In this section, we shall state and prove our main result. For this purpose, we define
$$\begin{aligned} J(t) =&\frac{1}{2} \biggl(a- \int_{0}^{t} g(s)\,ds \biggr)\bigl\| \nabla u(t ) \bigr\| ^{2} +\frac{b}{4} \bigl\| \nabla u(t)\bigr\| ^{4} + \frac{1}{2} \bigl\| \nabla u_{t} (t)\bigr\| ^{2} + \frac {1}{2} (g\circ\nabla u) (t) \\ &{} +\frac{\xi}{2} \int_{t-\tau(t) }^{t} \int_{\Gamma_{1}} e^{-\lambda(t-s) } u_{t}^{2} (s)\,d\Gamma \,ds+\frac{1}{2} \int_{\Gamma_{1}} m(x) h(x) y^{2} (t)\,d\Gamma \\ &{}- \frac{1}{p+2} \bigl\| u(t)\bigr\| _{p+2}^{p+2} \end{aligned}$$
(3.1)
and
$$ \begin{aligned}[b] I(t) ={}& \biggl(a- \int_{0}^{t} g(s)\,ds \biggr)\bigl\| \nabla u(t ) \bigr\| ^{2} +\frac {b}{2} \bigl\| \nabla u(t)\bigr\| ^{4} + \bigl\| \nabla u_{t} (t)\bigr\| ^{2} + (g\circ\nabla u) (t)\\ &{} +\xi \int_{t-\tau(t) }^{t} \int_{\Gamma_{1}} e^{-\lambda(t-s) } u_{t}^{2} (s)\,d\Gamma \,ds+ \int_{\Gamma_{1}} m(x) h(x) y^{2} (t)\,d\Gamma - \bigl\| u(t) \bigr\| _{p+2}^{p+2}, \end{aligned} $$
(3.2)
where \((g\circ\nabla u)(t)=\int_{\Omega}\int_{0}^{t} g(t-s) |u(t)-u(s)|^{2}\,ds \,dx\). We denote the modified energy functional \(E(t)\) associated with problem (2.13)-(2.20) by
$$\begin{aligned} E(t) =&\frac{1}{\rho+2} \bigl\| u_{t} (t) \bigr\| _{\rho+2}^{\rho+2}+\frac{1}{2} \biggl(a- \int_{0}^{t} g(s)\,ds \biggr)\bigl\| \nabla u(t ) \bigr\| ^{2} +\frac{b}{4} \bigl\| \nabla u(t)\bigr\| ^{4} \\ &{} +\frac{1}{2} \bigl\| \nabla u_{t} (t)\bigr\| ^{2} + \frac{1}{2} (g\circ\nabla u) (t) +\frac{\xi}{2} \int_{t-\tau(t) }^{t} \int_{\Gamma_{1}} e^{-\lambda (t-s) } u_{t}^{2} (s)\,d\Gamma \,ds \\ &{} +\frac{1}{2} \int_{\Gamma_{1}} m(x) h(x) y^{2} (t)\,d\Gamma- \frac {1}{p+2} \bigl\| u(t)\bigr\| _{p+2}^{p+2} \\ =&\frac{1}{\rho+2} \bigl\| u_{t} (t)\bigr\| _{\rho+2}^{\rho+2} + J(t), \end{aligned}$$
(3.3)
where \(\xi, \lambda\) are suitable positive constants.
Next, we will fix ξ such that
$$\begin{aligned}& 2\mu_{0} -\frac{|\mu_{1}|}{\sqrt{1-d}}-\xi>0, \qquad \xi- \frac{|\mu _{1}|}{\sqrt{1-d}}>0, \\& \lambda< \frac{1}{\bar{\tau}} \biggl| \log\frac{|\mu_{1} |}{\xi\sqrt {1-d}} \biggr|. \end{aligned}$$
(3.4)
Lemma 3.1
Let (2.8)-(2.11) be satisfied and
g
satisfy (2.5). Then for the solution of problem (2.13)-(2.20), the energy functional defined by (3.3) satisfies
$$\begin{aligned} E'(t) \leq& -\sigma \biggl(\frac{1}{2} \frac{d}{dt} \bigl\| \nabla u(t)\bigr\| ^{2} \biggr)^{2} + \frac{1}{2} \bigl(g' \circ\nabla u\bigr) (t) - \frac{1}{2} g(t) \bigl\| \nabla u(t)\bigr\| ^{2} \\ &{} - \int_{\Omega}\bigl|u_{t} (t)\bigr|^{q+2}\,dx -C_{1} \int_{\Gamma_{1}} \bigl[ u_{t}^{2} (x,t) +u_{t}^{2} \bigl(x, t-\tau(t)\bigr) \bigr]\,d\Gamma \\ &{} -\frac{\lambda\xi}{2} \int_{t-\tau(t)}^{t} \int_{\Gamma_{1}} e^{-\lambda(t-s)}u_{t}^{2} (s)\,d\Gamma \,ds - \int_{\Gamma_{1}} h(x) f(x) y_{t}^{2} (t)\,d\Gamma, \end{aligned}$$
(3.5)
for some positive constant
\(C_{1} \).
Proof
Differentiating (3.3) and using (2.14), we have
$$\begin{aligned} E'(t) =& \int_{\Omega} \bigl|u_{t} (t)\bigr|^{\rho+1} u_{tt} (t)\,dx -\frac{1}{2} g(t) \bigl\| \nabla u(t) \bigr\| ^{2} \\ &{} + \biggl(a- \int_{0}^{t} g(s)\,ds \biggr) \int_{\Omega}\nabla u(t) \nabla u_{t} (t)\,dx+ \frac{b}{2} \bigl\| \nabla u(t) \bigr\| ^{2} \int_{\Omega}\nabla u(t) \nabla u_{t} (t)\,dx \\ &{} + \int_{\Omega}\nabla u_{t} (t) \nabla u_{tt} (t)\,dx + \int_{0}^{t} g(t-s) \int_{\Omega}\nabla u_{t} (t) \bigl(\nabla u(t)-\nabla u(s)\bigr) \,dx\,ds \\ &{} +\frac{1}{2} \int_{0}^{t} g'(t-s) \int_{\Omega}\bigl|\nabla u(t)-\nabla u(s) \bigr|^{2} \,dx\,ds + \frac{\xi}{2} \int_{\Gamma_{1}} u_{t}^{2} (x,t)\,d\Gamma \\ &{} -\frac{\xi}{2} \int_{\Gamma_{1}} e^{-\lambda\tau(t)} u_{t}^{2} \bigl(x, t-\tau(t)\bigr) \bigl(1-\tau'(t)\bigr)\,d\Gamma - \frac{\lambda\xi}{2} \int_{t-\tau(t)}^{t} \int_{\Gamma_{1}} e^{-\lambda (t-s)} u_{t}^{2} (s)\,d\Gamma \,ds \\ &{} + \int_{\Gamma_{1}} h(x)m(x) y(t) y_{t} (t)\,d\Gamma- \int_{\Omega}\bigl|u(t)\bigr|^{p+1} u_{t} (t)\,dx \\ =& \int_{\Omega}u_{t} (t) \biggl[ \bigl( a+b\bigl\| \nabla u(t) \bigr\| ^{2} +\sigma \bigl(\nabla u(t), \nabla u_{t} (t)\bigr) \bigr)\Delta u(t) +\Delta u_{tt} (t) \\ &{} - \int_{0}^{t} g(t-s) \Delta u(s)\,ds - \int_{\Omega}\bigl|u_{t} (t)\bigr|^{q} u_{t} (t)\,dx + \int_{\Omega}\bigl|u(t)\bigr|^{p} u(t) \biggr]\,dx \\ &{} -\frac{1}{2} g(t) \bigl\| \nabla u(t)\bigr\| ^{2} + \biggl( a- \int_{0}^{t} g(s)\,ds \biggr) \int_{\Omega}\nabla u(t) \nabla u_{t} (t)\,dx \\ &{} +\frac{b}{2} \bigl\| \nabla u(t)\bigr\| ^{2} \int_{\Omega}\nabla u(t) \nabla u_{t} (t)\,dx + \int_{\Omega}\nabla u_{t} (t) \nabla u_{tt} (t)\,dx \\ &{} + \int_{0}^{t} g(t-s) \int_{\Omega}\nabla u_{t} (t) \bigl(\nabla u(t) -\nabla u(s)\bigr) \,dx\,ds \\ &{} +\frac{1}{2} \int_{0}^{t} g'(t-s) \int_{\Omega}\bigl|\nabla u(t)-\nabla u(s) \bigr|^{2} \,dx\,ds+ \frac{\xi}{2} \int_{\Gamma_{1}} u_{t}^{2} (t)\,d\Gamma \\ &{} -\frac{\xi}{2} \int_{\Gamma_{1} } e^{-\lambda\tau(t)} u_{t}^{2} \bigl(x, t-\tau(t)\bigr) \bigl(1-\tau'(t)\bigr)\,d\Gamma - \frac{\lambda\xi}{2} \int_{t-\tau(t)}^{t} \int_{\Gamma_{1}} e^{-\lambda (t-s)} u_{t}^{2} (s)\,d\Gamma \,ds \\ &{} + \int_{\Gamma_{1}} h(x) m(x) y(t) y_{t} (t)\,d\Gamma- \int_{\Omega}\bigl|u(t)\bigr|^{p+1} u_{t} (t)\,dx \\ = & \int_{\Gamma_{1}} h(x) y_{t} (t) u_{t} (t)\,d\Gamma- \mu_{0} \int_{\Gamma _{1}} u_{t} (x,t) u_{t} (t)\,d\Gamma - \mu_{1} \int_{\Gamma_{1}} u_{t} \bigl(x, t-\tau(t)\bigr) u_{t} (t)\,d\Gamma \\ &{} -\sigma \biggl(\frac{1}{2} \frac{d}{dt} \bigl\| \nabla u(t) \bigr\| ^{2} \biggr)^{2} - \int_{\Omega}\bigl|u_{t} (t)\bigr|^{q+2}\,dx - \frac{1}{2} g(t) \int_{\Omega}\bigl|\nabla u(t)\bigr|^{2}\,dx \\ &{} +\frac{1}{2} \int_{0}^{t} g'(t-s) \int_{\Omega}\bigl|\nabla u(t)-\nabla u(s) \bigr|^{2} \,dx\,ds+ \frac{\xi}{2} \int_{\Gamma_{1}} u_{t}^{2} (x,t)\,d\Gamma \\ &{} -\frac{\xi}{2} \int_{\Gamma_{1}} e^{-\lambda\tau(t) }u_{t}^{2} \bigl(x,t-\tau(t)\bigr) \bigl(1-\tau'(t)\bigr)\,d\Gamma- \frac{\lambda\xi}{2} \int_{t-\tau (t)}^{t} \int_{\Gamma_{1}} e^{-\lambda(t-s)}u_{t}^{2} (s)\,d\Gamma \,ds \\ &{} + \int_{\Gamma_{1}} h(x) m(x) y(t) y_{t} (t)\,d\Gamma. \end{aligned}$$
(3.6)
Applying Young’s inequality, we obtain
$$\begin{aligned} &{-}\mu_{1} \int_{\Gamma_{1} } u_{t} (x,t) u_{t} \bigl(x, t- \tau(t)\bigr)\,d\Gamma \\ &\quad\leq\frac{|\mu_{1}|}{2\sqrt{1-d}} \int_{\Gamma_{1}} u_{t}^{2} (x,t)\,d\Gamma+ \frac{|\mu_{1}|\sqrt{1-d}}{2} \int_{\Gamma_{1} } u_{t}^{2} \bigl(x, t-\tau (t) \bigr)\,d\Gamma, \end{aligned}$$
(3.7)
and using (2.7), we get
$$\begin{aligned} & \int_{\Gamma_{1}} h(x) y_{t} (t) u_{t} (t)\,d\Gamma+ \int_{\Gamma_{1}} f(x) h(x) y_{t}^{2} (t)\,d\Gamma =- \int_{\Gamma_{1}} h(x) m(x) y_{t} (t) y(t)\,d\Gamma. \end{aligned}$$
(3.8)
Thus, from (3.6)-(3.8) and assumptions (2.8), (2.9), and (3.4), we arrive at
$$\begin{aligned} E'(t) \leq&-\mu_{0} \int_{\Gamma_{1}} u_{t}^{2} (t)\,d\Gamma+ \frac{|\mu_{1} |}{2\sqrt{1-d}} \int_{\Gamma_{1}} u_{t}^{2} (t)\,d\Gamma+ \frac{|\mu_{1}|\sqrt {1-d}}{2} \int_{\Gamma_{1} } u_{t}^{2} \bigl(x, t-\tau(t) \bigr)\,d\Gamma \\ &{} -\sigma \biggl(\frac{1}{2} \frac{d}{dt} \bigl\| \nabla u(t) \bigr\| ^{2} \biggr)^{2} - \int_{\Omega}\bigl|u_{t} (t)\bigr|^{q+2}\,dx + \frac{1}{2} \bigl(g'\circ\nabla u\bigr) (t) - \frac{1}{2}g(t) \bigl\| \nabla u(t)\bigr\| ^{2} \\ &{}+\frac{\xi}{2} \int_{\Gamma_{1}} u_{t}^{2} (t)\,d\Gamma- \frac{\xi}{2} \int_{\Gamma_{1}} e^{-\lambda\tau(t)}u_{t}^{2} \bigl(x,t-\tau(t)\bigr) \bigl(1-\tau '(t)\bigr)\,d\Gamma \\ &{} -\frac{\lambda\xi}{2} \int_{t-\tau(t)}^{t} \int_{\Gamma _{1}}e^{-\lambda(t-s)} u_{t}^{2} (s)\,d\Gamma \,ds- \int_{\Gamma_{1}} f(x) h(x) y_{t}^{2} (t)\,d\Gamma \\ \leq&-\sigma \biggl(\frac{1}{2} \frac{d}{dt} \bigl\| \nabla u(t) \bigr\| ^{2} \biggr)^{2} - \int_{\Omega}\bigl|u_{t} (t)\bigr|^{q+2}\,dx + \frac{1}{2} \bigl(g'\circ\nabla u\bigr) (t) - \frac{1}{2}g(t) \bigl\| \nabla u(t)\bigr\| ^{2} \\ &{} - \biggl( \mu_{0} -\frac{|\mu_{1}|}{2\sqrt{1-d}}-\frac{\xi}{2} \biggr) \int_{\Gamma_{1} } u_{t}^{2} (t)\,d\Gamma \\ &{} - \biggl( \frac{\xi}{2}(1-d)e^{-\lambda\bar{\tau}} -\frac{|\mu _{1} |\sqrt{1-d}}{2} \biggr) \int_{\Gamma_{1}}u_{t}^{2} \bigl(x,t-\tau(t)\bigr)\,d\Gamma \\ &{} -\frac{\lambda\xi}{2} \int_{t-\tau(t)}^{t} \int_{\Gamma _{1}}e^{-\lambda(t-s)}u_{t}^{2} (s)\,d\Gamma \,ds - \int_{\Gamma_{1}} f(x) h(x) y_{t}^{2} (t)\,d\Gamma \\ \leq&-\sigma \biggl(\frac{1}{2} \frac{d}{dt} \bigl\| \nabla u(t) \bigr\| ^{2} \biggr)^{2} - \int_{\Omega}\bigl|u_{t} (t)\bigr|^{q+2}\,dx + \frac{1}{2} \bigl(g'\circ\nabla u\bigr) (t) - \frac{1}{2}g(t) \bigl\| \nabla u(t)\bigr\| ^{2} \\ &{} -C_{1} \int_{\Gamma_{1}} \bigl[ u_{t} (x,t) +u_{t}^{2} \bigl(x,t-\tau(t)\bigr) \bigr]\,d\Gamma-\frac{\lambda\xi}{2} \int_{t-\tau(t)}^{t} \int_{\Gamma _{1}}e^{-\lambda(t-s)}u_{t}^{2} (s)\,d\Gamma \,ds \\ &{} - \int_{\Gamma_{1}} f(x) h(x) y_{t}^{2} (t)\,d\Gamma, \end{aligned}$$
for some positive constant \(C_{1} \). □
Lemma 3.2
Let
\((u_{0} , u_{1}) \in(H^{2} (\Omega)\cap V) \times V, y_{0} \in L^{2} (\Gamma _{1} ), f_{0} \in L^{2} (\Gamma_{1} \times[-\tau(0), 0]) \), and
\((u(t), y(t), z(t))\)
be the solution of (2.13)-(2.20). If
\(I(0)>0\)
and
$$ \alpha=\frac{C_{*}^{p+2}}{l} \biggl( \frac{2(p+2)}{lp} E(0) \biggr)^{p/2} < 1 $$
(3.9)
then
\(I(t)>0 \)
for
\(t\in[0,T]\), where
\(I(t)\)
is defined in (3.2).
Proof
Since \(I(0)>0\), there exists (by continuity of \(u(t)\)) \(T^{*} < T\) such that
for all \(t\in[0, T^{*} ]\). Then (3.1), (3.2), and (3.10) give
$$\begin{aligned} J(t) =&\frac{p}{2(p+2)} \biggl[ \biggl( a- \int_{0}^{t} g(s)\,ds \biggr)\bigl\| \nabla u(t) \bigr\| ^{2} +\frac{b}{2} \bigl\| \nabla u(t)\bigr\| ^{4} +\bigl\| \nabla u_{t} (t)\bigr\| ^{2} +(g\circ\nabla u) (t) \\ &{} +\xi \int_{t-\tau(t)}^{t} \int_{\Gamma_{1} } e^{-\lambda(t-s)} u_{t}^{2} (s)\,d\Gamma \,ds + \int_{\Gamma_{1} }m(x) h(x) y^{2} (t)\,d\Gamma \biggr]+ \frac{1}{p+2}I(t) \\ \geq&\frac{p}{2(p+2)} \biggl[ \biggl(a- \int_{0}^{t} g(s)\,ds \biggr) \bigl\| \nabla u(t) \bigr\| ^{2} +\frac{b}{2} \bigl\| \nabla u(t)\bigr\| ^{4} +\bigl\| \nabla u_{t} (t)\bigr\| ^{2} +(g\circ\nabla u) (t) \\ &{} +\xi \int_{t-\tau(t)}^{t} \int_{\Gamma_{1} } e^{-\lambda(t-s)} u_{t}^{2} (s)\,d\Gamma \,ds + \int_{\Gamma_{1} }m(x) h(x) y^{2} (t)\,d\Gamma \biggr]. \end{aligned}$$
(3.11)
Hence from (2.5), (3.3), (3.11), and Lemma 3.1, we can deduce that
$$\begin{aligned} l\bigl\| \nabla u(t)\bigr\| ^{2} \leq& \biggl( a- \int_{0}^{t} g(s)\,ds \biggr)\bigl\| \nabla u(t) \bigr\| ^{2} \leq\frac{2(p+2)}{p} J(t) \\ \leq&\frac{2(p+2)}{p} E(t) \leq\frac{2(p+2)}{p} E(0),\quad \forall t\in \bigl[0,T^{*}\bigr]. \end{aligned}$$
(3.12)
Exploiting (2.1), (3.9), and (3.12), we obtain
$$\begin{aligned} \bigl\| u(t)\bigr\| _{p+2}^{p+2} \leq& C_{*}^{p+2} \bigl\| \nabla u(t) \bigr\| ^{p+2} \leq\frac {C_{*}^{p+2}}{l} \biggl( \frac{2(p+2)}{lp} E(0) \biggr)^{\frac{p}{2}} l\bigl\| \nabla u(t)\bigr\| ^{2} \\ \leq&\alpha l \bigl\| \nabla u(t)\bigr\| ^{2} \leq \biggl( a- \int_{0}^{t} g(s)\,ds \biggr) \bigl\| \nabla u(t) \bigr\| ^{2} ,\quad \forall t\in\bigl[0,T^{*}\bigr]. \end{aligned}$$
Consequently, we get
$$\begin{aligned} I(t) =& \biggl( a- \int_{0}^{t} g(s)\,ds \biggr)\bigl\| \nabla u(t) \bigr\| ^{2} +\frac {b}{2} \bigl\| \nabla u(t)\bigr\| ^{4} +\bigl\| \nabla u_{t} (t)\bigr\| ^{2} +(g\circ\nabla u) (t) \\ &{} +\xi \int_{t-\tau(t)}^{t} \int_{\Gamma_{1} } e^{-\lambda(t-s)} u_{t}^{2} (s)\,d\Gamma \,ds + \int_{\Gamma_{1} }m(x) h(x) y^{2} (t)\,d\Gamma \\ &{}- \bigl\| u(t) \bigr\| _{p+2}^{p+2} >0,\quad \forall t\in\bigl[0,T^{*}\bigr]. \end{aligned}$$
(3.13)
Repeat this procedure and use the fact that
$$\lim_{t\to T^{*}} \frac{C_{*}^{p+2}}{l} \biggl[ \frac{2(p+2)}{lp} E(t) \biggr]^{\frac{p}{2}}\leq\alpha< 1. $$
We can take \(T^{*} =T \). Thus the proof is complete. □
Theorem 3.1
Suppose that (2.1)-(2.4), (2.9)-(2.11), and (H1)-(H3) hold. If
\((u_{0} , u_{1} )\in(H^{2} (\Omega)\cap V) \times V, y_{0} \in L^{2} (\Gamma _{1} ), f_{0} \in L^{2} ( \Gamma_{1} \times[-\tau(0), 0] )\)
and
\((3.9 )\)
is satisfied, then the solution
\((u(t), y(t), z(t)) \)
of (2.13)-(2.20) is bounded and global in time.
Proof
It suffices to show that
$$\bigl\| \nabla u(t)\bigr\| ^{2} +\bigl\| \nabla u_{t} (t)\bigr\| ^{2} +\xi \int_{t-\tau(t)}^{t} \int _{\Gamma_{1}} e^{-\lambda(t-s)} u_{t}^{2} (s)\,d\Gamma \,ds + \int_{\Gamma_{1} } m(x) h(x) y^{2} (t)\,d\Gamma $$
is bounded independent of t. Under the hypotheses in Theorem 3.1, we see from Lemma 3.2 that \(I(t)>0\) for all \(t\geq0\). Therefore
$$\begin{aligned} J(t) =&\frac{p}{2(p+2)} \biggl[ \biggl( a- \int_{0}^{t} g(s)\,ds \biggr)\bigl\| \nabla u(t) \bigr\| ^{2} +\frac{b}{2} \bigl\| \nabla u(t)\bigr\| ^{4} +\bigl\| \nabla u_{t} (t)\bigr\| ^{2} +(g\circ \nabla u) (t) \\ &{} +\xi \int_{t-\tau(t)}^{t} \int_{\Gamma_{1} } e^{-\lambda(t-s)} u_{t}^{2} (s)\,d\Gamma \,ds + \int_{\Gamma_{1} }m(x) h(x) y^{2} (t)\,d\Gamma \biggr]+ \frac{1}{p+2}I(t) \\ \geq&\frac{p}{2(p+2)} \biggl[ \biggl(a- \int_{0}^{t} g(s)\,ds \biggr) \bigl\| \nabla u(t) \bigr\| ^{2} +\frac{b}{2} \bigl\| \nabla u(t)\bigr\| ^{4} +\bigl\| \nabla u_{t} (t)\bigr\| ^{2} \\ &{} +(g\circ\nabla u) (t) +\xi \int_{t-\tau(t)}^{t} \int_{\Gamma_{1} } e^{-\lambda(t-s)} u_{t}^{2} (s)\,d\Gamma \,ds + \int_{\Gamma_{1} }m(x) h(x) y^{2} (t)\,d\Gamma \biggr]. \end{aligned}$$
Hence by (H1) and the fact that \((g\circ\nabla u)(t)>0\), we can deduce that
$$\begin{aligned} & l\bigl\| \nabla u(t)\bigr\| ^{2} +\frac{b}{2} \bigl\| \nabla u(t) \bigr\| ^{4} +\bigl\| \nabla u_{t} (t)\bigr\| ^{2} \\ &\qquad{} +\xi \int_{t-\tau(t)}^{t} \int_{\Gamma_{1} } e^{-\lambda(t-s)} u_{t}^{2} (s)\,d\Gamma \,ds + \int_{\Gamma_{1} }m(x) h(x) y^{2} (t)\,d\Gamma \\ &\quad\leq \biggl( a- \int_{0}^{t} g(s)\,ds \biggr)\bigl\| \nabla u(t) \bigr\| ^{2} +\frac{b}{2} \bigl\| \nabla u(t)\bigr\| ^{4} +\bigl\| \nabla u_{t} (t)\bigr\| ^{2} \\ &\qquad{} +\xi \int_{t-\tau(t)}^{t} \int_{\Gamma_{1} } e^{-\lambda(t-s)} u_{t}^{2} (s)\,d\Gamma \,ds+ \int_{\Gamma_{1} }m(x) h(x) y^{2} (t)\,d\Gamma \\ & \quad\leq\frac{2(p+2)}{p} J(t) , \quad \forall t\in[0,T]. \end{aligned}$$
(3.14)
Using Lemma 3.1 and (3.14), it follows that
$$\begin{aligned} &\frac{1}{\rho+2}\bigl\| u_{t} (t)\bigr\| _{\rho+2}^{\rho+2}+ \frac{p}{2(p+2)} \biggl( l \bigl\| \nabla u(t)\bigr\| ^{2} +\frac{b}{2} \bigl\| \nabla u(t)\bigr\| ^{4} +\bigl\| \nabla u_{t} (t)\bigr\| ^{2} \\ &\qquad{}+\xi \int_{t-\tau(t)}^{t} \int_{\Gamma_{1} } e^{-\lambda(t-s)} u_{t}^{2} (s)\,d\Gamma \,ds + \int_{\Gamma_{1} }m(x) h(x) y^{2} (t)\,d\Gamma \biggr) \\ &\quad\leq\frac{1}{\rho+2} \bigl\| u_{t} (t)\bigr\| _{\rho+2}^{\rho+2} +J(t)=E(t)\leq E(0). \end{aligned}$$
Thus, there exists a constant \(C>0\) depending p and l such that
$$\begin{aligned} &\bigl\| \nabla u(t)\bigr\| ^{2} + \bigl\| \nabla u(t)\bigr\| ^{4} +\bigl\| \nabla u_{t} (t)\bigr\| ^{2} \\ &\qquad{} +\xi \int_{t-\tau(t)}^{t} \int_{\Gamma_{1} } e^{-\lambda(t-s)} u_{t}^{2} (s)\,d\Gamma \,ds + \int_{\Gamma_{1} }m(x) h(x) y^{2} (t)\,d\Gamma \\ & \quad \leq CE(t) \leq CE(0)< +\infty. \end{aligned}$$
This implies that the solution \((u(t), y(t), z(t))\) of (2.13)-(2.20) is bounded and global in time. □
Now, we define
$$ L(t)=ME(t) +\varepsilon\Psi(t) +\Phi(t), $$
(3.15)
where M and ε are positive constants which will be specified later and
$$\begin{aligned}& \begin{aligned}[b] \Psi(t)={}&\frac{1}{\rho+1} \int_{\Omega}\bigl|u_{t} (t) \bigr|^{\rho}u_{t} (t) u(t)\,dx +\frac{\sigma}{4} \bigl\| \nabla u(t)\bigr\| ^{4} + \int_{\Omega}\nabla u_{t} (t) \nabla u(t)\,dx \\ &{}+ \int_{\Gamma_{1} } h(x) u(t) y(t)\,d\Gamma+\frac{1}{2} \int _{\Gamma_{1} } h(x) f(x) y^{2} (t)\,d\Gamma, \end{aligned} \end{aligned}$$
(3.16)
$$\begin{aligned}& \begin{aligned}[b] \Phi(t) ={}&{-}\frac{1}{\rho+1} \int_{\Omega}\bigl|u_{t} (t) \bigr|^{\rho}u_{t} (t) \int _{0}^{t} g(t-s) \bigl(u(t)-u(s)\bigr)\,ds \,dx\\ &{}- \int_{\Omega}\nabla u_{t} (t) \int_{0}^{t} g(t-s) \bigl(\nabla u(t)-\nabla u(s) \bigr) \,ds\,dx. \end{aligned} \end{aligned}$$
(3.17)
Before we prove our main result, we need the following lemmas.
Lemma 3.3
Let
\(u\in L^{\infty}([0,T];H_{0}^{1} (\Omega))\), then for any
\(\rho\geq0\), we have
$$\begin{aligned} &\int_{\Omega}\biggl( \int_{0}^{t} g(t-s) \bigl(u(t)-u(s)\bigr)\,ds \biggr)^{\rho+2}\,dx \\ &\quad\leq(a-l)^{\rho+1} C_{*}^{\rho+2} \biggl( \frac{4(p+2)E(0)}{lp} \biggr)^{\rho/2} (g\circ\nabla u ) (t). \end{aligned}$$
(3.18)
Proof
By the Hölder inequality, (2.1), (2.5), and (3.12), we can deduce
$$\begin{aligned} & \int_{\Omega}\biggl( \int_{0}^{t} g(t-s) \bigl(u(t)-u(s)\bigr)\,ds \biggr)^{\rho+2}\,dx \\ &\quad \leq \int_{\Omega}\biggl( \int_{0}^{t} g(t-s)\,ds \biggr)^{\rho+1} \biggl( \int _{0}^{t} g(t-s) \bigl|u(t)-u(s)\bigr|^{\rho+2}\,ds \biggr)\,dx \\ &\quad \leq(a-l)^{\rho+1} C_{*}^{\rho+2} \int_{0}^{t} g(t-s) \bigl\| \nabla u(t)-\nabla u(s) \bigr\| ^{\rho+2}\,ds \\ &\quad \leq(a-l)^{\rho+1} C_{*}^{\rho+2} \biggl(\frac{4(p+2)E(0)}{lp} \biggr)^{\rho/2} (g\circ\nabla u ) (t). \end{aligned}$$
□
Lemma 3.4
Let
\((u(t), y(t), z(t))\)
be a solution of (2.13)-(2.20), then there exist two constants
\(\beta_{1} \)
and
\(\beta_{2} \)
such that
$$ \beta_{1} E(t) \leq L (t)\leq\beta_{2} E(t). $$
(3.19)
Proof
By using (2.1), (3.12), and Young’s inequality, we have
$$\begin{aligned}& \begin{aligned}[b] & \biggl| \frac{1}{\rho+1} \int_{\Omega}\bigl|u_{t} (t)\bigr|^{\rho}u_{t} (t) u(t)\,dx \biggr| \\ & \quad \leq\frac{1}{\rho+2}\bigl\| u_{t} (t)\bigr\| _{\rho+2}^{\rho+2}+ \frac {1}{(\rho+2)(\rho+1)}\bigl\| u(t)\bigr\| _{\rho+2}^{\rho+2} \\ & \quad \leq\frac{1}{\rho+2}\bigl\| u_{t} (t)\bigr\| _{\rho+2}^{\rho+2}+ \frac {C_{*}^{\rho+2}}{(\rho+2)(\rho+1)}\bigl\| \nabla u(t)\bigr\| ^{\rho+2} \\ & \quad \leq\frac{1}{\rho+2}\bigl\| u_{t} (t)\bigr\| _{\rho+2}^{\rho+2}+ \frac {C_{*}^{\rho+2}}{(\rho+2)(\rho+1)} \biggl( \frac{2(p+2)E(0)}{lp} \biggr)^{\frac{\rho}{2}}\bigl\| \nabla u(t) \bigr\| ^{2} \\ &\quad \leq\frac{1}{\rho+2}\bigl\| u_{t} (t)\bigr\| _{\rho+2}^{\rho+2}+ \frac {\alpha_{1}}{(\rho+2)(\rho+1)}\bigl\| \nabla u(t)\bigr\| ^{2}, \end{aligned} \end{aligned}$$
(3.20)
where \(\alpha_{1} =C_{*}^{\rho+2} ( \frac{2(p+2)E(0)}{lp} )^{\frac {\rho}{2}}\),
$$ \biggl| \int_{\Omega}\nabla u_{t} (t) \nabla u(t)\,dx \biggr| \leq \frac {1}{2} \bigl( \bigl\| \nabla u_{t} (t)\bigr\| ^{2} +\bigl\| \nabla u(t)\bigr\| ^{2} \bigr), $$
(3.21)
and by (2.2) and (2.7), we can deduce
$$\begin{aligned} \biggl| \int_{\Gamma_{1} } h(x) u(t) y(t)\,d\Gamma \biggr|&= \int_{\Gamma _{1}} \frac{m(x) h(x) u(t)}{m(x)} y(t)\,d\Gamma \\ & \leq\frac{\|h\|_{\infty}^{1/2} \|m\|_{\infty}^{1/2}}{m_{0}} \biggl( \int_{\Gamma_{1}} h(x) m(x) y^{2} (x)\,d\Gamma \biggr)^{1/2} \biggl( \int_{\Gamma_{1}} \bigl|u(t)\bigr|^{2}\,d\Gamma \biggr)^{1/2} \\ & \leq\frac{\|h\|_{\infty} \|m\|_{\infty}}{2m_{0}^{2}} \int_{\Gamma _{1}} h(x) m(x) y^{2} (x)\,d\Gamma+ \frac{\tilde{C}_{*}}{2}\bigl\| \nabla u(t)\bigr\| ^{2} . \end{aligned}$$
(3.22)
Similarly, we obtain
$$\begin{aligned} & \biggl| {-}\frac{1}{\rho+1} \int_{\Omega}\bigl|u_{t} (t)\bigr|^{\rho}u_{t} (t) \int _{0}^{t} g(t-s) \bigl(u(t)-u(s)\bigr)\,ds \,dx \biggr| \\ &\quad \leq\frac{1}{\rho+2} \bigl\| u_{t} (t)\bigr\| _{\rho+2}^{\rho+2} +\frac{C_{*}^{\rho +2}}{(\rho+2)(\rho+1)} (a-l)^{\rho+1} \int_{0}^{t} g(t-s) \bigl\| \nabla u(t)-\nabla u(s) \bigr\| ^{\rho+2}\,ds \\ &\quad \leq\frac{1}{\rho+2} \bigl\| u_{t} (t)\bigr\| _{\rho+2}^{\rho+2} +\frac {(a-l)^{\rho+1} 2^{\frac{\rho}{2}} \alpha_{1} }{(\rho+1)(\rho+2)} (g\circ \nabla u) (t) \end{aligned}$$
(3.23)
and
$$\begin{aligned} & \biggl|{-} \int_{\Omega}\nabla u_{t} (t) \int_{0}^{t} g(t-s) \bigl(\nabla u(t)-\nabla u(s) \bigr)\,ds \,dx \biggr| \\ &\quad \leq\frac{1}{2} \bigl\| \nabla u_{t} (t) \bigr\| ^{2} + \frac{a-l}{2} (g\circ \nabla u) (t). \end{aligned}$$
(3.24)
Combining (3.15)-(3.17) and (3.20)-(3.24), we arrive at
$$\begin{aligned} &\bigl|L(t) -ME(t)\bigr| \\ &\quad\leq\varepsilon\bigl|\Psi(t)\bigr|+\bigl|\Phi(t)\bigr| \\ &\quad\leq\frac{\varepsilon}{\rho+2} \bigl\| u_{t} (t) \bigr\| _{\rho+2}^{\rho+2} +\frac{\varepsilon\alpha_{1} }{(\rho+2)(\rho+1)}\bigl\| \nabla u(t)\bigr\| ^{2} +\frac {\varepsilon\sigma}{4} \bigl\| \nabla u(t)\bigr\| ^{4} +\frac{\varepsilon}{2} \bigl\| \nabla u_{t} (t) \bigr\| ^{2} \\ &\qquad{} +\frac{\varepsilon}{2}\bigl\| \nabla u(t)\bigr\| ^{2} + \biggl( \frac {\varepsilon}{2m_{0}} \frac{\|h\|_{\infty}\|m\|_{\infty}}{m_{0}} +\|f\| _{\infty} \biggr) \int_{\Gamma_{1} } h(x) m(x) y^{2} (t)\,d\Gamma \\ &\qquad{} +\frac{\varepsilon\tilde{C}_{*}}{2} \bigl\| \nabla u(t)\bigr\| ^{2} +\frac {1}{\rho+2} \bigl\| u_{t} (t)\bigr\| _{\rho+2}^{\rho+2} +\frac{(a-l)^{\rho+1} 2^{\frac {\rho}{2}} \alpha_{1} }{(\rho+1)(\rho+2)} (g\circ \nabla u) (t) \\ &\qquad{} +\frac{1}{2} \bigl\| \nabla u_{t} (t) \bigr\| ^{2} + \frac{a-l}{2} (g\circ \nabla u) (t) \\ &\quad =\frac{\varepsilon+1}{\rho+2} \bigl\| u_{t} (t)\bigr\| _{\rho+2}^{\rho+2} +\varepsilon \biggl( \frac{\alpha_{1}}{(\rho+2)(\rho+1)}+\frac{1}{2} +\frac {\tilde{C}_{*}}{2} \biggr) \bigl\| \nabla u(t)\bigr\| ^{2} \\ &\qquad{} +\frac{\varepsilon\sigma}{4b}b\bigl\| \nabla u(t)\bigr\| ^{4} +\frac {\varepsilon+1}{2} \bigl\| \nabla u_{t} (t)\bigr\| ^{2} + \biggl( \frac{(a-l)^{\rho +1}2^{\rho/2} \alpha_{1} }{(\rho+1)(\rho+2)}+ \frac{a-l}{2} \biggr) (g\circ \nabla u) (t) \\ &\qquad{} + \biggl( \frac{\varepsilon}{2m_{0} }\frac{\|h\|_{\infty}\|m\| _{\infty}}{m_{0} }+\|f\|_{\infty} \biggr) \int_{\Gamma_{1}} h(x) m(x) y^{2} (t)\,d\Gamma \\ &\quad \leq C E(t), \end{aligned}$$
where C is some positive constant. Choose \(M>0\) sufficiently large and ε small, there exist two positive constants \(\beta_{1} \) and \(\beta_{2} \) such that
$$\beta_{1} E(t)\leq L (t) \leq\beta_{2} E(t). $$
Thus the proof is complete. □
Now, we state our main result.
Theorem 3.2
Suppose that (2.1)-(2.4), (2.9)-(2.11), and (H1)-(H3) hold. If
\((u_{0} , u_{1} )\in(H^{2} (\Omega)\cap V) \times V, y_{2} \in L^{2} (\Gamma _{1} ), f_{0} \in ( L^{2} (\Gamma_{1}) \times[-\tau(0), 0] ) \)
and (3.9) is satisfied. Then for each
\(t>0\), there exist positive constants
K
and
ν
such that the energy of the solution for problem (2.13)-(2.20) satisfies
$$ E(t)\leq Ke^{-\nu\int_{t_{0}}^{t} \zeta(s)\,ds}, \quad \forall t\geq t_{0}. $$
(3.25)
Proof
In order to obtain the energy result of \(E(t)\), from Lemma 3.4, it suffices to prove that we have the estimate of \(L(t)\). To this end, we need the derivative of \(L (t)\). For this purpose, we estimate \(\Psi'(t)\). It follows from (3.16) and equations (2.13)-(2.17) that
$$\begin{aligned} \Psi' (t) = & \int_{\Omega}\bigl|u_{t} (t)\bigr|^{\rho}u_{tt}(t) u(t)\,dx+\frac {1}{\rho+1} \int_{\Omega}\bigl|u_{t} (t)\bigr|^{\rho+2}\,dx \\ &{} +\sigma \int_{\Omega}\nabla u(t) \nabla u_{t} (t) \int_{\Omega}\nabla u(t) \nabla u(t)\,dx + \int_{\Omega}\nabla u_{tt} (t) \nabla u(t)\,dx + \int_{\Omega}\bigl|\nabla u_{t} (t)\bigr|^{2}\,dx \\ &{} + \int_{\Gamma_{1}} h(x) u_{t} (t) y(t)\,d\Gamma+ \int_{\Gamma_{1}} h(x) u(t) y_{t} (t)\,d\Gamma+ \int_{\Gamma_{1}} h(x) f(x) y(t) y_{t} (t)\,d\Gamma \\ =& \int_{\Omega}\bigl[ \bigl|u_{t} (t)\bigr|^{\rho}u_{tt} (t) -\Delta u_{tt} (t) -\sigma\bigl(\nabla u_{t} (t) , \nabla u(t)\bigr)\Delta u(t) \bigr] u(t)\,dx \\ &{} + \int_{\Gamma_{1}} \biggl( \frac{\partial u_{tt} (t)}{\partial\nu } +\sigma\bigl(\nabla u_{t} (t) , \nabla u(t)\bigr)\frac{\partial u(t)}{\partial \nu} \biggr) u(t)\,d\Gamma \\ &{} +\frac{1}{\rho+1} \bigl\| u_{t} (t)\bigr\| _{\rho+2}^{\rho+2}+ \int_{\Omega}\bigl|\nabla u_{t} (t) \bigr|^{2}\,dx + \int_{\Gamma_{1}} h(x) u_{t} (t) y(t)\,d\Gamma \\ &{} + \int_{\Gamma_{1}} h(x) u(t) y_{t} (t)\,d\Gamma+ \int_{\Gamma_{1}} h(x) f(x) y(t) y_{t} (t)\,d\Gamma \\ &{} + \int_{\Gamma_{1}} h(x) f(x) y(t) y_{t} (t)\,d\Gamma \\ =&- \bigl( a+b\bigl\| \nabla u(t) \bigr\| ^{2} \bigr) \int_{\Omega}\bigl|\nabla u(t)\bigr|^{2}\,dx + \int_{\Omega}\nabla u(t) \int_{0}^{t} g(t-s) \nabla u(s)\,ds \,dx \\ &{} - \int_{\Omega}\bigl|u_{t} (t)\bigr|^{q} u_{t} (t) u(t)\,dx + \int_{\Omega}\bigl|u(t)\bigr|^{p+2}\,dx \\ &{} + \int_{\Gamma_{1}} \biggl[ \bigl( a+b\bigl\| \nabla u(t)\bigr\| ^{2}+ \sigma \bigl(\nabla u_{t} (t) , \nabla u(t)\bigr) \bigr) \frac{\partial u(t)}{\partial\nu } \\ &{} - \int_{0}^{t} g(t-s) \frac{\partial u(s) }{\partial\nu}\,dx + \frac {\partial u_{tt}(t)}{\partial\nu} \biggr] u(t)\,d\Gamma \\ &{} +\frac{1}{\rho+1} \bigl\| u_{t} (t)\bigr\| _{\rho+2}^{\rho+2} + \int_{\Omega}\bigl|\nabla u_{t} (t)\bigr|^{2}\,dx + \int_{\Gamma_{1}} h(x) u_{t} (t) y(t)\,d\Gamma \\ &{} + \int_{\Gamma_{1}} h(x) u(t) y_{t} (t)\,d\Gamma+ \int_{\Gamma_{1}} h(x) f(x) y(t) y_{t} (t)\,d\Gamma \\ =&- \bigl( a+b\bigl\| \nabla u(t) \bigr\| ^{2} \bigr) \int_{\Omega}\bigl|\nabla u(t)\bigr|^{2}\,dx + \int_{\Omega}\nabla u(t) \int_{0}^{t} g(t-s) \nabla u(s)\,ds \,dx \\ &{} - \int_{\Omega}\bigl|u_{t} (t)\bigr|^{q} u_{t} (t) u(t)\,dx + \int_{\Omega}\bigl|u(t)\bigr|^{p+2}\,dx-\mu_{0} \int_{\Gamma_{1}} u_{t} (x,t) u(t)\,d\Gamma \\ &{} -\mu_{1} \int_{\Gamma_{1}} u_{t} \bigl(x, t-\tau(t) \bigr)u(t)\,d\Gamma+\frac {1}{\rho+1} \bigl\| u_{t} (t) \bigr\| _{\rho+2}^{\rho+2} + \int_{\Omega}\bigl|\nabla u_{t} (t)\bigr|^{2}\,dx \\ & {} +2 \int_{\Gamma_{1}} h(x) u(t) y_{t} (t)\,d\Gamma- \int_{\Gamma_{1}} h(x) m(x) y^{2} (t)\,d\Gamma. \end{aligned}$$
(3.26)
Now, we estimate the right hand side of (3.26). By using (2.1), (2.2), (2.7), and Young’s inequality, for any \(\eta>0 \), we obtain
$$\begin{aligned} & \biggl| \int_{\Omega}\nabla u(t) \int_{0}^{t} g(t-s) \nabla u(s)\,ds \biggr| \\ & \quad = \int_{\Omega}\int_{0}^{t} g(t-s) \bigl(\nabla u(s) -\nabla u(t) \bigr) \nabla u(t)\,ds \,dx + \int_{0}^{t} g(s)\,ds \int_{\Omega}\bigl|\nabla u(t)\bigr|^{2}\,dx \\ & \quad \leq(1+\eta) \int_{0}^{t} g(s)\,ds \bigl\| \nabla u(t)\bigr\| ^{2} + \frac{1}{4\eta} (g\circ\nabla u) (t) \\ & \quad \leq(1+\eta) (a-l) \bigl\| \nabla u(t)\bigr\| ^{2} +\frac{1}{4\eta}(g \circ \nabla u) (t), \end{aligned}$$
(3.27)
$$\begin{aligned} & \biggl|- \int_{\Omega}\bigl|u_{t} (t)\bigr|^{q} u_{t} (t) u(t)\,dx \biggr| \\ & \quad \leq\eta C_{*}^{2} \int_{\Omega}\bigl|\nabla u(t)\bigr|^{2}\,dx +\frac{1}{4\eta} \bigl|u_{t} (t)\bigr|^{2(q+1)}\,dx \\ & \quad \leq\eta C_{*}^{2} \int_{\Omega}\bigl|\nabla u(t)\bigr|^{2}\,dx +\frac{\alpha_{2} }{4\eta} \int_{\Omega}\bigl|\nabla u_{t} (t)\bigr|^{2}\,dx, \end{aligned}$$
(3.28)
where \(\alpha_{2} =C_{*}^{2(q+1)} (\frac{2(p+2)E(0)}{p} )^{q} \),
$$\begin{aligned} & \biggl|{-}\mu_{0} \int_{\Gamma_{1}} u_{t} (x,t) u(t)\,d\Gamma \biggr| \\ &\quad\leq\eta \mu_{0} \tilde{C}_{*}^{2} \int_{\Omega}\bigl|\nabla u(t)\bigr|^{2}\,dx +\frac{\mu _{0}}{4\eta} \int_{\Gamma_{1}} u_{t}^{2} (t)\,d\Gamma, \end{aligned}$$
(3.29)
$$\begin{aligned} & \biggl|{-}\mu_{1} \int_{\Gamma_{1}} u_{t} \bigl(x,t-\tau(t)\bigr) u(t)\,d\Gamma \biggr| \\ & \quad \leq\eta|\mu_{1} | \tilde{C}_{*}^{2} \int_{\Omega}\bigl|\nabla u(t)\bigr|^{2}\,dx +\frac{|\mu_{1} |}{4\eta} \int_{\Gamma_{1}} u_{t}^{2} \bigl(x,t-\tau(t)\bigr)\,d\Gamma, \end{aligned}$$
(3.30)
and
$$\begin{aligned} \biggl| \int_{\Gamma_{1}} h(x) u(t) y_{t} (t)\,d\Gamma \biggr| & \leq\|h\|_{\infty}^{1/2} \biggl( \int_{\Gamma_{1}} h(x) y_{t}^{2} (t)\,d\Gamma \biggr)^{1/2} \biggl( \int_{\Gamma_{1}} \bigl|u(t)\bigr|^{2}\,d\Gamma \biggr)^{1/2} \\ & \leq\eta \tilde{C}_{*}^{2} \int_{\Omega}\bigl|\nabla u(t)\bigr|^{2}\,dx +\frac{\|h\| _{\infty}}{4\eta f_{0*}} \int_{\Gamma_{1}} h(x) f(x) y_{t}^{2} (t)\,d\Gamma. \end{aligned}$$
(3.31)
Substitution of (3.27)-(3.31) into (3.26) yields
$$\begin{aligned} \Psi'(t) \leq& - \bigl( a+b\bigl\| \nabla u(t) \bigr\| ^{2} \bigr) \int_{\Omega}\bigl|\nabla u(t)\bigr|^{2}\,dx+(1+\eta) (a-l) \int_{\Omega}\bigl|\nabla u(t)\bigr|^{2}\,dx \\ &{} +\frac{1}{4\eta}(g\circ\nabla u) (t)+\eta C_{*}^{2} \int_{\Omega}\bigl|\nabla u(t)\bigr|^{2}\,dx +\frac{\alpha_{2} }{4\eta} \int_{\Omega}\bigl|\nabla u_{t} (t)\bigr|^{2}\,dx \\ &{} + \int_{\Omega}\bigl|u(t)\bigr|^{p+2}\,dx+\eta\mu_{0} \tilde{C}_{*}^{2} \int _{\Omega}\bigl|\nabla u(t)\bigr|^{2}\,dx +\frac{\mu_{0}}{4\eta} \int_{\Gamma_{1}} u_{t}^{2} (t)\,d\Gamma \\ &{} +\eta|\mu_{1} | \tilde{C}_{*}^{2} \int_{\Omega}\bigl|\nabla u(t)\bigr|^{2}\,dx +\frac{|\mu_{1} |}{4\eta} \int_{\Gamma_{1}} u_{t}^{2} \bigl(x,t-\tau(t)\bigr)\,d\Gamma \\ &{} +\frac{1}{\rho+1} \bigl\| u_{t} (t) \bigr\| _{\rho+2}^{\rho+2} + \int_{\Omega}\bigl|\nabla u_{t} (t) \bigr|^{2}\,dx+\eta \tilde{C}_{*}^{2} \int_{\Omega}\bigl|\nabla u(t)\bigr|^{2}\,dx \\ &{} +\frac{\|h\|_{\infty}}{2\eta f_{0*}} \int_{\Gamma_{1}} h(x) f(x) y_{t}^{2} (t)\,d\Gamma- \int_{\Gamma_{1}} h(x) m(x) y^{2} (t)\,d\Gamma \\ =& \frac{1}{\rho+1} \bigl\| u_{t} (t)\bigr\| _{\rho+2}^{\rho+2} - \bigl[ \bigl( a+b\bigl\| \nabla u(t)\bigr\| ^{2} \bigr)-(1+\eta) (a-l )-\eta C_{*}^{2}-\eta\mu_{0} \tilde{C}_{*}^{2} \\ &{} -\eta|\mu_{1} |\tilde{C}_{*}^{2} -\eta \tilde{C}_{*}^{2} \bigr]\bigl\| \nabla u(t)\bigr\| ^{2} + \biggl( \frac{\alpha_{2} }{4\eta}+1 \biggr) \bigl\| \nabla u_{t} (t)\bigr\| ^{2} + \frac{1}{4\eta} (g\circ\nabla u) (t) \\ &{} +\bigl\| u(t)\bigr\| _{p+2}^{p+2}+\frac{\mu_{0} }{4\eta} \int_{\Gamma_{1} }u_{t}^{2} (t)\,d\Gamma+ \frac{|\mu_{1}|}{4\eta} \int_{\Gamma_{1}} u_{t}^{2} \bigl(x,t-\tau (t) \bigr)\,d\Gamma \\ &{} +\frac{\|h\|_{\infty}}{2\eta f_{0*}} \int_{\Gamma_{1}} h(x) f(x) y_{t}^{2} (t)\,d\Gamma- \int_{\Gamma_{1}} h(x) m(x) y^{2} (t)\,d\Gamma. \end{aligned}$$
(3.32)
Next, we would like to estimate \(\Phi' (t)\). Taking the derivative of \(\Phi(t)\) in (3.17) and using (2.13)-(2.17), we can deduce that
$$\begin{aligned} \Phi' (t) =&- \int_{\Omega}\bigl|u_{t} (t)\bigr|^{\rho}u_{tt} (t) \int_{0}^{t} g(t-s) \bigl(u(t)-u(s)\bigr)\,ds\,dx \\ &{} -\frac{1}{\rho+1} \int_{\Omega}\bigl|u_{t} (t)\bigr|^{\rho}u_{t} (t) \int_{0}^{t} g'(t-s) \bigl(u(t)-u(s) \bigr) \,ds\,dx \\ &{} -\frac{1}{\rho+1} \int_{\Omega}\bigl|u_{t} (t)\bigr|^{\rho}u_{t} (t) \int_{0}^{t} g(t-s) u_{t} (t) \,ds\,dx \\ &{} - \int_{\Omega}\nabla u_{tt}(t) \int_{0}^{t} g(t-s) \bigl(\nabla u(t)-\nabla u(s) \bigr)\,ds \,dx \\ &{} - \int_{\Omega}\nabla u_{t} (t) \int_{0}^{t} g'(t-s) \bigl(\nabla u(t)-\nabla u(s)\bigr)\,ds \,dx \\ &{} - \int_{\Omega}\nabla u_{t} (t) \int_{0}^{t} g(t-s) \nabla u_{t} (t) \,ds\,dx \\ =&- \int_{\Omega}\biggl[ \bigl( a+b\bigl\| \nabla u(t)\bigr\| ^{2} \bigr) \Delta u(t)+\sigma\bigl(\nabla u(t), \nabla u_{t} (t)\bigr)\Delta u(t) - \int_{0}^{t} g(t-s) u(s)\,ds \\ &{} -\bigl|u_{t} (t)\bigr|^{q} u_{t} (t) +\bigl|u(t)\bigr|^{p} u(t) \biggr] \int_{0}^{t} g(t-s) \bigl(u(t)-u(s)\bigr) \,ds\,dx \\ &{} - \int_{\Gamma_{1}} \frac{\partial u_{tt}(t)}{\partial\nu} \int_{0}^{t} g(t-s) \bigl(u(t)-u(s)\bigr) \,ds\,d\Gamma \\ &{} - \int_{\Omega}\nabla u_{t} (t) \int_{0}^{t} g'(t-s) \bigl(\nabla u(t)-\nabla u(s)\bigr)\,ds \,dx \\ &{} - \int_{\Omega}\nabla u_{t} (t) \int_{0}^{t} g(t-s) \nabla u_{t} (t) \,ds\,dx \\ &{} -\frac{1}{\rho+1} \int_{\Omega}\bigl|u_{t} (t)\bigr|^{\rho}u_{t} (t) \int_{0}^{t} g'(t-s) \bigl(u(t)-u(s) \bigr) \,ds\,dx \\ &{} -\frac{1}{\rho+1} \int_{\Omega}\bigl|u_{t} (t)\bigr|^{\rho}u_{t} (t) \int_{0}^{t} g(t-s) u_{t} (t) \,ds\,dx \\ =& \bigl( a+b\bigl\| \nabla u(t)\bigr\| ^{2} \bigr) \int_{\Omega}\nabla u(t) \int_{0}^{t} g(t-s) \bigl(\nabla u(t)-\nabla u(s) \bigr) \,ds\,dx \\ &{} +\sigma \int_{\Omega}\nabla u(t) \nabla u_{t} (t) \int_{\Omega}\nabla u(t) \int_{0}^{t} g(t-s) \bigl(\nabla u(t)-\nabla u(s) \bigr)\,ds\,dx \\ &{} - \int_{\Omega}\int_{0}^{t} g(t-s) \nabla u(s)\,ds \int_{0}^{t} g(t-s) \bigl(\nabla u(t)-\nabla u(s) \bigr)\,ds\,dx \\ &{} - \int_{\Gamma_{1}} \bigl( h(x) y_{t} (t)-\mu_{0} u_{t} (x,t)-\mu_{1} u_{t} \bigl(x, t-\tau(t)\bigr) \bigr) \int_{0}^{t} g(t-s) \bigl(u(t)-u(s)\bigr)\,ds \,d\Gamma \\ &{} + \int_{\Omega}\bigl|u_{t} (t)\bigr|^{q} u_{t} (t) \int_{0}^{t} g(t-s) \bigl(u(t)-u(s)\bigr)\,ds\,dx \\ &{} - \int_{\Omega}\bigl|u (t)\bigr|^{p} u (t) \int_{0}^{t} g(t-s) \bigl(u(t)-u(s)\bigr)\,ds\,dx \\ &{} - \int_{\Omega}\nabla u_{t} (t) \int_{0}^{t} g'(t-s) \bigl(\nabla u(t)-\nabla u(s)\bigr)\,ds \,dx \\ &{}- \int_{\Omega}\nabla u_{t} (t) \int_{0}^{t} g(t-s) \nabla u_{t} (t) \,ds\,dx \\ &{} -\frac{1}{\rho+1} \int_{\Omega}\bigl|u_{t} (t)\bigr|^{\rho}u_{t} (t) \int_{0}^{t} g'(t-s) \bigl(u(t)-u(s) \bigr) \,ds\,dx \\ &{} -\frac{1}{\rho+1} \int_{\Omega}\bigl|u_{t} (t)\bigr|^{\rho}u_{t} (t) \int_{0}^{t} g(t-s) u_{t} (t) \,ds\,dx. \end{aligned}$$
(3.33)
Now, we will estimate the right hand side of (3.33). From Lemma 3.3, (2.1), (2.2), (2.5), (3.5), (3.12), and Young’s inequality, for any \(\eta>0\), we have the following inequalities:
$$\begin{aligned} & \biggl| \int_{\Omega}\bigl( a+b\bigl\| \nabla u(t)\bigr\| ^{2} \bigr) \nabla u(t) \int_{0}^{t} g(t-s) \bigl(\nabla u(t)-\nabla u(s) \bigr) \,ds\,dx \biggr| \\ &\quad \leq \biggl| \int_{\Omega}\biggl( a+b\frac{2(p+2)}{lp} E(0) \biggr)\nabla u(t) \int_{0}^{t} g(t-s) \bigl(\nabla u(t) -\nabla u(s) \bigr) \,ds\,dx \biggr| \\ & \quad \leq\eta\bigl\| \nabla u(t)\bigr\| ^{2} +\frac{a-l}{4\eta} \biggl( a+b \frac {2(p+2)}{lp} E(0) \biggr)^{2} (g\circ\nabla u) (t), \end{aligned}$$
(3.34)
$$\begin{aligned} & \biggl| \sigma \int_{\Omega}\nabla u(t) \nabla u_{t} (t)\,dx \int_{\Omega}\nabla u(t) \int_{0}^{t} g(t-s) \bigl(\nabla u(t)-\nabla u(s) \bigr) \,ds\,dx \biggr| \\ & \quad \leq\sigma^{2} \biggl( \int_{\Omega}\nabla u(t) \nabla u_{t} (t)\,dx \biggr)^{2} \eta\bigl\| \nabla u(t)\bigr\| ^{2} \\ &\qquad{}+\frac{1}{4\eta} \int_{\Omega}\biggl( \int_{0}^{t} g(t-s) \bigl(\nabla u(t)-\nabla u(s) \bigr)\,ds \biggr)^{2}\,dx \\ & \quad \leq\frac{2(p+2)\sigma^{2}}{p} E(0) \biggl( \int_{\Omega}\nabla u(t)\nabla u_{t} (t)\,dx \biggr)^{2} +\frac{a-l}{4\eta} (g\circ\nabla u) (t) \\ &\quad \leq-\frac{2(p+2)\sigma}{p} E(0)E'(t) +\frac{a-l}{4\eta} (g \circ \nabla u) (t), \end{aligned}$$
(3.35)
$$\begin{aligned} & \biggl|{-} \int_{\Omega}\int_{0}^{t} g(t-s) \nabla u(s)\,ds \int_{0}^{t} g(t-s) \bigl(\nabla u(t)-\nabla u(s) \bigr)\,ds \,dx \biggr| \\ & \quad \leq \biggl(2\eta+\frac{1}{4\eta} \biggr) (a-l) (g\circ\nabla u) (t)+2\eta(a-l)^{2} \bigl\| \nabla u(t)\bigr\| ^{2}, \end{aligned}$$
(3.36)
$$\begin{aligned} & \biggl|{-} \int_{\Gamma_{1} } h(x) y_{t} (t) \int_{0}^{t} g(t-s) \bigl(u(t)-u(s)\bigr)\,ds \,d\Gamma \biggr| \\ & \quad \leq\frac{\eta\|h\|_{\infty}}{f_{0*}} \int_{\Gamma_{1}} h(x) f(x) y_{t}^{2} (t)\,d\Gamma+ \frac{(a-l)\tilde{C}_{*}^{2}}{4\eta} (g\circ\nabla u) (t), \end{aligned}$$
(3.37)
$$\begin{aligned} & \biggl| {-} \int_{\Gamma_{1}} \mu_{0} u_{t} (x,t) \int_{0}^{t} g(t-s) \bigl(u(t)-u(s)\bigr)\,ds\,d\Gamma \biggr| \\ &\quad \leq\eta\mu_{0} \int_{\Gamma_{1}} \bigl|u_{t} (x,t)\bigr|^{2}\,d\Gamma+ \frac{\mu_{0} (a-l)\tilde{C}_{*}^{2}}{4\eta} (g\circ\nabla u) (t), \end{aligned}$$
(3.38)
$$\begin{aligned} & \biggl| - \int_{\Gamma_{1}} \mu_{1} u_{t} \bigl(x,t-\tau(t) \bigr) \int_{0}^{t} g(t-s) \bigl(u(t)-u(s)\bigr)\,ds\,d\Gamma \biggr| \\ &\quad \leq\eta|\mu_{1} | \int_{\Gamma_{1}} u_{t}^{2} \bigl(x, t-\tau(t) \bigr)\,d\Gamma +\frac{|\mu_{1} | (a-l)\tilde{C}_{*}^{2}}{4\eta} (g\circ\nabla u) (t), \end{aligned}$$
(3.39)
$$\begin{aligned} & \biggl| \int_{\Omega} \bigl|u_{t} (t)\bigr|^{q} u_{t} (t) \int_{0}^{t} g(t-s) \bigl(u(t)-u(s)\bigr)\,ds\,dx\biggr| \\ & \quad \leq\eta\alpha_{2} \bigl\| \nabla u_{t} (t) \bigr\| ^{2} +\frac{ (a -l)\tilde {C}_{*}^{2}}{4\eta} (g\circ\nabla u) (t), \end{aligned}$$
(3.40)
where \(\alpha_{2} =C_{*}^{2(q+1)} ( \frac{2(p+2)E(0)}{p} )^{q} \),
$$\begin{aligned} & \biggl| - \int_{\Omega} \bigl|u_{t} (t)\bigr|^{p} u(t) \int_{0}^{t} g(t-s) \bigl(u(t)-u(s)\bigr)\,ds\,dx \biggr| \\ &\quad \leq\eta\alpha_{3} \bigl\| \nabla u (t)\bigr\| ^{2} + \frac{ (a -l)\tilde {C}_{*}^{2}}{4\eta} (g\circ\nabla u) (t), \end{aligned}$$
(3.41)
where \(\alpha_{3} =C_{*}^{2(p+1)} ( \frac{2(p+2)E(0)}{lp} )^{p} \),
$$\begin{aligned} &\begin{aligned}[b] & \biggl| {-} \int_{\Omega} \nabla u_{t} (t) \int_{0}^{t} g'(t-s) \bigl(\nabla u(t)-\nabla u(s)\bigr)\,ds\,dx \biggr| \\ &\quad \leq\eta \bigl\| \nabla u_{t} (t)\bigr\| ^{2} -\frac{ g(0)}{4\eta} \bigl(g'\circ \nabla u\bigr) (t), \end{aligned} \end{aligned}$$
(3.42)
$$\begin{aligned} & \biggl|-\frac{1}{\rho+1} \int_{\Omega}\bigl|u_{t} (t)\bigr|^{\rho}u_{t} (t) \int_{0}^{t} g'(t-s) \bigl(u(t)-u(s) \bigr)\,ds \,dx \biggr| \\ & \quad \leq\frac{\eta}{\rho+1}C_{*}^{2(\rho+1)} \biggl( \frac {2(p+2)E(0)}{p} \biggr)^{\rho}\bigl\| \nabla u_{t} (t)\bigr\| ^{2} - \frac{ g(0) C_{*}^{2} }{4\eta(\rho+1)} \bigl(g'\circ\nabla u\bigr) (t) \\ &\quad =\frac{\eta\alpha_{4} }{\rho+1} \bigl\| \nabla u_{t} (t)\bigr\| ^{2} - \frac{ g(0) C_{*}^{2} }{4\eta(\rho+1)} \bigl(g'\circ\nabla u\bigr) (t), \end{aligned}$$
(3.43)
where \(\alpha_{4} =C_{*}^{2(\rho+1)} ( \frac{2(p+2)E(0)}{p} )^{\rho}\). Thus from (3.33)-(3.43) we arrive at
$$\begin{aligned} \Phi' (t) \leq& \eta\bigl\| \nabla u(t)\bigr\| ^{2}+ \frac{a-l}{4\eta} \biggl( a+b\frac{2(p+2)}{lp} E(0) \biggr)^{2} (g \circ\nabla u) (t) \\ &{} -\frac{2(p+2)\sigma}{p} E(0)E'(t) +\frac{a-l}{4\eta} (g\circ \nabla u) (t) \\ &{} + \biggl(2\eta+\frac{1}{4\eta} \biggr) (a-l) (g\circ\nabla u) (t)+2\eta (a-l)^{2} \bigl\| \nabla u(t)\bigr\| ^{2} \\ &{} +\frac{\eta\|h\|_{\infty}}{f_{0}} \int_{\Gamma_{1}} h(x) f(x) y_{t}^{2} (t)\,d\Gamma+ \frac{(a-l)\tilde{C}_{*}^{2}}{4\eta} (g\circ\nabla u) (t) \\ &{} +\eta\mu_{0} \bigl\| u_{t} (x,t)\bigr\| ^{2} + \frac{\mu_{0} (a-l)\tilde{C}_{*}^{2}}{4\eta } (g\circ\nabla u) (t) \\ &{} +\eta|\mu_{1} | \int_{\Gamma_{1}} u_{t}^{2} \bigl(x, t-\tau(t) \bigr)\,d\Gamma+\frac {|\mu_{1} | (a-l)\tilde{C}_{*}^{2}}{4\eta} (g\circ\nabla u) (t) \\ &{} +\eta\alpha_{2} \bigl\| \nabla u_{t} (t)\bigr\| ^{2} + \frac{ (a -l)\tilde {C}_{*}^{2}}{4\eta} (g\circ\nabla u) (t) +\eta\alpha_{3} \bigl\| \nabla u (t) \bigr\| ^{2} +\frac{ (a -l)\tilde{C}_{*}^{2}}{4\eta} (g\circ\nabla u) (t) \\ &{} +\eta \bigl\| \nabla u_{t} (t)\bigr\| ^{2} -\frac{ g(0)}{4\eta} \bigl(g'\circ\nabla u\bigr) (t)- \biggl( \int_{0}^{t} g(s)\,ds \biggr) \bigl\| \nabla u_{t} (t)\bigr\| ^{2} \\ &{} +\frac{\eta\alpha_{4} }{\rho+1} \bigl\| \nabla u_{t} (t)\bigr\| ^{2} - \frac{ g(0) C_{*}^{2} }{4\eta(\rho+1)} \bigl(g'\circ\nabla u\bigr) (t) - \biggl( \int_{0}^{t} g(s)\,ds \biggr)\frac{1}{\rho+1} \bigl\| u_{t} (t)\bigr\| _{\rho+2}^{\rho+2} \\ =&- \biggl( \int_{0}^{t} g(s)\,ds \biggr)\frac{1}{\rho+1} \bigl\| u_{t} (t)\bigr\| _{\rho +2}^{\rho+2}+\eta \bigl( 1+ \alpha_{3} +2(a-l)^{2} \bigr) \bigl\| \nabla u(t)\bigr\| ^{2} \\ &{} - \biggl[ \int_{0}^{t} g(s)\,ds -\eta \biggl( \alpha_{2} +\frac{\alpha_{4}}{\rho +1} \biggr) \biggr]\bigl\| \nabla u_{t} (t)\bigr\| ^{2} \\ &{} +\frac{(a-l)}{4\eta} \biggl[ \biggl(a+b\frac{2(p+2)}{lp} E(0) \biggr)^{2}+2+3\tilde{C}_{*}^{2} +8\eta^{2} + \mu_{0} \tilde{C}_{*}^{2} +|\mu_{1} |\tilde {C}_{*}^{2} \biggr] \\ &{}\times (g\circ\nabla u) (t) -\frac{g(0)}{4\eta} \biggl( 1+\frac{C_{*}^{2}}{\rho+1} \biggr) \bigl(g'\circ \nabla u\bigr) (t)-\frac{2(p+2)\sigma}{p} E(0)E'(t) \\ &{} +\frac{\eta\|h\|_{\infty}}{f_{0*}} \int_{\Gamma_{1}} h(x) f(x) y_{t}^{2} (t)\,d\Gamma+ \eta\mu_{0} \int_{\Gamma_{1}} \bigl|u_{t} (x,t)\bigr|^{2}\,d\Gamma \\ &{}+\eta | \mu_{1} | \int_{\Gamma_{1}} u_{t}^{2} \bigl(x,t-\tau(t) \bigr)\,d\Gamma. \end{aligned}$$
(3.44)
Since \(g(t)\) is positive and continuous, for any \(t_{0} >0\) we see that
$$\int_{0}^{t} g(s)\,ds \geq \int_{0}^{t_{0}} g(s)\,ds \equiv g_{0} , \quad \forall t\geq t_{0}. $$
Hence from (3.5), (3.15), (3.32), and (3.44), we conclude that for any \(t\geq t_{0} >0\),
$$\begin{aligned} L'(t) =&ME'(t) +\varepsilon\Psi'(t)+ \Phi'(t) \\ \leq&-\frac{1}{\rho+1} (g_{0} -\varepsilon)\bigl\| u_{t} (t) \bigr\| _{\rho+2}^{\rho +2} \\ &{} - \biggl[ \frac{M g(0)}{2}-\varepsilon(1+\eta) (a -l)-\varepsilon \eta \bigl(C_{*}^{2} +\mu_{0} \tilde{C}_{*}^{2} + \tilde{C}_{*}^{2} +|\mu_{1} |\tilde{C}_{*}^{2} \bigr) \\ &{} -\eta(1+\alpha_{3} ) -2\eta(a -l)^{2} \biggr] \bigl\| \nabla u(t)\bigr\| ^{2}-\varepsilon \bigl(a+b\bigl\| \nabla u(t)\bigr\| ^{2} \bigr)\bigl\| \nabla u(t)\bigr\| ^{2} \\ &{} - \biggl[ g_{0} -\eta \biggl( \alpha_{2} + \frac{\alpha_{4}}{\rho+1} \biggr) -\varepsilon \biggl(\frac{\alpha_{2}}{4\eta}+1 \biggr) \biggr]\bigl\| \nabla u_{t} (t) \bigr\| ^{2} \\ &{}+\frac{1}{4\eta} \biggl[\varepsilon+(a-l) \biggl( a+b\frac{2(p+2)}{lp} E(0) \biggr)^{2} \\ &{} +(a-l) (2+ 3\tilde{C}_{*}^{2}+8\eta^{2} + \mu_{0} \tilde{C}_{*}^{2} +|\mu_{1} | \tilde{C}_{*}^{2} \biggr](g\circ\nabla u) (t) \\ &{} + \biggl[\frac{M}{2} -\frac{g(0)}{4\eta} \biggl(1+ \frac{C_{*}^{2} }{\rho +1} \biggr) \biggr] \bigl(g'\circ\nabla u\bigr) (t)-M\bigl\| u_{t} (t)\bigr\| _{q+2}^{q+2} +\varepsilon\bigl\| u(t) \bigr\| _{p+2}^{p+2} \\ &{} -M\sigma \biggl( \frac{1}{2} \frac{d}{dt} \bigl\| \nabla u(t) \bigr\| ^{2} \biggr)^{2} - \biggl( M-\frac{\varepsilon\|h\|_{\infty}}{4\eta f_{0} } - \frac{\eta\|h\| _{\infty}}{f_{0}} \biggr) \int_{\Gamma_{1}} h(x) f(x) y_{t}^{2} (t)\,d\Gamma \\ &{} -\varepsilon \int_{\Gamma_{1}} h(x) m(x) y^{2} (t)\,d\Gamma- \biggl( MC_{1} -\frac{\varepsilon|\mu_{1}|}{4\eta}-\eta|\mu_{1}| \biggr) \int_{\Gamma_{1}} u_{t}^{2} \bigl(x,t-\tau(t)\bigr)\,d\Gamma \\ &{} -\frac{M\lambda\xi}{2} \int_{t-\tau(t)}^{t} \int_{\Gamma _{1}}e^{-\lambda(t-s)} u_{t}^{2} (s)\,d\Gamma \,ds- \biggl( MC_{1} -\frac {\varepsilon\mu_{0} }{4\eta}-\eta\mu_{0} \biggr) \int_{\Gamma_{1}} u_{t}^{2} (x,t)\,d\Gamma \\ &{} -\frac{2(p+2)\sigma}{p} E(0)E'(t). \end{aligned}$$
At this point, we choose \(\varepsilon>0 \) small enough and we pick \(\eta>0 \) sufficiently small such that
$$\begin{aligned}& K_{1} =\frac{1}{\rho+1} (g_{0} -\varepsilon)>0, \qquad K_{2} =g_{0} -\eta \biggl(\alpha_{2} + \frac{\alpha_{4}}{\rho+1} \biggr)-\varepsilon \biggl( \frac{\alpha_{2}}{4\eta}+1 \biggr) >0, \end{aligned}$$
and then we choose M so large that
$$\begin{aligned}& K_{3} =\frac{M g(0)}{2}-\varepsilon(1+\eta) (a -l)-\varepsilon\eta \bigl(C_{*}^{2} +\mu_{0} \tilde{C}_{*}^{2} + \tilde{C}_{*}^{2} +|\mu_{1} |\tilde{C}_{*}^{2} \bigr)\\& \hphantom{K_{3}=}{}-\eta(1+\alpha_{3} ) -2\eta(a -l)^{2} >0, \\& K_{4} =\frac{M}{2} -\frac{g(0)}{4\eta} \biggl(1+ \frac{C_{*}^{2} }{\rho +1} \biggr)>0, \\& K_{5} =M-\frac{\varepsilon\|h\|_{\infty}}{4\eta f_{0*} } -\frac{\eta \|h\|_{\infty}}{f_{0*}} >0, \\& K_{6} =MC_{1} -\frac{\varepsilon|\mu_{1}|}{4\eta}-\eta| \mu_{1}|>0, \\& K_{7} =MC_{1} -\frac{\varepsilon\mu_{0} }{4\eta}-\eta\mu_{0} >0. \end{aligned}$$
Hence, for any \(t\geq t_{0} \), we arrive at
$$\begin{aligned} L'(t) \leq&-K_{1} \bigl\| u_{t} (t) \bigr\| _{\rho+2}^{\rho+2}- K_{3} \bigl\| \nabla u(t)\bigr\| ^{2}- \varepsilon \bigl(a+b\bigl\| \nabla u(t)\bigr\| ^{2} \bigr)\bigl\| \nabla u(t) \bigr\| ^{2} \\ &{} -K_{2} \bigl\| \nabla u_{t} (t)\bigr\| ^{2} +K_{8} (g\circ\nabla u) (t)+K_{4} \bigl(g' \circ \nabla u\bigr) (t) -M\bigl\| u_{t} (t)\bigr\| _{q+2}^{q+2} + \varepsilon\bigl\| u(t)\bigr\| _{p+2}^{p+2} \\ &{} -M\sigma \biggl( \frac{1}{2} \frac{d}{dt}\bigl\| \nabla u(t) \bigr\| ^{2} \biggr)^{2}-K_{5} \int_{\Gamma_{1}} h(x) f(x) y_{t}^{2} (t)\,d\Gamma- \varepsilon \int _{\Gamma_{1}} h(x) m(x) y_{t}^{2} (t)\,d\Gamma \\ &{} -K_{6} \int_{\Gamma_{1}} u_{t}^{2} \bigl(x,t-\tau(t) \bigr)\,d\Gamma-\frac{M\lambda\xi }{2} \int_{t-\tau(t)}^{t} \int_{\Gamma_{1}}e^{-\lambda(t-s)} u_{t}^{2} (s)\,d\Gamma \,ds\\ &{}-K_{7} \int_{\Gamma_{1}} u_{t}^{2} (x,t)\,d\Gamma -\frac{2(p+2)}{p} E(0)E'(t), \end{aligned}$$
where
$$K_{8} =\frac{1}{4\eta} \biggl[\varepsilon+(a-l) \biggl( a+b \frac {2(p+2)}{lp} E(0) \biggr)^{2} +(a-l) (2+ 3\tilde{C}_{*}^{2}+8 \eta^{2} +\mu_{0} \tilde{C}_{*}^{2} +| \mu_{1} |\tilde {C}_{*}^{2} \biggr]. $$
It follows that
$$L'(t) \leq-K_{9} E(t)+K_{10} (g\circ\nabla u) (t)-\frac{2(p+2)}{p} E(0)E'(t), $$
where \(K_{9} \) and \(K_{10}\) are some positive constants. Multiplying the above inequality by \(\zeta(t) \) and using (2.6) and (3.5), we obtain, for any \(t\geq t_{0}\),
$$ \zeta(t)L'(t)\leq-K_{9} \zeta(t) E(t)- \bigl(2K_{10} +K_{11} \zeta(t)\bigr)E'(t), $$
(3.45)
where \(K_{11} = \frac{2(p+2)}{p} E(0)\).
Now, we define
$$G(t)=\zeta(t) L(t) +\bigl(2K_{10}+K_{11}\zeta(t)\bigr) E(t). $$
As ξ is non-increasing positive function, by using Lemma 3.4, the function \(G(t)\) is equivalent to \(E(t)\). Using the fact that \(\xi'(t)\leq0\), (3.45) implies that
$$G'(t)\leq-K_{9} \zeta(t) E(t) \leq-\nu\zeta(t) G(t), $$
where ν is a positive constant.
The integration of the inequality between \(t_{0} \) and t gives the following estimation for the function \(G(t)\):
$$G(t)\leq G(t_{0} ) e^{-\nu\int_{t_{0}}^{t} \zeta(s)\,ds} ,\quad \forall t\geq t_{0}. $$
Again, employing that G is equivalent to E, we get
$$E(t)\leq K e^{-\nu\int_{t_{0}}^{t} \zeta(s)\,ds} , \quad \forall t\geq t_{0}, $$
where K is a positive constant. Thus the proof of Theorem 3.2 is completed. □