Some properties of solutions for an isothermal viscous Cahn–Hilliard equation with inertial term

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.

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.

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. □

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. (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. (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. □

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

Using (4.4)–(4.7), we deduce

By Lemma 4.1, we conclude that

This completes the proof. □

Acknowledgements

The authors would like to express their deep thanks to the referee for valuable suggestions for the revision and improvement of the manuscript.

Availability of data and materials

Not applicable.

Funding

This work is supported by the Jilin Scientific and Technological Development Program [number 20170101143JC].

Affiliations

1. Department of Mathematics, Jilin University, Changchun, China

   • Changchun Liu
   •  & Jiaojiao Wang

Contributions

All authors contributed equally to the manuscript and read and approved the final manuscript.

Corresponding author

Correspondence to Changchun Liu.

Competing interests

The authors declare that they have no competing interests.

Abbreviations

Not applicable.

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