Global existence and decay to equilibrium for some crystal surface models

In this paper we study the large time behavior of the solutions to the following nonlinear fourth-order equations $$ \partial_t u=\Delta e^{-\Delta u}, $$ $$ \partial_t u=-u^2\Delta^2(u^3). $$ These two PDE were proposed as models of the evolution of crystal surfaces by J. Krug, H.T. Dobbs, and S. Majaniemi (Z. Phys. B, 97, 281-291, 1995) and H. Al Hajj Shehadeh, R. V. Kohn, and J. Weare (Phys. D, 240, 1771-1784, 2011), respectively. In particular, we find explicitly computable conditions on the size of the initial data (measured in terms of the norm in a critical space) guaranteeing the global existence and exponential decay to equilibrium in the Wiener algebra and in Sobolev spaces.


INTRODUCTION
Crystal films are important in many modern electronic devices (as mobile phone antennae) and nanotechnology.Thus, the evolution of a crystal surface is an interesting topic with important applications.The purpose of this paper is to find explicitly computable conditions guaranteeing the global existence and exponential decay to equilibrium of the solutions of certain PDE models of crystal surfaces.
The first problem we deal with is where the initial datum u 0 satisfies 2) and 0 < T < ∞.This model was suggested by Krug, Dobbs, & Majaniemi [11] (equations (4.4) and (4.5) in [11]) (see also Marzuola & Weare [15]) as a description (under certain physical assumptions and simplifications) of the large scale evolution of the crystal surface.
This equation has been previously studied in the mathematical literature.In particular, Liu & Xu [12] obtained the existence of weak solutions starting from arbitrary initial data.Moreover, they define a stationary solution having a curvature singularity evidencing that the solution may develop singularities.Gao, Liu, & Lu [9] used the gradient flow approach to study the solution of (1.1).In their paper, the solution (which exists globally in time and emanates from an initial data with arbitrary size) is allowed to have a singularity at the level of second derivatives (the laplacian of the solution is allowed to be a Radon measure).Let us also mention that when the exponential in (1.1) is linearized and the Laplacian is replaced by the p−Laplacian, the resulting equation has been studied by Giga & Kohn [10] (see also the recent preprint by Xu [19]).
The second problem we deal with is with periodic boundary conditions and where u 0 > 0. Using that we can rescale variables and, without lossing generality, assume that T d 1/u(x, t) dx = 1.This equations was proposed by Shehadeh, Kohn & Weare [17] (equation (1.1) in [17]) as a continuous description of the slope of the crystal surface (as a function of height and time) in the so-called Attachment-Detachment-Limited regime (see Appendix A in [17] and the references therein for further explanation on the physics behind the model).
When the extra term −αu 2 ∆ 2 u is added (1.2) reads This problem has been studied in the mathematical literature by Gao, Ji, Liu & Witelski [7].In particular, when the domain is one-dimensional and α > 0, by using the two Lyapunov functionals 2 dx, Gao, Ji, Liu & Witelski proved the global existence of positive solutions to (1.3).Furthermore, these authors also proved that u(t) tends (at an unspecified rate) towards to certain constant steady state (which depends on the initial data).
When α = 0, Gao, Liu, & Lu [8] proved the global existence of weak solutions to (1.2) for non-negative initial data such that u 3 0 ∈ H 2 .Moreover, in [8] the convergence of the weak solution towards a constant steady state is also obtained (at an unspecified rate).
We observe that (1.2) is equivalent to in T d , (1.4) under the change of variables ∆w = 1/u.Equation (1.4) was studied by Liu & Xu [13].In this paper, the authors proved the existence of global weak solutions.
Under the change of variable v = ∆u, (1.1) becomes Due to the definition of v and the periodic boundary conditions, we have that Using the formulations (1.5) and (1.6) allow us to find two Lyapunov functionals L 1 (for problem (1.5)) and L 2 (for problem (1.6)): Let us remark that (1.5) and (1.6) (when the domain is R d to simplify the exposition) are invariant by the scaling (1.7) In the rest of the paper we will use the formulations (1.5) and (1.6) to prove several global existence results.The main contribution is a global existence and decay in the Wiener algebra A 0 (see (1.8) below for the proper definition), for initial data satisfying certain explicit size restrictions.We note that the scaling (1.7) leaves the Wiener algebra's norm invariant.Indeed, in the case where the domain is Thus, our Theorem 2.2 and Theorem 2.6 are global existence results in a critical space for problems (1.5) and (1.6).Using the decay in the Wiener algebra, we also provide a global existence and decay for initial data in Sobolev spaces.The results for the original problems (1.1) and (1.2) follow easily.
Our approach is very adaptable and can lead to advances in other systems of PDE.For instance, it has been used Bruell & Granero-Belinchón to study the evolution of thin films in Darcy and Stokes flows [2] by Córdoba and Gancedo [5], Constantin, Córdoba, Gancedo, Rodriguez-Piazza, & Strain [4] for the Muskat problem (see also [6] and [16]), by Burczak & Granero-Belinchón [3] to analyze the Keller-Segel system of PDE with diffusion given by a nonlocal operator and by Bae, Granero-Belinchón & Lazar [1] to prove several global existence results (with infinite L p energy) for nonlocal transport equations.
After the completion of this work, Liu and Strain posted the paper [14] where the DL problem (1.1) was studied using a similar approach.

Notation & Basic Tools. Recalling the expression of the
we have the Fourier series representation Note that, dropping t from the notation, we have the following well-known facts: Let n ∈ Z d + and denote by being ∂ a differential operator of order 1 with respect to a spatial variable.Then, we define the In a similar way, we consider the Wiener spaces A α (T d ) as We note that We recall the following inequalities: We also introduce the space of Radon measures from an interval [0, T] to a Banach space X, M(0, T; X).
We simplify the notation rewriting the Lebesgue, Sobolev and Wiener norms as Similarly we set We denote with c, C positive constants which may vary from line to line during the proofs.Finally, for j = 1, . . ., d, we write u, j = ∂u ∂x j and we adopt Einstein convention for summation.

MAIN RESULTS & DISCUSSION
2.1.Main results of problem (1.5).We consider the following definition of weak solutions for problem (1.5).
Definition 2.1.We say that the function v ∈ L ∞ (0, T; L ∞ (T d )) is a weak solution of (1.5) with initial data v 0 if We prove that Theorem 2.2.Let v 0 ∈ A 0 (T d ) be such that the value satisfies 0 < δ(|v 0 | 0 ).Then, there exist at least one global weak solution in the sense of Definition 2.1 to equation (1.5) having the regularity for any T > 0. Furthermore, the solution satisfies Furthermore, beyond (2.3), the solution obeys v(t) H r ≤ c 3 e −c 4 t for all 0 ≤ r < 2, (2.6) where c i = c i (v 0 ).

Main results of problem (1.6).
First we state our definition of weak solution for problem (1.6): We prove Theorem 2.6.Let v 0 ∈ A 0 (T d ) be such that the value Then there exists at least one global weak solution in the sense of Definition 2.5 to equation (1.6) having the regularity Furthermore, the solution satisfies for δ(|v 0 | 0 ) satisfying (2.7).
Remark 2.7.In order v 0 satisfies (2.7), it is enough to have As for problem (1.5), for initial data having certain Sobolev regularity, we have that Theorem 2.8.Let v 0 ∈ A 0 (T d ) ∩ H 2 (T d ) be a function satisfying the smallness condition in (2.7) and Then, the solution constructed in Theorem 2.4 also satisfies Furthermore, beyond (2.9), the solution obeys where c i = c i (v 0 ).

Discussion.
In this paper we prove the global existence of weak solution for (1.5) and (1.6) for initial data satisfying a size restriction in the Wiener algebra.One of the main advantages is that the size restriction is explicit and concerns a lower order norm (it does not impose any requirement on the size of derivatives of v).Furthermore, when the domain is R d , the norm in the Wiener algebra is invariant by the scaling (1.7).This makes the Wiener algebra a critical space for problems (1.5) and (1.6).
In terms of the original problem (1.1), our results state that if the Laplacian of the initial data satisfies certain explicit size restriction in the Wiener algebra, then the solution u exists and its Laplacian remains globally bounded.In particular, the Laplacian of these solutions u can not be a singular measure (compare with [12, 9] and note that the u reconstructed from our solutions v satisfies u ∈ W 2,∞ ).
Concerning (1.2), our results imply that if the appropriate transformation of the initial data satisfies certain hypotheses, then the solution u exists and remains bounded and positive.Furthermore, we can also quantify the convergence rate towards the steady state (compare with [7,8]).

THE PROBLEM (1.5)
We consider the following weakly nonlinear regularized analog of (1.5) For this problem, we can construct a solution v N following a standard Galerkin approach.
3.1.Estimates for the regularized problem in the Wiener algebra A 0 .Proposition 3.1 (Estimates in the Wiener algebra).Let v 0 ∈ A 0 (T d ) be a function satisfying the condition in (2.1).Then, every approximating sequence of solutions

1).
Proof.To simplify the notation, we write v when referring to v N .We compute In Fourier variables and omitting the time variable, the above equality reads where the nonlinearities are We want to obtain an estimate for |v| 0 .Reintroducing the time variable and since we have that Then, using Tonelli's Theorem, we estimate N j 1 as Using the trivial inequality The interpolation inequality (1.9) provides us with We obtain that We compute As a consequence, we find that To prove that t * = ∞ we argue by contradiction.In the case where t * < ∞, necessarily we have that which is a contradiction with t * begin finite.
Then we have that where 0 < δ(|v 0 | 0 ).Furthermore, using a standard Poincaré-like inequality, we find that The inequality 3.2.Convergence of the approximate problems.Proposition 3.2 (Compactness results).Let v 0 ∈ A 0 (T d ) be a function satisfying the condition in (2.1) and ∂ be any differential operator of order one.Then, up to subsequences, we have that every approximating sequence of solutions {v N } N of (3.1) verifies ) ) ) ) Proof.We need to prove the compactness of the sequence of approximate solutions in appropriate spaces.Thanks to Proposition 3.1, the approximate solutions are uniformly bounded in Due to Banach-Alaoglu Theorem, this boundedness is enough to have (3.4).
Similarly, we have the uniform bound where ∂ 4 is a differential operator of order fourth.Using Banach-Alaoglu Theorem, we find (3.5).
The interpolation inequality in (1.9) gives us ) so (3.6) follows.Furthermore (3.9) implies both Let us begin with the proof of (3.11).Using the inequality (3.10) with r = 2 and the finiteness of the domain, we have that concluding (3.11).As far as (3.12) is concerned, being then we have We compute that As a consequence, we obtain that and the boundedness of H −2 dt follows thanks to (3.13).The fact that v ∈ C ([0, T]; L 2 (T d )) follows straightforwardly from (3.13) and the finiteness of H −2 dt.Then, up to a subsequence, we have both (3.7) and Invoking [18,Corollary 4], with the choice we obtain that Using interpolation in Sobolev spaces and the uniform boundedness in L 2 (0, T; H 2 (T d )), we have that for every 0 ≤ r < 2. Thus (3.8) follows.
3.3.Proof of Theorem 2.2.We have to pass to the limit in N to conclude the existence of the weak solution.First, let us remark that Using Propositions 3.1 and 3.2, together with the previous equality, we have that and we can pass to the limit in the weak formulation.The regularity in (2.2) follows from Proposition 3.2 as well.Finally, the weakly- * lower semicontinuity of the norm guarantees that so we recover (2.3).

3.4.
Estimates for the regularized problem in the Sobolev space H 2 .Proposition 3.3 (Estimates in Sobolev spaces).Let v 0 ∈ A 0 (T d ) ∩ H 2 (T d ) be a function satisfying the condition in (2.1).Then, every approximating sequence of solutions {v N } N of (3.1) is uniformly bounded in . Furthermore, we have that Proof.We omit the time variable when not needed.We consider the solutions of (3.1).We want to obtain appropriate bounds on Sobolev spaces such that we can pass to the limit in N. As before, to simplify the notation, we write v instead v N .We multiply the equation in (3.1) by ∆ 2 v N and integrate over (0, T) × T d , obtaining We have that We begin dealing with the I m terms.It holds We are left with Integrating by parts, Thus, We recall the following inequality Now we can estimate this terms using (3.15) and (3.16) as Putting all together, recovering the N and the t in the notation and using Theorem 2.2, we have that Finally,taking 0 < ε ≪ 1 small enough, we have that Thus, we deduce that v N ∈ L 2 (0, T; H 4 (T d )).

Proposition 3.4 (Compactness results
).Let v 0 ∈ A 0 (T d ) ∩ H 2 (T d ) be a function satisfying the condition in (2.1) and ∂ be any differential operator of order one.Then, up to subsequences, we have that every approximating sequence of solutions {v N } N of (3.1) verifies (3.17) Furthermore, the limit function v satisfies Proof.The proof of (3.17) is similar to Proposition 3.2.The continuity is obtained from the fact that Then, the limit procedure can be achieved as in the proof of Theorem 2.2 and (2.5) follows.Finally, the decay in (2.6) can be obtained using interpolation in Sobolev spaces

THE PROBLEM (1.6)
We recall the definition of the binomial coefficient: Then, the equation (1.6) becomes The approximating problem we consider is the following: For this problem, we can construct a solution v N following a standard Galerkin approach (one can also use the mollifier approach).
4.1.Estimates for the regularized problem in the Wiener algebra A 0 .Proposition 4.1 (Estimates in the Wiener algebra).Let v 0 ∈ A 0 (T d ) be a function satisfying the condition in (2.7).Then, every approximating sequence of solutions {v N } N of (4.1) is uniformly bounded in . Furthermore, we have that with δ(|v 0 | 0 ) defined in (2.7).
Proof.To simplify the notation, we write v instead of v N .We rewrite ∆ 2 v j (computed in (3.2)) as and set where thanks to the binomial coefficient identity Again, we omit to write the time variable when not necessary.The previous equality (4.3) in Fourier variables reads as where We recall (3.3), reintroduce the time variable in the notation and estimate (4.4) as follows: We are left with the estimates of |N j m (t)| 0 .We first apply Tonelli's Theorem and then the interpolation inequality (1.9) to obtain that Collecting the previous steps, we have We recall the following identities for power series We set The uniform boundedness W and ∂ be any differential operator of order one.Then, up to subsequences, we have that every approximating sequence of solutions {v N } N of (4.1) verifies ) ) ) Proof.Being the approximating solutions {v N } N uniformly bounded in thanks to Proposition 4.1, we can reason as in Proposition 3.2 in order to prove that (4.6), (4.7), (4.8) and (4.9) hold.As far as (4.10) is concerned we need In this way, we can reason as in Proposition 3.2 and prove a uniform bound for {∂ t v N } N in L 2 (0, T, H −2 (T d )) so the assertion follows from [18,Corollary 4].We compute with ϕ ∈ H 2 (T d ) and such that ϕ H 2 ≤ 1.We recall the following expressions (valid for for 0 < w < 1) for finite sums ≤ 24 (1 − w) 5 .
We come back to (4.11) and, letting w = |v N (t)|, we deduce (3.15) too.Finally, we get from which we deduce that Proof.Again, we write v instead of v N and omit the time variable if not needed.We multiply the main equation in (1.6) by ∆ 2 v obtaining 1 2 Again, we have that Recalling that |v 0 | 0 < 1 and using also (3.15), we estimate Here, we have also used the computations contained in Proposition 4.1.We are left with I 1 : Integrating by parts, we compute We just deal with since the other terms can be studied in the same way.We have that The uniform boundedness in L 2 (0, T; H 4 (T d )) follows from (4.12) and from the inequality which holds for N large and 0 < ε ≪ 1, since ).Let v 0 ∈ A 0 (T d ) ∩ H 2 (T d ) be a function satisfying the condition in (2.7) and ∂ be any differential operator of order one.Then, up to subsequences, we have that every approximating sequence of solutions {v N } N of (4.1) verifies Furthermore, the limit function v satisfies v ∈ C ([0, T]; H 2 (T d )).
Proof.The proof of (4.13) reasoning as in Proposition 3.4 (see also Proposition 3.2).More precisely, an interpolation in Sobolev spaces and the uniform boundedness in L 2 (0, T; H 4 (T d )) lead to and passing to the limit as in Subsection 4.3.Finally, the exponential decay (2.12) follows as in Subsection 3.6.

3. 6 . 4 .
Proof of Theorem 2.The regularity (2.4) follows from Propositions 3.3 and 3.4 and we are left with the proof of (2.5) and (2.6).The equality in (3.14) and the bound of the I m (t) terms in Proposition 3.3 provide us with

Convergence of the approximate problems. Proposition 4.2
1,1(0, T; A 0 (T d )) follows from the inequality (Compactness results).Let v 0 ∈ A 0 (T d ) be a function satisfying the condition in(2.7)