1. Introduction
In the study of a thin, solid film grown on a solid substrate, in order to describe the continuum evolution of the film free surface, there arise a classical surface-diffusion equation (see [1])
where vn is the normal surface velocity, D = DSS0Ω0V0=(RT) 23 (Ds is the surface diffusivity, S0 is the number of atoms per unit area on the surface, Ω0 is the atomic volume, V0 is the molar volume of lattice cites in the film, R is the universal gas constant and T is the absolute temperature), ΔS is the surface Laplace operator, v is the regularization coefficient that measures the energy of edges and corners, Cαβ is the surface curvature tensor and μw being an exponentially decaying function of u that has a singularity at u → 0 (see [1]).
In the small-slop approximation, in the particular cases of high-symmetry orientations of a crystal with cubic symmetry, neglect the exponentially decaying, consider the 1D case, then the evolution equation (1) for the film thickness can be written in the following form
(see [1]). Moreover, from a mathematical point of view, we will consider the nonlinear parabolic problem
where QT = (0, 1) × (0, T) and p > 2, γ, k > 0, α are constants
In this paper, we consider some properties of solutions for problem (3). This paper is organized as follows. In the next section, we establish the existence of global weak solution in the space H6,1(QT ). In Section 3, we consider the regularity of the solution for problem (3). In the last section, we consider the blow-up of solutions for the above problem.
In the following, the letters C, Ci, (i = 0, 1, 2, · · · ) will always denote positive constants different in various occurrences.
2. Global weak solution
In this section, we consider the existence and uniqueness of global weak solutions of the problem (3).
Theorem 2.1. Assume that α > 0, p ≥ 4, u0 ∈ H3(0, 1) with Diu0(0, t) = Diu0(1, t) = 0 (i = 0, 2)), then for all t ∈ (0, T), there exists a unique solution u(x, t) such that
Proof. Multiplying the equation of (3) by u and integrating with respect to x over (0, 1), we obtain
Noticing that
Hence, a simple calculation shows that
Gronwall’s inequality implies that
The energy function is
Integrations by parts and (3) yield
Therefore
That is
It then follows from (5) that
Summing the above two inequalities together, we get
By (5), (6) and Sobolev’s embedding theorem, we have
Again multiplying the equation of (3) by D6u and integrating with respect to x over (0; 1), we obtain
By Nirenberg’s inequality, we get
and
Using (8) and above four inequalities, we derive that
On the other hand, we also have
Then, summing up, we get
Hence
Therefore, by (9), (10) and (12), we immediately obtain
The a priori estimates (5)-(6) and (12)-(13) complete the proof of global existence of a u(x, t) ∈ L2(0, T;H6(0, 1)) ∩ L∞(0, T;H3(0, 1)).
Since the proof of uniqueness of global solution is so easy, we omit it here. Then, we complete the proof.
3. Regularity
The following Lemma (see [4]) will be used to prove the main result of this section.
Lemma 3.1. Assume that sup |ƒ| < +∞, O < α < 1, and there exist two constants a0, b0, A0,B0 such that 0 < a0 ≤ a(x, t) ≤ A0, 0 < b0 ≤ b(x, t) ≤ B0 for all (x, t) ∈ QT. If u is a smooth solution for the following linear problem
then, for any there is a constant C depending on a0, b0, A0, B0, δ, T, ∫∫QT u2dxdt and ∫∫QT |D3u|2dxdt, such that
Now, we turn our discussion to the regularity of solutions.
Theorem 3.2. Assume that p ≥ 6, u0 ∈ C6+k[0, 1], (0 < k < 1), then for any smooth initial value u0, problem (3) admits a unique classical solution u(x, t) ∈
Proof. By (5) and (8), we have
Integrating the equation of (3) with respect to x over where 0 < t1 < t2 < T, Δt = t2 − t1, we see that
where For simplicity, set
Then, (14) is converted into
Integrating the above equality with respect to y over we derive that
Here, we have used the mean value theorem, where θ∗ ∈ (0, 1). Then, by Hölder’s inequality and (8), (13), we end up with
Similar to the discussion above , we have
and
We shall consider the Hölder estimate of D2u based on Lemma 3.1. Suppose that w = D2u − D2u0, then w satisfies the following problem
where a(x, t) = γ, b(x, t) = k and
Define the linear spaces
and the associated operator T : X → X, u → v, where v is determined by the following linear problem
From the classical parabolic theory (see[3,5]), we know that the above problem admits a unique solution in the space Thus, the operator T is well defined. It follows from the embedding theorem that the operator T is a compact operator. If u = σTu holds for some σ ∈ (0, 1], then by the previous arguments, we know that there exists a constant C which is independent of u and σ, such that Then, it follows from the Leray-Schauder fixed point theorem that the operator T admits a fixed point u, which is the desired solution of problem (3). Furthermore, by the above arguments, we know that u is a classical solution.
4. Blow-up
In the previous section, we have seen that the solution of problem (3) is globally classical, provided that α > 0. The following theorem shows that the solution of the problem (3) blows up at a finite time for α < 0 and F(0) ≤ 0.
Theorem 4.1. Assume u0 ≢ 0, p > 2, α < 0 and F(0) ≤ 0, then the solution of problem (3) must blow up at a finite time, namely, for some T∗ > 0,
Proof. Without loss of generality, we assume that Otherwise, we may replace u by v = u − M, where For the energy functional F(t),a direct calculation yields that F′(t) ≤ 0, which implies that F(t) ≤ F(0). Let ω be the unique solution of the problem
Based on the equation of (3), we immediately obtain then, such function as ω is exists, which satisfies
Multiplying the equation of (3) by ω and integrating with respect to x over (0, 1), integrating by parts and using the boundary value conditions, we deduce that
By Poincaré’s inequality and the embedding of Lp space, we get
It then follows from (20) that
A direct integration of (21), we obtain
where Noticing that u0 ≢ 0, then Combining (19) and above inequality, setting we get u must blow up in a finite time T∗.
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