We study the instability of the traveling waves of
a sixth-order parabolic equation which arises naturally as a continuum model
for the formation of quantum dots and their faceting. We prove that some
traveling wave solutions are nonlinear unstable under perturbations. These
traveling wave solutions converge to a constant as .
In this paper, we consider the following sixth-order parabolic equation
Equation (1.1) arises naturally as a continuum model for the formation of quantum dots and their faceting; see . Here denotes the surface slope. The high-order derivatives are a result of the additional regularization energy which is required to form an edge between two-plane surfaces with different orientations.
During the past years, only a few works have been devoted to the sixth-order parabolic equation [2–7]. Barrett et al.  considered the above equation with . A finite element method is presented which proves to be well posed and convergent. Numerical experiments illustrate the theory.
Recently, Jüngel and Milišić  studied the sixth-order nonlinear parabolic equation
They proved the global-in-time existence of weak nonnegative solutions in one space dimension with periodic boundary conditions.
Evans et al. [3, 4] considered the sixth-order thin film equation containing an unstable (backward parabolic) second-order term
By a formal matched expansion technique, they show that, for the first critical exponent for , where is the space dimension, the free-boundary problem with zero-height, zero-contact-angle, zero-moment, and zero-flux conditions at the interface admits a countable set of continuous branches of radially symmetric self-similar blow-up solutions , , where is the blow-up time.
Korzec et al.  considered the sixth-order equation
New type of stationary solutions is derived by an extension of the method of matched asymptotic expansion.
In this paper, we study instability of the traveling waves of (1.1). Our main result is as follows.
Theorem 1.1. All the traveling waves of (1.1) satisfying , are nonlinearly unstable in the space , where denotes derivative of .
The stability and instability of special solutions for the higher-order parabolic equation are very important in the applied fields. Carlen et al.  proved the nonlinear stability of fronts for the Cahn-Hilliard, under perturbations. Gao and Liu  prove that it is nonlinearly unstable under perturbations, for some traveling wave solution of the convective Cahn-Hilliard equation. The relevant equations have also been studied in [11, 12]. The main difficulties for treating (1.1) are caused by the principal part and the lack of the Lyapunov functional. Our proof is based on the principle of linearization. We invoke a general theorem that asserts that linearized instability implies nonlinear instability.
This paper is organized as follows. In the next section, we find an exact traveling wave solution for (1.1). In Section 3, we give the proof of our main result.
2. Exact Traveling Wave Solutions
In this section, we construct an exact traveling wave which satisfies all conditions of Theorem 1.1.
If is a traveling wave solution of (1.1), then satisfies the ordinary differential equation
Let . Then
Substituting the above equations into (2.1), we have
Then comparing the order of , we obtain
A simple calculation shows that , . Hence, we get
We easily proved that
and satisfies the conditions of the Theorem 1.1.
3. Proof of The Result
To prove the Theorem 1.1, we first consider an evolution equation
where is a linear operator that generates a strongly continuous semigroup on a Banach space , and is a strongly continuous operator such that . In , authors considered the whole problem only on space X, that is, the nonlinear operator maps to . However, many equations posses nonlinear terms that include derivatives and therefore, maps into a large Banach space . Hence, they again got the following lemma.
Lemma 3.1 (see ). Assume the following.(i)X, Z are two Banach spaces with and for .(ii)L generates a strongly continuous semigroup on the space Z, and the semigroup maps Z into X for and .(iii)The spectrum of on meets the right half-plane, .(iv) is continuous and such that , for . Then the zero solution of (3.1) is nonlinearly unstable in the space .
In this paper, we are going to use Lemma 3.1 for the proof of Theorem 1.1.
Definition 3.2. A traveling wave solution of (1.1) is said to be nonlinearly unstable in the space X, if there exist positive and , a sequence of solutions of (1.1) and a sequence of time such that but .
If is a traveling wave solution of (1.1), then letting , we have
with the initial value
So the stability of traveling wave solutions of (1.1) is translated into the stability of the zero solution of (3.4). In order to prove Theorem 1.1, taking , , we need to prove that the four conditions of Lemma 3.1 are satisfied by the associated equation (3.4). The condition (i) is satisfied by our choice of and .
Denote the linear partial differential operator in (3.4) by = with . Then (3.4) may be rewritten in the form of (3.1)
Note the F maps into , using the Sobolev embedding theorem, we have
So, the condition (iv) is satisfied.
To prove condition (ii) in Lemma 3.1, we need the following two lemmas.
Lemma 3.3. Let . Then
Proof. We write . By Fourier transformation
On the other hand, letting , we have
with , . Elementary computation shows that
since . Thus, Lemma 3.3 has been proved.
Lemma 3.4. Let + with , . Then
Proof. Consider the initial value problem
Then , , , thus
Denote , , , , and
Then, we have
where we use to denote . By iteration,
Let . Then
Multiplying both sides of the above inequality by , we have
Integrating the above inequality with respect to over , we obtain
Observing that is bounded and substituting the above inequality into (3.22), we get
thus (3.17) has been proven. Next, we prove the inequality (3.16). Clearly, we have
where is defined in Lemma 3.3, and we use to denote . By iteration,
The second term on the right of (3.29) is
where . By exchanging the order of integration, we get from the third term on the right side of (3.29),
Therefore (3.28)–(3.32) imply
From (3.17), we know . Then
Therefore, there exists a , such that
So, we proved the inequality (3.16). Hence (3.17) is proven and proof of Lemma 3.4 is finished.
We now proceed to verify condition (iii) of Lemma 3.1. Observing that if satisfies
then also satisfies the above equation. By uniqueness of solution, we know that generates a strongly continuous semigroup on the Banach space (see  p.344). By Fourier transformation, the essential spectrum of on is
The curve meets the vertical lines for because .
We now prove that the same curve belongs to the essential spectrum of .
Lemma 3.5. The essential spectrum of on contains that of .
Proof. Let and let . Following Schechter , if there exists a sequence with
and does not have a strongly convergent subsequence in . Here we use the definition if and only if is Fredholm with index zero. Now let be a function with compact support in . Define
where is chosen so that . In fact,
for some positive constant . Hence . Since but is bounded away from zero, can have no convergent subsequence in . It remains to show that . We write
A simple calculation shows that
Moreover, for any positive integer , , as , we have
From the assumptions on , we obtain
Similarly, we have