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A further study on the coupled Allen–Cahn/Cahn–Hilliard equations
Boundary Value Problems volume 2019, Article number: 54 (2019)
Abstract
In this paper, we will show that solutions of the initial boundary value problem for the coupled system of Allen–Cahn/Cahn–Hilliard equations continuously depend on parameters of the system, and under some restrictions on the parameters all solutions of the initial boundary value problem for Allen–Cahn/Cahn–Hilliard equations tend to zero with an exponential rate as \(t\rightarrow\infty\).
1 Introduction
In this paper, we consider the following Allen–Cahn/Cahn–Hilliard system:
where \(\mu=f'(h)-\gamma\partial_{x}^{2}h\), \(f'(h)=h^{3}-h\), and \(\vec {0}\) is a zero vector of \(R^{N}\), \(mk>\frac{5}{4}\), \(m, k, \alpha>0\) are given numbers, \(A(x,t)=(A_{1}(x,t),\ldots,A_{N}(x,t))\) is the unknown vector function, \(h(x,t)\) is the unknown scalar function, \(A_{0}(x)\) and \(h_{0}(x)\) are given initial data.
System (1.1)–(1.4) was introduced to model simultaneous order-disorder and phase separation in binary alloys on a BCC lattice in the neighborhood of the triple point [1]. Here, h denotes the concentration of one of the components, while A is an order parameter. The Allen–Cahn equation and the Cahn–Hilliard equation have been intensively studied [2,3,4,5]. Miranville, Saoud, and Talhouk [5] studied the long time behavior, in terms of finite-dimensional attractors, of a coupled Allen–Cahn/Cahn–Hilliard system. In particular, they proved the existence of an exponential attractor and, as a consequence, the existence of a global attractor with finite fractal dimension. Çelebi and Kalantarov [6] proved the decay of solutions and structural stability for the coupled Kuramoto–Sivashinsky–Ginzburg–Landau equations.
The large time behavior and the structural stability of solutions are important for the study of a higher-order parabolic system. Many papers have already been published to study the decay and the structural stability of solutions [7,8,9]. In this paper, we consider the asymptotic behavior of solutions and the continuous dependence of solutions for system (1.1)–(1.4). We are going to show the continuous dependence when the coefficient changes, which helps us to know whether a coefficient in the system can cause a large change in the solution.
The following is the main result of the paper.
Theorem 1.1
If
then all solutions of problem (1.1)–(1.4) tend to zero with an exponential rate as \(t\rightarrow\infty\).
Theorem 1.1 implies that the concentration of one of the components and the order parameter will tend to zero as \(t\rightarrow \infty\). Hence one of the components will disappear and the system will become disorder in a background point of view.
To prove Theorem 1.1, the basic a priori estimates are the \(L^{2}\) norm estimates on h and \(\partial_{x} h\). The main difficulties are caused by the nonlinearity of both the diffusive and the convective factors in equation (1.2). To overcome such difficulty, we establish two new functionals \(E_{1}(t)\) and \(E_{2}(t)\). Our method is based on the global energy estimates and require some delicate local integral estimates.
This paper is arranged as follows. We first study a priori estimates in Sect. 2, and then establish the exponential decay of solution in Sect. 3. Subsequently, we discuss the continuous dependence results in Sect. 4.
2 A priori estimates
Similar to [10], we know that problem (1.1)–(1.4) has a unique global solution. The first step is to obtain a priori estimates of solutions of system (1.1), (1.2). Applying the operator \(P^{2}\) to both sides of equation (1.2), here \(P^{2}\) is the inverse operator of the operator \(L=-\frac{d^{2}}{dx^{2}}\) with the domain of definition \(D(L)=H^{2}(0,l)\cap H_{0}^{1}(0,l)\), we get the following problem:
Multiplying equation (2.1) by A and (2.2) by h shows
and
Adding the two resulting equations together, we obtain
that is,
Due to the Cauchy–Schwarz inequality, we have
Choosing \(\varepsilon=\frac{\gamma}{3}-\frac{1}{3m\lambda_{1}^{2}}\) and \(C_{1}(\varepsilon)=\frac{1}{4\varepsilon}\) for the last inequality and using it in (2.8), we get
Thanks to (2.9), it is true that
and
where \(D_{1}(t)=[\|A_{0}\|^{2}+\|Ph_{0}\|^{2}]e^{(2\alpha+C_{2})t}\), \(D_{2}(t)=D_{1}(t)+\|A_{0}\|^{2}+\|Ph_{0}\|^{2}\).
Based on (2.9) and (2.10), we use the standard Faedo–Galerkin method to prove the existence of a global weak solution \([A,h]\) of problem (2.1)–(2.4) with the following properties:
Multiplying (1.2) by h in \(L^{2}(0,l)\), we have
By Young’s inequality, we see that
Applying (2.11), integrating inequality (2.14) in \((0,t)\) gives the following estimate:
where \(D_{3}(t):= (\frac{1}{4mk}+\frac{1}{4m^{2}}+\frac{3}{2\gamma} )D_{2}(t) +\frac{1}{2}\|h_{0}\|^{2}\). Therefore, we deduce
Multiplying equation (1.1) by \(\partial_{t}A\) in \(L^{2}(0,l)\), we see
Depending on Sobolev’s imbedding theorem and estimates (2.11), (2.16), the term on the right-hand side of (2.17) can be estimated as follows:
Thus, according to (2.17),
It is easy to obtain the estimate
where
Remark 2.1
If \(\lambda_{1} (\gamma m-\frac{1}{2\lambda_{1}^{2}} )-\frac {3m}{2}=r_{0}>0\), the following uniform estimate holds true:
where \(\gamma_{1}:=2\lambda_{1}\min{\{1,r_{0}\}}\).
Proof
We deduce from (2.6) the inequality
From (2.5), we know that
Adding the above two inequalities, we derive
that is,
Hence
Taking \(\gamma_{1}=2\lambda_{1}\min{\{1,r_{0}\}}\), we have
We get (2.19) by integrating the last inequality. □
3 Exponential decay of solution
In this section, we are going to prove the exponential decay of solution.
Proof of Theorem 1.1.
Multiplying in \(L^{2}(0,l)\) (2.1) by A, (2.2) by h, and adding the obtained relations, we get
that is,
which implies
where \(d_{0}:=\lambda_{1}-\alpha\), \(r_{0}:=\lambda_{1} (\gamma m-\frac{1}{2\lambda_{1}^{2}} )-\frac{3m}{2}\), and \(\gamma_{0}=\min{\{d_{0},r_{0}\lambda_{1}\}}\). Hence, we have
We conclude from (3.1) that
Integrating this inequality over the interval \((0,t)\) and employing estimate (3.3), we obtain
We can know that if \(A_{0}, h_{0}\in H_{0}^{1}(0,l)\) then problem (1.1)–(1.4) has a unique weak solution such that
Taking the inner product of (2.2) with \(-\partial _{x}^{2}h\), we have
Besides, we know
Let us multiply (3.6) by a positive parameter \(\varepsilon_{1}\) and add it with the above inequality
Choosing \(\varepsilon_{1}=2md_{0}\), we obtain the inequality
where
and \(\delta_{1}:=\min{\{d_{0},\lambda_{1}r_{0},\frac {r_{0}}{\varepsilon_{1}}\}}\). Then we obtain
Taking the inner product in \(L^{2}(0,l)\) of (2.1) with \(\partial_{t}A\), we get
According to [6],
Multiplying (3.4) by \(\frac{1}{2}\) and adding with the inequality, we obtain
where
We choose in the last inequality \(\varepsilon_{2}=\frac{k}{4}\), \(\varepsilon_{3}=\frac{r_{0}}{2\lambda _{1}}\) and obtain the equality
where
Using estimate (3.8), we have
where \(A_{2}=A_{1}E_{1}^{3}(0)+A_{0}E_{1}(0)\). Integrating (3.9) and by Gronwall’s inequality, we get
Combining with (3.3), we have
Multiplying (2.2) by \(\partial_{t}h\), we obtain
Adding \(\frac{m}{2}\) (3.6) to (3.12), we get
thus
Note that
Hence, we know that
Finally, we integrate (3.13) and get
where \(c_{1}=-\frac{E_{2}(0)}{2m\delta_{1}}e^{-\delta_{1}t}+\frac {m}{2}\|\partial_{x}h_{0}\|^{2}+ \frac{m}{4}\int_{0}^{l}|h_{0}|^{4}\,dx-\frac{E_{2}(0)}{2m\delta_{1}}\). The theorem is true. □
4 Continuous dependence results
Assume that \([\tilde{A},\tilde{h} ]\) is the weak solution of the problem
where \(\tilde{\mu}=f'(\tilde{h})-\gamma\partial_{x}^{2}\tilde{h}\), \(f'(\tilde{h})=\tilde{h}^{3}-\tilde{h}\).
Theorem 4.1
Assume that \([A,h]\) is a solution of problem (1.1)–(1.4) and \([\tilde{A},\tilde{h} ]\) is a solution of problem (4.1)–(4.4). Let \([a,H]= [A-\tilde{A},h-\tilde{h} ]\), we have
where \(q_{0}=\frac{2k_{1}^{\frac{4}{3}}C_{0}^{4}}{(k_{2}b_{0})^{\frac{1}{3}}}+ \frac{C_{0}^{4}}{4m^{3}\lambda_{1}^{4}\gamma^{2}}\),
Proof
Note that \([a,H]= [A-\tilde{A},h-\tilde{h} ]\) is a solution of the following problem:
By
and
we see that \([a,H]\) satisfies the following system:
where \(k_{2}=\tilde{k}\), \(k_{1}=\tilde{k}-k\).
On the other hand, we know that
Multiplying (4.7) by a and using inequality (4.9), we obtain
We are going to estimate the first integral on the right-hand side of (4.10) by the Nirenberg inequality as follows:
We can infer from the Nirenberg inequality and the Friedrichs inequality that the following estimate of the second term on the right-hand side of (4.10) is true:
Employing Young’s inequality and Sobolev’s inequality, we have
Then employing the last inequality and (4.11), (4.12), we deduce the following inequality from (4.10):
Multiplying (4.8) in \(L^{2}(0,l)\) by H, we get
and
Employing Sobolev’s imbedding theorem and the Nirenberg inequality, we get
Taking \(\varepsilon=\frac{\gamma m}{8}\) in (4.13) and adding it to (4.17), we get
where
From (4.18) we derive the estimate
where \(q_{0}=\frac{2k_{1}^{\frac{4}{3}}C_{0}^{4}}{(k_{2}b_{0})^{\frac{1}{3}}}+ \frac{C_{0}^{4}}{4m^{3}\lambda_{1}^{4}\gamma^{2}}\). The proof is complete. □
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This work is supported by the Jilin Scientific and Technological Development Program [number 20170101143JC].
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Yang, J., Liu, C. A further study on the coupled Allen–Cahn/Cahn–Hilliard equations. Bound Value Probl 2019, 54 (2019). https://doi.org/10.1186/s13661-019-1166-4
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DOI: https://doi.org/10.1186/s13661-019-1166-4