and a consequence of epitaxial roughening, J. Cryst. Growth, 163 (1996), 8–21. Qiang Liu School of Mathematical Sciences, Shenzhen University, Shenzhen, 518060, China Email address: matliu@szu.edu.cn Wanyu Zhu School of Mathematical Sciences, Shenzhen University, Shenzhen, 518060, China Email address: 2200201025@email.szu.edu.cn Hailong Ye (corresponding author) School of Mathematical Sciences, Shenzhen University, Shenzhen, 518060, China Email address: yhl@szu.edu.cn 1. Introduction and main result 2. A priori estimates 3. Proof of Theorem 1.2 Acknowledgements References