On global solutions of the nonlinear Moore-Gibson-Thompson equation

1. Introduction

The object of this work is to study the global solution to the following boundary value problem for the Moore-Gibson-Thompson equation

where \(\Omega \) is a bounded domain in \(R^n(n\ge 1)\) with sufficiently smooth boundary \(\partial \Omega\), \(u_0(x),u_1(x)\) and \(u_2(x)\) are given functions and \(f\) is a given nonlinear function. All the parameters \(\alpha,\beta,c^2,r\) are assumed to be positive constants.

In recent years, increasing attention has been paid to the well-posedness and asymptotic behavior of the Moore-Gibson-Thompson (MGT) equation, see [1,2,3,4,5,6,7]. The MGT model is considered through third-order (in time), strictly hyperbolic partial differential equation as follows

it is one of the nonlinear acoustic models describing the propagation of acoustics wave in gases and liquid, it has a wide range of applications in medical and industry. In the physical context of the acoustic waves, \(u \) is the velocity potential of the acoustic phenomena, \(\alpha\) denotes the thermal relaxation time, \(c\) denotes the speed of sound, \(\beta\) denotes friction, and \(b\) denotes a parameter of diffusivity.

It is often convenient to write MGT equation as an abstract form

and it has been shown [8,9] that the linear part of Eq. (5) generates a strongly continuous semigroup as long as \(r>0\). In [10], the authors provided a brief overview of well-posedness results, both local and global, pertinent to various configurations of MGT equations. Especially, the authors in [11] considered the following model with nonlinear control feedback where the parameter \(\beta >0\), \(p(u)\) denotes an active force and the operator \(A\) is strictly positive. By semigroup method, it was proved in [11] we that (6) with initial data of arbitrary size in \(H\) is locally and globally well-posed under the following assumption: \(p\in C^1(R)\) and its derivative satisfies \(-\delta\le p'(s)\le m \) for some positive constants \(\delta \) and \(m\). Kaltenbacher et al., [12] established the well-posedness by Galerkin approximations and then employ fixed-point arguments for well-posedness of the Jordan-Moore-Gibson-Thompson (JMGT) equation More recently, Boulaaras et al., [13] proved the existence and uniqueness of the weak solution of the Moore-Gibson-Thompson equation with the integral condition by applying the Galerkin method.

In this paper, we extend the results in [11] to Problem (1)-(3) by applying the Galerkin method and compact method. The contents of this paper are organized as follows; In §2, we prepare some materials needed for our proof. Finally, in §3, we give the main result and the proof.

2. Preliminaries

Throughout this paper, the domain \(\Omega\) is assumed to be sufficiently smooth to admit integration by parts and second-order elliptic regularity. We use \(C\) to denote a universal positive constant that may have different values in different places. \(W^{m,2}(\Omega)=H^m(\Omega)\) and \(W^{m,2}_0(\Omega)=H^m_0(\Omega)\) denote the well-known Soblev space. We denote by \(||.||_p\) the \(L^p(\Omega)\) norm and by \(||\nabla .||\) the norm in \(H^1_0(\Omega)\). In particular, we denote \(||.||=||.||_2\)

By a weak solution \(u(x,t)\) of Problem (1)-(3) on \(\Omega\times [0,T]\) for any \(T>0\), we mean \(u\in L^{\infty}((0,T);H^2(\Omega)\cap H^1_0(\Omega))\cap W^{1,\infty}((0,T);H^1_0(\Omega))\cap W^{2, \infty}((0,T);L^2(\Omega))\), \(\Delta u_t, u_{ttt}\in L^{\infty}((0,T); H^{-1}(\Omega))\) such that \(u(x,0)=u_0(x)\) a.e. in \(\Omega\), \(u_t(x,0)=u_1(x)\) a.e. in \(\Omega\), \(u_{tt}(x,0)=u_2(x)\) a.e. in \(\Omega\), and

\begin{align} & \alpha (u_{ttt}+\beta u_{tt}- c^2\Delta u-r \Delta u_t+f(u),v)=0\nonumber \end{align} for any \(v\in H_0^1(\Omega)\), a.e. \(t\in[0,T]\).

In this paper, we assume \(\alpha,\beta,c^2,r>0\) and

Lemma 1.[14] Let \(\Omega \in R^n\) be a bounded domain and \(w_j\) be a base of \(L^2(\Omega)\). Then for any \(\epsilon>0\) there exist a positive constant \(N_{\epsilon}\), such that \[||u||\le (\sum_{j=1}^{N_{\epsilon}}(u,w_j))^{\frac{1}{2}}+\epsilon ||u||_{1,p}\] for any \(u\in W_0^{1,p}(\Omega)(2\le p< \infty)\), where \(N_{\epsilon}\) is independent on \(u\).

Lemma 2.[15] Let \(G(z_1,z_2,...z_h)\) be the function of the variables \(z_1,z_2,...z_h\) and suppose that \(G\) is continuous differentiable for k-times \((k\ge 1)\) with respect to every variable. Let \(z_i(x,t)\in L^\infty([0,T];H^k(\Omega))(i=1,2,...h)\), then the estimation \[\int_\Omega|D_x^kG(z_1(x,t),z_2(x,t),...,z_h(x,t))|^2dx< C(M,k,h)\sum_{i=1}^h||z_i||_{H^k(\Omega)}\] holds, where \(D_x=\frac{\partial}{\partial x}, M=\max_{i=1,2,...,h}\max_{0\le t\le T,x\in \Omega}|z_i(x,t)|\).

3. Solvability of the problem

In this section, by using Galerkin's method and compactness method, we shall prove the existence of global solutions of Problem (1)-(3).

Let \(\{w_j(x)\}_{j\in N}\) be the eigenfunctions of the following boundary problem

corresponding to the eigenvalue \(\lambda_j(j=1,2,3,...)\). Then \(\{w_j(x)\}_{j\in N}\) can be normalized to from an orthogonal basis of \(H^2(\Omega)\cap H_0^1(\Omega)\) and to be orthnormal with respect to the \(L^2(\Omega)\) scalar product.

Now, we seek an approximate solution of Problem (1)-(3) in the form of

where the constants \(T_{jN}\) are defined by the conditions \(T_{jN}(t)=(u^N(x, t),w_j (x))\) and can be determined from the relation

Lemma 3. Assume (8) holds, \(u_0 \in H^2(\Omega)\cap H_0^1(\Omega)\), \(u_1 \in H_0^1(\Omega)\), and \(u_2 \in L^2(\Omega)\), then for any \(T>0\), Problem (11)-(12) possesses a solution \(u^N\) on \([0,T]\), and the following estimate holds in the class

Proof. Problem (11)-(12) leads to a system of ODEs for unknown functions \(T_{jN} (t)\). Based on standard existence theory for ODE, one can obtain functions \( T_{jN} (t):[0, t_k) \rightarrow R , j = 1, 2, ..., k,\) which satisfy approximate Problem (11)-(12) in a maximal interval \([0, t_k), t_k\in(0, T]\). This solution is then extended to the closed interval \([0, T]\) by using the estimate below.

Multiplying (11) by \(T_{jNtt}(t)\), summing up the products for \(j=1,2,...,N\) and integrating by parts, we get

Integrating (14) with respect to \(t\) from 0 to \(t\), we obtain We observe that and Adding \(2[(u^N,u^N_t)+(u^N_{t},u^N_{tt})+( \nabla u^N, \nabla u^N_{t})]\) to both sides of (15) and a substitution of the equalities (16) and (17) in (15) gives Then, by Hölder inequality and the fact \(|f(s)|=|\int_0^tf'(s)ds|\le C_1|s|\) by (A1), we arrive at Taking into account that \[||u^N_{tt}(0)||^2+||\nabla u^N_t(0)||^2+||\nabla u^N(0)||^2\rightarrow ||u_2||^2 +||\nabla u_0||^2+||\nabla u_1||^2\] and \[||u^N(0)||^2+ ||u^N_t(0)||^2\rightarrow ||u_0||^2+||u_1||^2\] as \(N \rightarrow \infty\), then applying the Gronwall inequality to (19) and then integrating from 0 to \(t\) appears that Multiplying (11) by \(\lambda_j T_{jN}(t)\) summing up the products for \(j=1,2,...N\), integrating by parts and integrating with respect to \(t\), we get Combining Cauchy inequality, the fact \(||\Delta u^N(0)||^2\rightarrow ||\Delta u_0||^2\), and \(|f(s)|\le C_1|s|\), and making use of the following inequality \begin{align} \int_0^t(u^N_{ttt},\Delta u^N)d\tau &=(u^N_{tt},\Delta u^N)|_0^t -\int_0^t(u^N_{tt},\Delta u^N_t)d\tau\nonumber\\ &=(u^N_{tt},\Delta u^N)-(u^N_{tt}(0),\Delta u^N(0)) +\frac{1}{2}||\nabla u^N_t||^2-\frac{1}{2}||\nabla u^N_t(0)||^2, \nonumber \end{align} we have Choosing \(\epsilon_1\) sufficiently small and \(\epsilon_2\) sufficiently large such that \(\epsilon_2>2c^2\), then it follows from (22) and (20) that Thus, applying Gronwall's inequality to (23), we deduce Combining (20) and (24), we get Furthermore, by (25), we have that (11)-(12) possesses a global solution.

Theorem 1. Assume (8) holds, \(u_0 \in H^2(\Omega)\cap H_0^1(\Omega)\), \(u_1 \in H_0^1(\Omega)\), and \(u_2 \in L^2(\Omega)\), then for any \(T>0\), Problem (1)-(3) possesses a unique global solution.

Proof. For any \(v\in H_0^1(\Omega)\), it follows that

Thus, using Lemma 3, it follows that Similarly, we have From Lemma 3, (27) and (28), there exist a subsequence of \(\{u^N\}\), still denoted by \(\{u^N\}\), and a function \(u,\xi,\eta\), such that and for any \(t\in [0,T]\) Since \(f\in C^1\) and \(||f(u^N)||\le C||u^N||\le C\), for any \(v\in H_0^1(\Omega)\) and any \(t\in [0,T]\), we have as \(N\rightarrow \infty\). Then we get \(\xi=f(u), \eta=\nabla u_t\), combining this with (35)-(40), we have \[u\in L^{\infty}((0,T);H^2(\Omega)\cap H^1_0(\Omega))\cap W^{1,\infty}((0,T);H^1_0(\Omega))\cap W^{2, \infty}((0,T);L^2(\Omega)),\] \[\Delta u_t, u_{ttt}\in L^{\infty}((0,T); H^{-1}(\Omega)).\] By using Lemma 3 and (27), we observe that where \(u^N_{t^k}=\frac{\partial^k u^N}{\partial t^k}\). Then, by Ascoli-Arcela theorem, we can select from \(\{u^N\}\) a subsequence, still denoted by \(\{u^N\}\), such that as \(N\rightarrow \infty\), the subsequence In particular, we take \(t=0\) and we note that \(\{w_j(x)\}_{j\in N}\) are an orthogonal basis of \(L^2(\Omega)\), we know that By (29)-(34),(44) and Lemma 2.1, we have Thanks to (29)-(42), letting \(N\rightarrow \infty\) in (11), leads to for any \(v\in H_0^1(\Omega)\). Altogether, we conclude that \(u\) is a solution of the initial boundary Problem (1)-(3).

Now, suppose that there exist two different solutions \(u_1,u_2\) for Problem (1)-(3), then the difference \(w=u_1-u_2\) satisfies

Integrating (48) for \(t\) from 0 to \(t\), we have Multiplying the Eq. (51) by \(w_t\), integrating over \(\Omega\), adding up \((w,w_t)\), we obtain where we have used mean value theorem and \(|\theta|\le 1\). By applying Gronwall inequality, we deduce that This implies that \(w=0\) for all \(t\in[0,T]\). Thus the uniqueness is proved.

Acknowledgments :

The authors would like to thank the referee for his/her valuable comments that resulted in the present improved version of the article.

Author Contributions:

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Conflicts of Interest:

''The authors declare no conflict of interest.''