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
\begin{align}
&
\alpha u_{ttt}+\beta u_{tt}- c^2\Delta u-r \Delta u_t+f(u)=0, ~
\text{in}~ \Omega \times (0,+\infty),\label{1.1}\\
\end{align}
(1)
\begin{align}
&
u(x,t)=0~\text{on}~ \partial \Omega, \label{1.2}\\
\end{align}
(2)
\begin{align}
&
u(x, 0) = u_0(x), u_t(x, 0) = u_1(x),u_{tt}(x, 0) = u_2(x),~x \in \Omega, \label{1.3}
\end{align}
(3)
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
\begin{align}
\alpha u_{ttt}+\beta u_{tt}- c^2\Delta u-r \Delta u_t=f(x), \label{1.4}
\end{align}
(4)
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
\begin{align}
\alpha u_{ttt}+\beta u_{tt}+ c^2A u+rA u_t=f(u,u_t,u_{tt}), \label{1.5}
\end{align}
(5)
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
\begin{align}
\tau u_{ttt}+\alpha \beta u_{tt}+ c^2A u+bA u_t+\beta u_{t}^3=2k u_{t}^2+p(u), \label{1.6}
\end{align}
(6)
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
\begin{equation}
\alpha u_{ttt}+\beta u_{tt}-b \Delta u_t- c^2\Delta u
=(\frac{1}{c^2}\frac{B}{2A}u^2_t+|\nabla u|^2)_t.\label{1.7}
\end{equation}
(7)
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
\begin{align}
&f\in C^1~ and ~|f'(s)|\le C_1.\label{2.1}
\end{align}
(8)
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
\begin{align}
&-\Delta w=\lambda w, x\in \Omega; w=0, x\in \partial \Omega,\label{3.1}
\end{align}
(9)
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
\begin{align}
u^N(x, t) =\sum_{j=1}^{N} T_{jN} (t)w_j (x),\label{3.2}
\end{align}
(10)
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
\begin{align}
&\alpha (u^N_{ttt},w_j)+\beta (u^N_{tt},w_j)-c^2( \Delta u^N, w_j)
– r( \Delta u^N_t, w_j)+(f( u^N),w_j)=0 , \label{3.3} \\
\end{align}
(11)
\begin{align}
&
(u^N(0), w_j)=(u_0, w_j)= u_{0j}, (u^N_t (0),w_j) = (u_1,w_j)
= u_{1j},(u^N_{tt} (0),w_j) = (u_2,w_j) = u_{2j}.\label{3.4}
\end{align}
(12)
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
\begin{align}
&
||u^N||^2+||u^N_t||^2+||\nabla u^N||^2+\alpha ||u^N_{tt}||^2+r||\nabla u^N_t||^2
+ \int_0^t||\nabla u^N_t||^2d\tau +\beta ||\nabla u^N_{tt}||^2d\tau\le C . \label{3.5}
\end{align}
(13)
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
\begin{align}&
\alpha (u^N_{ttt},u^N_{tt})+\beta (u^N_{tt},u^N_{tt})+c^2( \nabla u^N, \nabla u^N_{tt})
+ r( \nabla u^N_t,\nabla u^N_{tt})+(f( u^N),u^N_{tt})=0. \label{3.6}
\end{align}
(14)
Integrating (14) with respect to \(t\) from 0 to \(t\), we obtain
\begin{align}
\alpha ||u^N_{tt}||^2+&2\beta \int_0^t||u^N_{tt}||^2d\tau +r||\nabla u^N_t||^2
+2\int_0^t(f( u^N),u^N_{tt})d\tau\nonumber\\
&
=-c^2\int_0^t( \nabla u^N, \nabla u^N_{tt})d\tau
+\alpha ||u^N_{tt}(0)||^2+ r||\nabla u^N_t(0)||^2. \label{3.7}
\end{align}
(15)
We observe that
\begin{align}
&
\int_0^t(f( u^N),u^N_{tt})d\tau
=(f( u^N),u^N_{tt})|_0^t-\int_0^t\int_{\Omega}f'( u^N)(u^{N}_t)^2dxd\tau \label{3.8}
\end{align}
(16)
and
\begin{align}
&
\int_0^t( \nabla u^N, \nabla u^N_{tt})d\tau
=(\nabla u^N,\nabla u^N_t)|_0^t-\int_0^t||\nabla u^N_t||^2d\tau. \label{3.9}
\end{align}
(17)
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
\begin{align}
\frac{d}{dt}&[||u^N||^2+||u^N_t||^2+||\nabla u^N||^2]+\alpha ||u^N_{tt}||^2
+2\beta \int_0^t||u^N_{tt}||^2d\tau +r||\nabla u^N_t||^2\nonumber\\
&
=2[(u^N,u^N_t)+(u^N_{t},u^N_{tt})+( \nabla u^N, \nabla u^N_{t})]
+\alpha ||u^N_{tt}(0)||^2+ r||\nabla u^N_t(0)||^2-2(f( u^N),u^N_{tt})|_0^t\nonumber\\
&
\;\;\;+2\int_0^t\int_{\Omega}f'( u^N)(u^{N}_t)^2dxd\tau
-2c^2(\nabla u^N,\nabla u^N_t)|_0^t+2c^2\int_0^t||\nabla u^N_t||^2d\tau. \label{3.10}
\end{align}
(18)
Then, by Hölder inequality and the fact \(|f(s)|=|\int_0^tf'(s)ds|\le C_1|s|\) by (A1), we arrive at
\begin{align}
\frac{d}{dt}&[||u^N||^2+||u^N_t||^2+||\nabla u^N||^2]+\alpha ||u^N_{tt}||^2
+2\beta \int_0^t||u^N_{tt}||^2d\tau +r||\nabla u^N_t||^2\nonumber\\
&
\le 2||u^N||||u^N_t||+2||u^N_t||||u^N_{tt}||+2||\nabla u^N||||\nabla u^N_t||
+\alpha ||u^N_{tt}(0)||^2+ r||\nabla u^N_t(0)||^2\nonumber\\
&\;\;\;+2C_1||u^N||||u^N_t||+2C_1||u^N(0)||||u^N_t(0)||+2C_1\int_0^t||u^N_t||^2d\tau\nonumber\\
&\;\;\;+2c^2||\nabla u^N||||\nabla u^N_t|| +2c^2||\nabla u^N(0)||||\nabla u^N_t(0)||
+2c^2\int_0^t||\nabla u^N_t||^2d\tau\nonumber\\
&\le\frac{1}{2}(\alpha ||u^N_{tt}||^2+r||\nabla u^N_t||^2)+C_2(||u^N||^2
+||u^N_t||^2+||\nabla u^N||^2)\nonumber\\
&\;\;\;+2C_1\int_0^t||u^N_t||^2d\tau+2c^2\int_0^t||\nabla u^N_t||^2d\tau
+\alpha ||u^N_{tt}(0)||^2+ r||\nabla u^N_t(0)||^2\nonumber\\
&\;\;\;+C_3||u^N(0)||^2+ C_4||u^N_t(0)||^2+c^2||\nabla u^N(0)||^2+c^2||\nabla u^N_t(0)||^2. \label{3.11}
\end{align}
(19)
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
\begin{align}
&
||u^N||^2+||u^N_t||^2+||\nabla u^N||^2+\alpha ||u^N_{tt}||^2+r||\nabla u^N_t||^2
+ \int_0^t||\nabla u^N_t||^2d\tau +\beta ||\nabla u^N_{tt}||^2d\tau\le C . \label{3.12}
\end{align}
(20)
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
\begin{align}
&
r||\Delta u^N||^2 + c^2\int_0^t||\Delta u^N||^2d\tau
=2\alpha \int_0^t(u^N_{ttt},\Delta u^N)d\tau
+2\beta \int_0^t(u^N_{tt},\Delta u^N)d\tau
+\int_0^t(f( u^N),\Delta u^N)d\tau +r||\Delta u^N(0)||^2. \label{3.13}
\end{align}
(21)
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
\begin{align}
r||\Delta u^N||^2 &+ c^2\int_0^t||\Delta u^N||^2d\tau\notag\\
&\le 2\alpha ||u^N_{tt}||||\Delta u^N||+2\alpha ||u^N_{tt}(0)||||\Delta u^N(0)||
+\alpha(||\nabla u^N_t||^2-||\nabla u^N_t(0)||^2)
+2\beta \int_0^t||u^N_{tt}||||\Delta u^N||d\tau\nonumber\\
&
\;\;\;+\int_0^t||f( u^N)||||\Delta u^N||d\tau +r||\Delta u^N(0)||^2\nonumber\\
&\le \epsilon_1 ||\Delta u^N||^2+C_6(||u^N_{tt}||^2+||\nabla u^N_t||^2)
+C_7(||u^N_{tt}(0)||^2+||\Delta u^N(0)||^2+||\nabla u^N_t(0)||^2)\nonumber\\
&\;\;\;+\epsilon_1 \int_0^t||\Delta u^N||^2d\tau+C_8\int_0^t||u^N_{tt}||^2d\tau
+C_9\int_0^t||u^N||^2d\tau . \label{3.14}
\end{align}
(22)
Choosing \(\epsilon_1\) sufficiently small and \(\epsilon_2\) sufficiently large
such that \(\epsilon_2>2c^2\), then it follows from (22) and (20) that
\begin{align}
&||\Delta u^N||^2 \le C_{10} \int_0^t||\Delta u^N||^2d\tau+C_{11}. \label{3.15}
\end{align}
(23)
Thus, applying Gronwall’s inequality to (23), we deduce
\begin{align}
&||\Delta u^N||^2 \le C. \label{3.16}
\end{align}
(24)
Combining (20) and (24), we get
\begin{align}
&
||u^N||^2+||u^N_t||^2+||\nabla u^N||^2+ ||u^N_{tt}||^2+||\nabla u^N_t||^2
+ \int_0^t||\nabla u^N_t||^2d\tau +\beta ||\nabla u^N_{tt}||^2d\tau\le C . \label{3.17}
\end{align}
(25)
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
\begin{align}
&\alpha |(u^N_{ttt}, v)|\le (\beta ||u^N_{tt}||+c^2||\Delta u^N||
+||\nabla u^N_t||+C_1||u^N||)||v||_{H_0^1}. \label{3.18}
\end{align}
(26)
Thus, using Lemma 3, it follows that
\begin{align}
&||u^N_{ttt}||_{H^{-1}(\Omega)}\le M. \label{3.19}
\end{align}
(27)
Similarly, we have
\begin{align}
&||\Delta u^N_{t}||_{H^{-1}(\Omega)}\le M. \label{3.20}
\end{align}
(28)
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
\begin{align}&
u^N~\rightarrow u ~ weak * ~in~L^{\infty}(0,T,H^2(\Omega)\cap H^1_0(\Omega)),\label{3.21}\\
\end{align}
(29)
\begin{align}
&
u^N_t~\rightarrow u_t ~ weak * ~in ~L^{\infty}(0,T,H^1_0(\Omega)),\label{3.22}\\
\end{align}
(30)
\begin{align}&
u^N_{tt}~\rightarrow u_{tt} ~ weak * ~in ~L^{\infty}(0,T,L^2(\Omega)),\label{3.23}\\
\end{align}
(31)
\begin{align}
&
u^N_{ttt}~\rightarrow u_{ttt} ~ weak * ~in ~L^{\infty}(0,T,H^{-1}(\Omega)),\label{3.24}\\
\end{align}
(32)
\begin{align}&
f(u^N)~\rightarrow \xi ~ weak * ~in ~L^{\infty}(0,T,H^{-1}(\Omega)),\label{3.25}\\
\end{align}
(33)
\begin{align}
&
\Delta u^N_t~\rightarrow \eta ~ weak * ~in ~L^{\infty}(0,T,H^{-1}(\Omega)).\label{3.26}
\end{align}
(34)
and for any \(t\in [0,T]\)
\begin{align}&
u^N~\rightarrow u ~ weakly ~in~H^2(\Omega)\cap H^1_0(\Omega),\label{3.27}\\
\end{align}
(35)
\begin{align}
&
u^N_t~\rightarrow u_t ~ weakly ~in ~H^1_0(\Omega),\label{3.28}\\
\end{align}
(36)
\begin{align}
&
u^N_{tt}~\rightarrow u_{tt} ~ weakly ~in ~L^2(\Omega),\label{3.29}\\
\end{align}
(37)
\begin{align}
&
u^N_{ttt}~\rightarrow u_{ttt} ~ weakly ~in ~H^{-1}(\Omega),\label{3.30}\\
\end{align}
(38)
\begin{align}
&
f(u^N)~\rightarrow \xi ~ weak * ~in ~H^{-1}(\Omega)),\label{3.31}\\
\end{align}
(39)
\begin{align}&
\Delta u^N_t~\rightarrow \eta ~ weak * ~in ~H^{-1}(\Omega)).\label{3.32}
\end{align}
(40)
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
\begin{align}&
(\Delta u^N_t,v)=-(\nabla u^N_t,\nabla v)\rightarrow -(\nabla u_t,\nabla v)
=(\Delta u_t,v),\label{3.33}\\
\end{align}
(41)
\begin{align}
&
f(u^N)~\rightarrow f(u)\label{3.34}
\end{align}
(42)
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
\begin{align}
&|( u^N,w_j)|+\sum_{k=1}^{3}|( u^N_{t^k},w_j)|\le M, \label{3.35}
\end{align}
(43)
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
\begin{align}&
(u^N,w_j)~\rightarrow~ (u,w_j),~(u^N_{t^k},w_j)~\rightarrow ~(u_{t^k},w_j),~
k=1,2,3,~j=1,2…..\label{3.36}
\end{align}
(44)
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
\begin{align}
&
u(x,0)=u_0(x),~u_t(x,0)=u_1(x),~u_{tt}(x,0)=u_2(x) ~a.e.~ in~\Omega.\label{3.37}
\end{align}
(45)
By (29)-(34),(44) and Lemma 2.1, we have
\begin{align}&
u^N~\rightarrow u,~u^N_{t}~\rightarrow~ u_{t}~ in~ C([0,T],L^2(\Omega)).\label{3.38}
\end{align}
(46)
Thanks to (29)-(42), letting \(N\rightarrow \infty\) in (11), leads to
\begin{align}&
\alpha (u_{ttt},v)+\beta (u_{tt},v)-c^2( \Delta u, v)
– r( \Delta u_t, v)+(f( u),v)=0 \label{3.39}
\end{align}
(47)
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
\begin{align}
&
\alpha w_{ttt}+\beta w_{tt}- c^2\Delta w-r \Delta w_t+f(u_1)-f(u_2)=0, ~
in~ \Omega \times (0,+\infty),\label{3.40}\\
\end{align}
(48)
\begin{align}
&
w(x,t)=0~on~ \partial \Omega, \label{3.41}\\
\end{align}
(49)
\begin{align}
&
w(x, 0) = 0, w_t(x, 0) = 0,w_{tt}(x, 0) = 0,~x \in \Omega, \label{3.42}
\end{align}
(50)
Integrating (48) for \(t\) from 0 to \(t\), we have
\begin{align}
&
\alpha w_{tt}+\beta w_{t}-r\Delta w=\int_0^t(c^2\Delta w+f(u_2)-f(u_1))d\tau. \label{3.43}
\end{align}
(51)
Multiplying the Eq. (51) by \(w_t\), integrating over \(\Omega\), adding up \((w,w_t)\),
we obtain
\begin{align}
\frac{1}{2}(\alpha ||w_{t}||^2+r||\nabla w||^2+||w||^2)+\beta ||w_{t}||^2
&=2\int_0^t(c^2\Delta w+f(u_2)-f(u_1))w_td\tau\nonumber\\
&
=2c^2(||\nabla w||^2-||\nabla w_0||^2)+2\int_0^t\int_{\Omega}\theta ww_tdxd\tau\nonumber\\
&\le C(||\nabla w||^2+||w_t||^2), \label{3.44}
\end{align}
(52)
where we have used mean value theorem and \(|\theta|\le 1\).
By applying Gronwall inequality, we deduce that
\begin{align}
&
\alpha ||w_{t}||^2+r||\nabla w||^2+||w||^2=0. \label{3.45}
\end{align}
(53)
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.”
