SOME PROPERTIES OF SOLUTIONS FOR A SIXTH-ORDER PARABOLIC EQUATION IN ONE SPATIAL DIMENSION

Abstract

In this paper, we consider the existence and uniqueness of global weak solution for a sixth-order classical surface-diffusion equation in one spatial dimension. Moreover, the regularity and blow-up of solutions are also studied.

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∗.