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SOME PROPERTIES OF SOLUTIONS FOR A SIXTH-ORDER PARABOLIC EQUATION IN ONE SPATIAL DIMENSION
SOME PROPERTIES OF SOLUTIONS FOR A SIXTH-ORDER PARABOLIC EQUATION IN ONE SPATIAL DIMENSION
Journal of Applied Mathematics & Informatics. 2014. Sep, 32(3_4): 547-554
Copyright © 2014, Korean Society of Computational and Applied Mathematics
  • Received : October 05, 2013
  • Accepted : March 17, 2014
  • Published : September 28, 2014
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XIAOPENG ZHAO

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. AMS Mathematics Subject Classification : 35K55, 49A22.
Keywords
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] )
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where vn is the normal surface velocity, D = DS S 0 Ω 0 V 0 =( RT ) 23 ( Ds is the surface diffusivity, S 0 is the number of atoms per unit area on the surface, Ω 0 is the atomic volume, V 0 is the molar volume of lattice cites in the film, R is the universal gas constant and T is the absolute temperature), Δ S 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
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(see [1] ). Moreover, from a mathematical point of view, we will consider the nonlinear parabolic problem
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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 H 6,1 ( 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, u 0 H 3 (0, 1) with Diu 0 (0, t ) = Diu 0 (1, t ) = 0 ( i = 0, 2)), then for all t ∈ (0, T ), there exists a unique solution u ( x , t ) such that
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Proof . Multiplying the equation of (3) by u and integrating with respect to x over (0, 1), we obtain
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Noticing that
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Hence, a simple calculation shows that
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Gronwall’s inequality implies that
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The energy function is
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Integrations by parts and (3) yield
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Therefore
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That is
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It then follows from (5) that
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Summing the above two inequalities together, we get
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By (5), (6) and Sobolev’s embedding theorem, we have
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Again multiplying the equation of (3) by D 6 u and integrating with respect to x over (0; 1), we obtain
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By Nirenberg’s inequality, we get
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and
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Using (8) and above four inequalities, we derive that
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On the other hand, we also have
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Then, summing up, we get
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Hence
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Therefore, by (9), (10) and (12), we immediately obtain
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The a priori estimates (5)-(6) and (12)-(13) complete the proof of global existence of a u ( x, t ) ∈ L 2 (0, T ; H 6 (0, 1)) ∩ L (0, T ; H 3 (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 | < +∞,
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O < α < 1, and there exist two constants a 0 , b 0 , A 0 , B 0 such that 0 < a 0 a ( x, t ) ≤ A 0 , 0 < b 0 b ( x, t ) ≤ B 0 for all ( x, t ) ∈ QT . If u is a smooth solution for the following linear problem
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then, for any
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there is a constant C depending on a 0 , b 0 , A 0 , B 0 , δ , T , ∫∫QT u 2 dxdt and ∫∫QT | D 3 u | 2 dxdt, such that
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Now, we turn our discussion to the regularity of solutions.
Theorem 3.2. Assume that p ≥ 6, u 0 ∈ C 6+k [0, 1], (0 < k < 1), then for any smooth initial value u 0 , problem (3) admits a unique classical solution u ( x, t ) ∈
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Proof . By (5) and (8), we have
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Integrating the equation of (3) with respect to x over
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where 0 < t 1 < t 2 < T , Δ t = t 2 t 1 , we see that
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where
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For simplicity, set
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Then, (14) is converted into
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Integrating the above equality with respect to y over
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we derive that
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Here, we have used the mean value theorem, where
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θ ∈ (0, 1). Then, by Hölder’s inequality and (8), (13), we end up with
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Similar to the discussion above , we have
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and
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We shall consider the Hölder estimate of D 2 u based on Lemma 3.1. Suppose that w = D 2 u D 2 u 0 , then w satisfies the following problem
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where a ( x, t ) = γ , b ( x, t ) = k and
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Define the linear spaces
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and the associated operator T : X X, u v , where v is determined by the following linear problem
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From the classical parabolic theory (see [3 , 5] ), we know that the above problem admits a unique solution in the space
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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
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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 u 0 ≢ 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,
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Proof . Without loss of generality, we assume that
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Otherwise, we may replace u by v = u M , where
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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
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Based on the equation of (3), we immediately obtain
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then, such function as ω is exists, which satisfies
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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
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By Poincaré’s inequality and the embedding of Lp space, we get
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It then follows from (20) that
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A direct integration of (21), we obtain
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where
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Noticing that u 0 ≢ 0, then
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Combining (19) and above inequality, setting
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we get u must blow up in a finite time T .
BIO
Xiaopeng Zhao received M.Sc. from Jilin University and Ph.D at Jilin University. Since 2013 he has been at Jiangnan University. His research interests include properties of higherorder nonlinear parabolic equations and numerical solutions for PDE.
School of Science, Jiangnan University, Wuxi 214122, China.
e-mail: zhaoxiaopeng@sina.cn
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