In this paper, we consider the existence and uniqueness of global weak solution for a sixthorder classical surfacediffusion equation in one spatial dimension. Moreover, the regularity and blowup of solutions are also studied.
AMS Mathematics Subject Classification : 35K55, 49A22.
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 surfacediffusion equation (see
[1]
)
where
v_{n}
is the normal surface velocity,
D
=
D_{S}
S
_{0}
Ω
_{0}
V
_{0}
=(
RT
)
^{23}
(
D_{s}
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,
C_{αβ}
is the surface curvature tensor and
μ_{w}
being an exponentially decaying function of
u
that has a singularity at
u
→ 0 (see
[1]
).
In the smallslop approximation, in the particular cases of highsymmetry 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
Q_{T}
= (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}
(
Q_{T}
). In Section 3, we consider the regularity of the solution for problem (3). In the last section, we consider the blowup of solutions for the above problem.
In the following, the letters
C
,
C_{i}
, (
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
D^{i}u
_{0}
(0,
t
) =
D^{i}u
_{0}
(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
D
^{6}
u
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
) ∈
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
 < +∞,
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
) ∈
Q_{T}
.
If u is a smooth solution for the following linear problem
then, for any
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
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
) ∈
Proof
. By (5) and (8), we have
Integrating the equation of (3) with respect to x over
where 0 <
t
_{1}
<
t
_{2}
<
T
, Δ
t
=
t
_{2}
−
t
_{1}
, 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
D
^{2}
u
based on Lemma 3.1. Suppose that
w
=
D
^{2}
u
−
D
^{2}
u
_{0}
, 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 LeraySchauder 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. Blowup
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,
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
L^{p}
space, we get
It then follows from (20) that
A direct integration of (21), we obtain
where
Noticing that
u
_{0}
≢ 0, then
Combining (19) and above inequality, setting
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.
email: zhaoxiaopeng@sina.cn
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