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Some properties of solutions for an isothermal viscous Cahn–Hilliard equation with inertial term
Boundary Value Problems volume 2018, Article number: 62 (2018)
Abstract
In this paper, we study the global existence and blow-up of solutions for an isothermal viscous Cahn–Hilliard equation with inertial term, which arises in isothermal fast phase separation processes. Based on the Galerkin method and the compactness theorem, we establish the existence of the global generalized solution. Using a lemma on the ordinary differential inequality of second order, we prove the blow-up of the solution for the initial-boundary problem.
1 Introduction
In this paper, we are concerned with the following initial-boundary problem:
where \(\Omega\subset{\mathbb {R}}^{n}\) (\(n\leq3\)) is a bounded domain with smooth boundary, \(\delta> 0\) is an inertial parameter, \(k \geq0\) is a viscosity coefficient, and \(f (s)\) is a given nonlinear function.
Equation (1.1) was proposed in [1] to model rapid spinodal decompositions in a binary alloy. Zheng and Milani [2] proved that the dynamical systems generated by problem (1.1)–(1.3) admit exponential attractors and inertial manifolds. Zheng and Milani [3] show that the dynamical systems admit global attractors and that these global attractors are at least upper-semicontinuous with respect to the vanishing of the perturbation parameter. Gatti et al. [4] considered problem (1.1)–(1.3). Their result is the construction of a robust family of exponential attractors, whose common basins of attraction are the whole phase-space. They [5] also considered the same problem in the three-dimensional setting.
Grasselli et al. [6] studied a differential model describing nonisothermal fast phase separation processes taking place in a three-dimensional bounded domain.
where \(\sigma\in[0,1]\). This model consists of a viscous Cahn–Hilliard equation characterized by the presence of an inertial term \(\chi_{tt}\), χ being the order parameter, which is linearly coupled with an evolution equation for the (relative) temperature ϑ.
The blow-up of solutions for the fourth order equation has been intensively studied. Chen and Lu [7] considered the initial-boundary value problem for the nonlinear wave equation
They obtained the blow-up of the solution and the energy decay of the solutions. Wang [8] studied the equation
He gave necessary and sufficient conditions for global existence and finite time blow-up of solutions. Escudero et al. [9] discussed a fourth order parabolic equation involving the Hessian
The authors proved the global existence versus blow-up results. Qu and Zhou [10] studied the following:
By using the method of potential wells, they obtained a threshold result of global existence and blow-up for the sign-changing weak solutions and the conditions under which the global solutions become extinct in finite time. In this paper, we consider the global existence and blow-up of solutions for problem (1.1)–(1.3). To prove the blow-up of solutions, we establish a new functional and consider the solution of the Bernoulli type equation. Basing on the required estimates and using a lemma on the ordinary differential inequality of second order, we prove the blow-up of the solution for the initial-boundary problem. The main method is nontrivial because of both the nonlinearity of \(\Delta f(u)\) and more delicate estimates which are necessary to overcome some delicate technical points.
The plan of this paper is as follows. In Sect. 2, we prove the existence and uniqueness of the global generalized solution for the initial-boundary value problems (1.1)–(1.3) by the Galerkin method. We also give some sufficient conditions of the blow-up of the solutions for the initial-boundary value problems (1.1)–(1.3) in Sect. 3. Finally, in Sect. 4, we discussed the decay rate of energy. For simplicity, we set \(\delta= 1\) in this paper.
2 Existence of the global solution
We are going to prove the existence and uniqueness for problems (1.1)–(1.3) by the Galerkin method and the compactness theorem in this section.
Let \({y_{i}(x)}\) be the orthonormal basis in \(L^{2}(\Omega)\) composed of the eigenfunctions of the eigenvalue problem
corresponding to eigenvalue \(\lambda_{i}\) (\(i=1,2,\ldots\)).
Let
be the Galerkin approximate solution for problem (1.1)–(1.3), where \(\gamma_{Ni}(t)\) are the undetermined functions and N is a natural number. Suppose that the initial value functions \(\varphi(x)\) may be expressed as
where \(\mu_{i}\) and \(\nu_{i}\) (\(i=1,2,\ldots\)) are constants.
Substituting the approximate solution \(u_{N}(x,t)\) into Eq. (1.1), multiplying both sides by \(y_{s}(x)\), we obtain
where \((\cdot,\cdot)\) denotes the inner product of \(L^{2}(\Omega)\).
Substituting the approximate solution \(u_{N}(x,t)\) and the approximations
of the initial value functions into the initial condition (1.3), we get
where \(\dot{\gamma}_{Ns}(t)=\frac{d}{dt}\gamma_{Ns}(t)\).
In order to prove the existence of the global generalized solution for problem (1.1)–(1.3), we make a series of estimations for the approximate solution \(u_{N}(x,t)\).
Lemma 2.1
Suppose that \(\varphi\in H^{2}(\Omega)\) and \(\psi\in L^{2}(\Omega)\) satisfy the boundary condition (1.2), \(f \in C^{1}(R)\), \(0\leq F(s)=\int_{0}^{s}f(\eta)\,d\eta\), and \(|f'(s)|\leq C_{1}|s|^{2}+C_{2}\), where \(C_{1}>0\), \(C_{2}>0\) are constants. Then the following estimate holds:
where and in the sequel \(C>0\) is a constant which only depends on T.
Proof
Let \(w_{n}\) be the unique solution of the problem
Substituting the approximate solution \(u_{N}(x,t)\) into Eq. (1.1), multiplying both sides by \(2w_{Nt}\), we obtain
Integrating by parts with respect to x on Ω, we have
Hence, we know
By the Sobolev imbedding theorem, it follows from (2.7) and (2.8) that
Multiplying both sides of (2.4) by \(2\gamma_{Nst}(t)\), summing up for \(s=1,2,\ldots, N\), we have
Integrating by parts with respect to x on Ω, we get
By \(|f'(s)|\leq C_{1}|u|^{2}+C_{2}\), hence
On the other hand, by the Gagliardo–Nirenberg inequality, (2.9), and (2.10), we see
Therefore, by (2.11),
Then, integrating (2.12) on \([0,t]\) and using the Gronwall inequality, we deduce
Immediately, we get (2.6) from (2.13). The proof is completed. □
Lemma 2.2
Suppose that the conditions of Lemma 2.1 hold. If \(f\in C^{3}(R)\), \(\varphi\in H^{4}(\Omega)\), \(\psi\in H^{2}(\Omega)\), and \(|f''(s)|\leq C_{3}|s|+C_{4}\), \(|f'''(s)|\leq C\), then the approximate solution for problem (1.1)–(1.3) satisfies the following estimate:
Proof
Multiplying both sides of (2.4) by \(2\lambda_{s}^{2}\gamma _{Nst}(t)\), summing up for \(s=1,2,\ldots, N\), we have
Integrating by parts with respect to x, we get
On the other hand, we know
and
By (2.13), we know that \(\|u\|_{\infty}\leq C\), hence
Thus,
By (2.13) and the Sobolev imbedding theorem, we see that
Using the Gagliardo–Nirenberg inequality, we conclude
On the other hand, by boundary conditions (1.2), we obtain
Substituting the above inequalities into (2.15), we get
Integrating the above inequality, and using the Gronwall inequality, we have
Similarly, multiplying both sides of (2.4) by \(\gamma_{Nstt}(t)\), summing up for \(s=1,2,\ldots, N\), we deduce
Integrating by parts with respect to x and using the Cauchy inequality, we have
Therefore, we conclude
Immediately, we get (2.14) from (2.16) and (2.17). This completes the proof. □
Theorem 2.1
Suppose that \(\varphi\in H^{4}(\Omega)\) and \(\psi\in H^{2}(\Omega)\) satisfy the boundary conditions (1.2), \(f\in C^{3}(R)\), \(0\leq F(s)=\int_{0}^{s}f(\eta)\,d\eta\), \(|f'(s)|\leq C_{1}|s|^{2}+C_{2}\), \(|f''(s)|\leq C_{3}|s|+C_{4}\), and \(|f'''(s)|\leq C\). Then problem (1.1)–(1.3) has a unique global generalized solution
Proof
From (2.14) we know that \(u_{N}\in C ([0,T];H^{4}(\Omega) )\), \(u_{Nt}\in C ([0,T];H^{2}(\Omega) )\), \(u_{Ntt}\in C ([0,T];L^{2}(\Omega) )\). Using the Sobolev imbedding theorem, we have \(D^{k}u_{N}\in C ([0,T]\times\Omega )\), \(0\leq k\leq2\). It follows from the above two relations and the Ascoli–Arzelá theorem that there exist a function \(u(x,t)\) and a subsequence of \({u_{N}(x,t)}\), still denoted by \({u_{N}(x,t)}\), such that as \(N\rightarrow\infty\), \({u_{N}(x,t)}\) uniformly converges to \(u(x,t)\) in \([0,T]\times\Omega\). The corresponding subsequence of \(\Delta u_{N}(x,t)\) also uniformly converges to \(\Delta u(x,t)\) in \([0,T]\times\Omega\). According to the compactness theorem, the subsequence \(D^{k}u_{N}(x,t)\) (\(0\leq k\leq4\)), \(D^{k}u_{Nt}(x,t)\) (\(0\leq k\leq2\)), and \(u_{Ntt}(x,t)\) weakly converge to \(D^{k}u(x,t)\) (\(0\leq k\leq4\)), \(D^{k}u_{t}(x,t)\) (\(0\leq k\leq2\)), and \(u_{tt}(x,t)\) in \(L^{2} ([0,T]\times\Omega )\), respectively. Hence, we know that \(u(x,t)\) satisfies (2.18). Therefore \(u(x,t)\) is the generalized solution for problem (1.1)–(1.3). It is easy to prove the uniqueness of the solutions for problem (1.1)–(1.3). This completes the proof of the theorem. □
3 Blow-up of solutions
In the previous sections, we have seen that the solution of problem (1.1)–(1.3) is globally existent, provided that \(F(s)\geq0\). In this section, we will prove the blow-up of the solution for \(F(s)<0\). For this purpose, we need the following lemma.
Lemma 3.1
([7])
Assume that \(u'=G(t, u)\), \(v'\geq G(t, v)\), \(G\in C([0,\infty)\times (-\infty, \infty))\), and \(u(t_{0})=v(t_{0})\), \(t_{0}\geq0\), then when \(t\geq t_{0}\), \(v(t)\geq u(t)\), where \(u'=\frac{d}{dt}u(t)\).
Let w be the unique solution of the problem
We have the following theorem.
Theorem 3.1
Suppose that
-
(1)
\(f(s)s\leq\gamma F(s)\), \(F(s)\leq-\alpha|s|^{p+1}\), where \(F(s)=\int_{0}^{s}f(u)\,du\), \(\gamma>2\), \(\alpha>0\), and \(p>1\) are constants.
-
(2)
\(\varphi\in H^{4}(\Omega)\), \(\psi\in H^{2}(\Omega)\) and
$$\begin{aligned} E(0)={}&\big\| \nabla w_{t}(0)\big\| ^{2}+\|\nabla\varphi \|^{2}+2 \int_{\Omega}F\bigl(\varphi (x)\bigr)\,dx \\ \leq{}& \frac{-2^{\frac{2p}{p-1}}}{ (\frac{\alpha(\gamma -2)}{p+3} )^{\frac{2}{p-1}} (1-e^{-\frac{p-1}{4}} )^{\frac{4}{p-1}}}< 0,\end{aligned} $$then the generalized solution u(x, t) of problem (1.1)–(1.3) blows up in finite time, i.e.,
$$\|\nabla w\|^{2}+ \int_{0}^{t}\big\| u(\cdot,\tau)\big\| ^{2}\,d\tau+ \int_{0}^{t}\big\| \nabla w(\cdot ,\tau) \big\| ^{2}\,d\tau+ \int_{0}^{t} \int_{0}^{\tau}\|u\|^{2}\,ds\,d\tau\to \infty,\quad \textit{as } t\to T^{*}. $$
Proof
Let
A simple calculation shows that
Noticing equation (1.1), we know that
which implies
Moreover, we easily see
Now, we define
It is obvious that
Further, we have
Integrating (3.5), we conclude that
Integrating (3.6), we deduce
Combining (3.5, (3.6) with (3.7), we derive
Substituting (3.4) into (3.8), we get
Recalling \(H''(t)>0\), \(H(t)\geq0\), and
therefore, from (3.9) we obtain
On the other hand, the Hölder inequality implies that
Thus
Substituting the above inequalities into (3.10), and by the fact \((x+y+z)^{n}\leq2^{2(n-1)}(x^{n}+y^{n}+z^{n})\), \(x,y,z>0\), \(n>1\), we know that
In addition, note from (3.6) and (3.7) that \(H'(t)\to+\infty \) and \(H(t)\to+\infty\) as \(t\to\infty\). Therefore, we see that there is \(t_{0}\geq1\) such that when \(t\geq t_{0}\), \(H'(t) > 0\) and \(H(t) > 0\). Multiplying (3.11) by \(2H'(t)\), we get
where
It follows from (3.12) that
Integrating the above inequality over \((t_{0}, t)\), we easily see
Note that when \(t\to\infty\), the right-hand side of (3.13) approaches positive infinity, hence, there is \(t_{1} > t_{0}\) such that when \(t\geq t_{1}\), the right-hand side of (3.13) is larger than or equal to zero. We thus have
that is,
where \(M_{1}=M^{1/2}\).
Now, we consider the initial value problem of the ordinary differential equation
Therefore, we conclude that
where
It is obvious that \(J(t_{1})=1>0\), and
By (3.7), we can take \(t_{1}\) sufficiently large such that
Condition (2) of Theorem 3.1 implies
Noticing the continuity of \(J (t)\), we know that there is a constant \(T^{*}\) (\(t_{1} < T^{*}< t_{1} + 1\)) such that \(J (T^{*} ) = 0\). Hence \(S(t)\to\infty\), as \(t\to T^{*}\). It follows from Lemma 3.1 that when \(t\geq t_{1}\), \(H(t)\geq S(t)\). Thus, \(H(t)\to \infty\) as \(t\to T^{*}\). Theorem 3.1 is proved. □
4 Decay rate of energy
In this section, we are going to discuss the decay rate of energy for problem (1.1)–(1.3). We need the following lemma.
Lemma 4.1
([11])
Suppose that \(J : [0,\infty)\to[0,\infty)\) is a non-increasing function and assume that there is a constant \(L > 0\) such that
Then
Theorem 4.1
Suppose that the assumptions of Theorem 2.1 hold and \(2F(s)\leq f(s)s\). Let \(u(x, t)\) be a global generalized solution for problem (1.1)–(1.3). Then we have
where \(G(t)=\int_{\Omega}|\nabla w_{t}|^{2}\,dx+\int_{\Omega}|\nabla u|^{2}\, dx+2\int_{\Omega}F(u)\,dx\).
Proof
Recalling (3.1), we derive
A simple computation gives, for any \(0\leq t_{1}\leq t_{2}<\infty\),
which shows that \(G(t)\) is non-increasing.
Multiplying (1.1) by \(w(x, t)\), integrating over \((t_{1},t_{2})\times \Omega\), and integrating by parts, we have
which implies
Recalling the assumption \(f(s)s\leq2F(s)\), we know
Using the Poincaré inequality, we obtain
The Cauchy inequality yields
On the other hand, by the non-increasing property of \(G(t)\), we get
By Lemma 4.1, we conclude that
This completes the proof. □
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The authors would like to express their deep thanks to the referee for valuable suggestions for the revision and improvement of the manuscript.
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This work is supported by the Jilin Scientific and Technological Development Program [number 20170101143JC].
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Liu, C., Wang, J. Some properties of solutions for an isothermal viscous Cahn–Hilliard equation with inertial term. Bound Value Probl 2018, 62 (2018). https://doi.org/10.1186/s13661-018-0982-2
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DOI: https://doi.org/10.1186/s13661-018-0982-2