In this section, we derive general decay results for both cases when \(m_{1} \geq 2\) and when \(1 < m_{1} < 2\) by following the ideas in [20] and [23] with some necessary modification.
First, we let
$$ k_{ \eta }(t) = \eta k(t) - k'(t) \quad \text{and} \quad {\mathcal{C}}_{\eta } = \int ^{\infty }_{0} \frac{ k^{2}(s) }{ k_{\eta }(s) } \,ds \quad \text{for } 0 < \eta < 1 , $$
then, by the arguments of [14, 23], we have the following lemma.
Lemma 4.1
Let \((A_{3})\) be satisfied. Then, for \(v \in L^{2}_{\mathrm{loc}} ( [0, \infty ), L^{2}(\Omega ) ) \), it holds
$$\begin{aligned} \int _{\Omega } \biggl( \int ^{t}_{0} k ( t-s ) \bigl( v (t) - v (s) \bigr) \,ds \biggr)^{2} \,dx \leq {\mathcal{C}}_{\eta } ( k_{\eta } \Box v ) (t). \end{aligned}$$
(4.1)
Now, we define
$$ L(t) = N E(t) + N_{1} \Phi (t) + N_{2} \Psi (t) + \Upsilon (t) , $$
where \(N > 0 \), \(N_{i} >0 \), \(i = 1, 2 \),
$$\begin{aligned}& \Phi (t) = \int _{\Omega } u u_{t} \,dx , \\& \Psi (t) = - \int ^{t}_{0} k (t-s) \int _{\Omega } \bigl( u(t) -u(s) \bigr) u_{t} \,dx \,ds , \end{aligned}$$
and
$$ \Upsilon (t) = \tau \int _{\Omega } \int _{0}^{1} e^{ - \rho \tau } \bigl\vert y (x, \rho , t ) \bigr\vert ^{ m (x)} \,d\rho \,dx. $$
Lemma 4.2
Assume that \((A_{1})\), \((A_{2})\), \((A_{3})\), and (3.12) hold. Then \(L(t)\) is equivalent to \(E(t)\).
Proof
From (3.6), (3.13), and (3.7), we have
$$\begin{aligned}& \bigl\vert L(t) - N E(t) \bigr\vert \\& \quad \leq \frac{ N_{1} + N_{2} }{2} \Vert u_{t} \Vert _{2}^{2} + \frac{ N_{1} B_{2}^{2} }{2} \Vert \Delta u \Vert _{2}^{2} + \frac{ N_{2} B_{2}^{2} ( 1 - k_{l}) }{2} ( k \Box \Delta u ) \\& \qquad {}+ \tau \int _{\Omega } \int _{0}^{1} \bigl\vert y (x, \rho , t ) \bigr\vert ^{ m (x)} \,d\rho \,dx \\& \quad \leq \frac{ N_{1} + N_{2} }{2} \Vert u_{t} \Vert _{2}^{2} + \frac{ N_{2} B_{2}^{2} ( 1 - k_{l}) }{2} ( k \Box \Delta u ) \\& \qquad {}+ \frac{ N_{1} B_{2}^{2} p_{1} }{ k_{l} ( p_{1} -2 ) } \biggl\{ J(t) - \frac{1}{ p_{1} } I(t) - \frac{ p_{1} -2 }{ 2 p_{1} } ( k \Box \Delta u) \biggr\} + \tau \int _{\Omega } \int _{0}^{1} \bigl\vert y (x, \rho , t ) \bigr\vert ^{ m (x)} \,d\rho \,dx \\& \quad \leq \frac{ N_{1} + N_{2} }{2} \Vert u_{t} \Vert _{2}^{2} + \frac{ N_{2} B_{2}^{2} ( 1 - k_{l} ) }{2} ( k \Box \Delta u ) + \frac{ N_{1} B_{2}^{2} p_{1} }{ k_{l} ( p_{1} -2 ) } J(t) \\& \qquad {}+ \tau \int _{\Omega } \int _{0}^{1} \bigl\vert y (x, \rho , t ) \bigr\vert ^{ m (x)} \,d\rho \,dx \\& \quad \leq \frac{ N_{1} + N_{2} }{2} \Vert u_{t} \Vert _{2}^{2} + \biggl( \frac{ 2 N_{2} B_{2}^{2} ( 1 - k_{l} ) p_{1} }{ 2 ( p_{1} -2 ) } + \frac{ N_{1} B_{2}^{2} p_{1} }{k_{l} ( p_{1} -2 ) } \biggr) J(t) \\& \qquad {}+ \tau \int _{\Omega } \int _{0}^{1} \bigl\vert y (x, \rho , t ) \bigr\vert ^{ m (x)} \,d\rho \,dx \\& \quad \leq \max \biggl\{ N_{1} + N_{2} , \frac{ 2 N_{2} B_{2}^{2} ( 1 - k_{l} ) p_{1} }{ 2 ( p_{1} -2 ) } + \frac{ N_{1} B_{2}^{2} p_{1} }{ k_{l} ( p_{1} -2 ) } , \frac{ 2 }{ \xi } \biggr\} E(t) . \end{aligned}$$
Taking \(N > \max \{ N_{1} + N_{2} , \frac{ 2 N_{2} B_{2}^{2} ( 1 - k_{l} ) p_{1} }{ 2 ( p_{1} -2 ) } + \frac{ N_{1} B_{2}^{2} p_{1} }{ k_{l} ( p_{1} -2 ) } , \frac{ 2 }{ \xi } \} \), we finish the proof. □
Lemma 4.3
The function ϒ satisfies
$$\begin{aligned} \Upsilon ' (t) \leq & \int _{\Omega } \vert u_{t} \vert ^{ m (x) } \,dx - \tau e^{ - \tau } \int _{\Omega } \int ^{1}_{0} \bigl\vert y (x, \rho , t ) \bigr\vert ^{ m (x) } \,d\rho \,dx . \end{aligned}$$
(4.2)
Proof
Using (2.9) and \(y (x, 0, t ) = u_{t}(x,t) \), we get
$$\begin{aligned} \Upsilon ' (t) = & \tau \int _{\Omega } \int ^{1}_{0} e^{ - \rho \tau } m (x) \bigl\vert y (x, \rho , t ) \bigr\vert ^{ m (x) -2} y (x, \rho , t ) y_{t} (x, \rho , t ) \,d\rho \,dx \\ = & - \int _{\Omega } \int ^{1}_{0} e^{ - \rho \tau } \frac{ \partial }{\partial \rho } \bigl\vert y (x, \rho , t ) \bigr\vert ^{ m (x) } \,d\rho \,dx \\ = & - \int _{\Omega } e^{ - \tau } \bigl\vert y (x, 1, t ) \bigr\vert ^{ m (x) } \,dx + \int _{\Omega } \bigl\vert y (x, 0 , t ) \bigr\vert ^{ m (x) } \,dx \\ &{}- \tau \int _{\Omega } \int ^{1}_{0} e^{ - \rho \tau } \bigl\vert y (x, \rho , t ) \bigr\vert ^{ m (x) } \,d\rho \,dx \\ \leq & \int _{\Omega } \vert u_{t} \vert ^{ m (x) } \,dx - \tau e^{ - \tau } \int _{\Omega } \int ^{1}_{0} \bigl\vert y (x, \rho , t ) \bigr\vert ^{ m (x) } \,d\rho \,dx . \end{aligned}$$
□
Lemma 4.4
Let \({ g(t) = \int ^{\infty }_{t} k(s) \,ds } \). The following function
$$ \Lambda (t) = \int ^{t}_{0} g(t-s) \bigl\Vert \Delta u (s) \bigr\Vert _{2}^{2} \,ds $$
satisfies
$$\begin{aligned} \Lambda '(t) \leq 2 (1- k_{l} ) \Vert \Delta u \Vert _{2}^{2} - \frac{1}{2}(k \Box \Delta u) . \end{aligned}$$
(4.3)
Proof
Noting \(g'(t) = - k (t) \) and using Young’s inequality, we see
$$\begin{aligned} \Lambda '(t) = & g(0) \Vert \Delta u \Vert _{2}^{2} - \int ^{t}_{0} k(t-s) \bigl\Vert \Delta u (s) \bigr\Vert _{2}^{2} \,ds \\ = & \biggl( \int ^{\infty }_{0} k(s) \,ds \biggr) \Vert \Delta u \Vert _{2}^{2} - ( k \Box \Delta u) - \int ^{t}_{0} k(t-s) \bigl\Vert \Delta u (t) \bigr\Vert _{2}^{2} \,ds \\ & {}- 2 \int _{\Omega } \int ^{t}_{0} k(t-s) \Delta u(t) \bigl( \Delta u(s) - \Delta u(t) \bigr) \,ds \,dx \\ \leq & \biggl( \int ^{\infty }_{t} k(s) \,ds \biggr) \Vert \Delta u \Vert _{2}^{2} - \frac{1}{2} ( k \Box \Delta u) + 2 \biggl( \int ^{t}_{0} k (s) \,ds \biggr) \Vert \Delta u \Vert _{2}^{2} \\ \leq & 2 ( 1 - k_{l} ) \Vert \Delta u \Vert _{2}^{2} - \frac{1}{2} ( k \Box \Delta u) . \end{aligned}$$
□
From here, c and \(C_{i}\) denote generic constants, \(c_{\delta } > 0 \) denotes a generic constant depending on \(\delta >0 \), and \(c_{\delta } (x) = \frac{ m (x) -1}{ \delta ^{ \frac{1}{ m (x) -1 } } ( m (x))^{ \frac{ m (x)}{ m (x) -1 } } } \). We note that \(c_{\delta } (x) \) is bounded on Ω for fixed \(\delta >0 \), that is, \(| c_{\delta } (x) | \leq c_{\delta } \) for all \(x \in \Omega \).
General decay for the case \({m_{1} \geq 2 } \)
In this subsection, we derive a general decay result for the case \(m_{1} \geq 2\).
Lemma 4.5
Assume that \((A_{1})\), \((A_{2})\), \((A_{3})\), and (3.12) hold. If \(m_{1} \geq 2 \), then Φ satisfies
$$\begin{aligned} \Phi '(t) \leq & \Vert u_{t} \Vert _{2}^{2} - \frac{ k_{l} }{4} \Vert \Delta u \Vert _{2}^{2} - \bigl\Vert \Delta \chi (u) \bigr\Vert _{2}^{2} + \gamma \int _{\Omega } \vert u \vert ^{ p (x)} \,dx \\ &{} + \frac{ {\mathcal{C}}_{\eta } }{ 2 k_{l} } ( k_{\eta } \Box \Delta u ) + \alpha C_{1} \int _{\Omega } \vert u_{t} \vert ^{ m (x) } \,dx + \vert \beta \vert C_{1} \int _{\Omega } \bigl\vert y (x, 1, t ) \bigr\vert ^{ m (x)} \,dx . \end{aligned}$$
(4.4)
Proof
Using (2.7)–(2.12), we get
$$\begin{aligned} \Phi '(t) = & \Vert u_{t} \Vert _{2}^{2} - \biggl( 1- \int ^{t}_{0} k(s) \,ds \biggr) \Vert \Delta u \Vert _{2}^{2} - \bigl\Vert \Delta \chi (u) \bigr\Vert _{2}^{2} + \gamma \int _{\Omega } \vert u \vert ^{ p (x)} \,dx \\ & {}+ \int ^{t}_{0} k(t-s) \int _{\Omega } \bigl( \Delta u(s) - \Delta u(t) \bigr) \Delta u (t) \,dx \,ds \\ &{} - \alpha \int _{\Omega } u \vert u_{t} \vert ^{ m (x) -2 } u_{t} \,dx -\beta \int _{\Omega } u \bigl\vert y (x, 1, t) \bigr\vert ^{ m (x) -2 } y (x, 1, t) \,dx . \end{aligned}$$
(4.5)
Using Young’s inequality and (4.1), we have
$$\begin{aligned}& \int ^{t}_{0} k(t-s) \int _{\Omega } \bigl( \Delta u(s) - \Delta u(t) \bigr) \Delta u (t) \,dx \,ds \\& \quad \leq \frac{ k_{l} }{2} \Vert \Delta u \Vert _{2}^{2} + \frac{ 1 }{ 2 k_{l} } \int _{\Omega } \biggl( \int ^{t}_{0} k ( t-s ) \bigl( \Delta u(s) - \Delta u(t) \bigr) \,ds \biggr)^{2} \,dx \\& \quad \leq \frac{ k_{l} }{2} \Vert \Delta u \Vert _{2}^{2} + \frac{ {\mathcal{C}}_{\eta } }{ 2 k_{l} } ( k_{\eta } \Box \Delta u ) . \end{aligned}$$
(4.6)
From (4.5) and (4.6), one sees
$$\begin{aligned} \Phi '(t) \leq & \Vert u_{t} \Vert _{2}^{2} - \frac{ k_{l} }{2} \Vert \Delta u \Vert _{2}^{2} - \bigl\Vert \Delta \chi (u) \bigr\Vert _{2}^{2} + \gamma \int _{\Omega } \vert u \vert ^{ p (x)} \,dx + \frac{ {\mathcal{C}}_{\eta } }{ 2 k_{l}} ( k_{\eta } \Box \Delta u ) \\ & {}- \underbrace{ \alpha \int _{\Omega } u \vert u_{t} \vert ^{ m (x) -2 } u_{t} \,dx }_{I_{1}} - \underbrace{ \beta \int _{\Omega } u \bigl\vert y (x, 1, t) \bigr\vert ^{ m (x) -2 } y (x, 1, t) \,dx }_{I_{2}} . \end{aligned}$$
(4.7)
Using Young’s inequality with \(\frac{1}{ m (x) } + \frac{ m (x) -1 }{ m (x) } =1 \), (3.1), and (3.17), we find
$$\begin{aligned} - I_{1} \leq & \alpha \delta _{1} \int _{\Omega } \vert u \vert ^{ m (x) } \,dx + \alpha \int _{\Omega } c_{\delta _{1}} (x) \vert u_{t} \vert ^{ m (x) } \,dx \\ \leq & \alpha \delta _{1} C_{E(0)} \Vert \Delta u \Vert _{2}^{2} + \alpha \int _{\Omega } c_{\delta _{1} }(x) \vert u_{t} \vert ^{ m (x) } \,dx \end{aligned}$$
(4.8)
for \(\delta _{1} >0 \), where \(C_{E(0)} = B_{ m_{1} }^{ m_{1} } ( \frac{ 2 p_{1} E(0) }{ k_{l} ( p_{1} -2 ) } )^{ \frac{ m_{1} -2 }{2} } + B_{ m_{2}}^{ m_{2} } ( \frac{ 2 p_{1} E(0) }{ k_{l} ( p_{1} -2 ) } )^{ \frac{ m_{2} -2 }{2} } \).
Similarly, using (2.5), we have
$$\begin{aligned} - I_{2} \leq & \vert \beta \vert \delta _{1} C_{E(0)} \Vert \Delta u \Vert _{2}^{2} + \vert \beta \vert \int _{\Omega } c_{\delta _{1} }(x) \bigl\vert y (x,1, t) \bigr\vert ^{ m (x) } \,dx \\ \leq & \alpha \delta _{1} C_{E(0)} \Vert \Delta u \Vert _{2}^{2} + \vert \beta \vert \int _{\Omega } c_{\delta _{1} }(x) \bigl\vert y (x,1, t) \bigr\vert ^{ m (x) } \,dx . \end{aligned}$$
(4.9)
Combining (4.7), (4.8), (4.9) and taking \(\delta _{1} = \frac{ k_{l} }{ 8 \alpha C_{E(0)}} \), we obtain (4.4). □
Lemma 4.6
Assume that \((A_{1})\), \((A_{2})\), \((A_{3})\), and (3.12) hold. If \(m_{1} \geq 2 \), then Ψ satisfies
$$\begin{aligned} \Psi '(t) \leq & - \biggl( \int ^{t}_{0} k(s) \,ds - \delta \biggr) \Vert u_{t} \Vert _{2}^{2} + \delta C_{3} \Vert \Delta u \Vert _{2}^{2} + \biggl( \frac{ C_{4} ( 1+ {\mathcal{C}}_{\eta }) }{ \delta } + {\mathcal{C}}_{ \eta } \biggr) (k_{\eta } \Box \Delta u) \\ &{} + \delta C_{5} ( k \Box \Delta u ) + \alpha \int _{\Omega } c_{ \delta }(x) \vert u_{t} \vert ^{ m (x) } \,dx + \vert \beta \vert \int _{\Omega } c_{ \delta }(x) \bigl\vert y (x,1, t) \bigr\vert ^{ m (x) } \,dx \end{aligned}$$
(4.10)
for any \(\delta >0 \).
Proof
Using (2.7)–(2.12), we have
$$\begin{aligned} \Psi '(t) = & - \biggl( \int ^{t}_{0} k(s) \,ds \biggr) \Vert u_{t} \Vert _{2}^{2} - \int _{\Omega } u_{t} \int ^{t}_{0} k'(t-s) \bigl( u(t) - u(s) \bigr) \,ds \,dx \\ & {}+ \biggl( 1 - \int ^{t}_{0} k(s) \,ds \biggr) \int _{\Omega } \Delta u \int ^{t}_{0} k(t-s) \bigl( \Delta u(t) - \Delta u(s) \bigr) \,ds \,dx \\ & {}+ \int _{\Omega } \biggl( \int ^{t}_{0} k(t-s) \bigl( \Delta u(t) - \Delta u(s) \bigr) \,ds \biggr)^{2} \,dx \\ & {}- \int _{\Omega } \bigl[ u, \chi (u)\bigr] \int ^{t}_{0} k(t-s) \bigl( u(t) - u(s) \bigr) \,ds \,dx \\ & {}- \gamma \int _{\Omega } \vert u \vert ^{ p (x) -2} u \int ^{t}_{0} k(t-s) \bigl( u(t) - u(s) \bigr) \,ds \,dx \\ &{} + \alpha \int _{\Omega } \vert u_{t} \vert ^{ m (x) -2} u_{t} \int ^{t}_{0} k(t-s) \bigl( u(t) - u(s) \bigr) \,ds \,dx \\ &{} + \beta \int _{\Omega } \bigl\vert y (x,1,t) \bigr\vert ^{ m (x) -2} y (x,1,t) \int ^{t}_{0} k(t-s) \bigl( u(t) - u(s) \bigr) \,ds \,dx \\ : = & - \biggl( \int ^{t}_{0} k(s) \,ds \biggr) \Vert u_{t} \Vert _{2}^{2} + \sum _{i =1}^{5} J_{i} \\ & {}+ \underbrace{ \alpha \int _{\Omega } \vert u_{t} \vert ^{ m (x) -2} u_{t} \int ^{t}_{0} k(t-s) \bigl( u(t) - u(s) \bigr) \,ds \,dx }_{J_{6}} \\ &{} + \underbrace{ \beta \int _{\Omega } \bigl\vert y (x,1,t) \bigr\vert ^{ m (x) -2} y (x,1,t) \int ^{t}_{0} k(t-s) \bigl( u(t) - u(s) \bigr) \,ds \,dx }_{J_{7}} . \end{aligned}$$
(4.11)
Using Young’s inequality, \(k ' = \eta k - k_{\eta } \), (4.1) we get
$$\begin{aligned} \vert J_{1} \vert \leq & \delta \Vert u_{t} \Vert _{2}^{2} + \frac{1}{4 \delta } \biggl\Vert \int ^{t}_{0} k'(t-s) \bigl( u(t) - u(s) \bigr) \,ds \biggr\Vert _{2}^{2} \\ \leq &\delta \Vert u_{t} \Vert _{2}^{2} + \frac{1}{2 \delta } \biggl( \biggl\Vert \int ^{t}_{0} k_{\eta }(t-s) \bigl( u(t) - u(s) \bigr) \,ds \biggr\Vert _{2}^{2} + \eta ^{2} \biggl\Vert \int ^{t}_{0} k(t-s) \bigl( u(t) - u(s) \bigr) \,ds \biggr\Vert _{2}^{2} \biggr) \\ \leq& \delta \Vert u_{t} \Vert _{2}^{2} + \frac{1}{2 \delta } \biggl\{ \biggl( \int ^{t}_{0} k_{\eta }(s) \,ds \biggr) ( k_{\eta } \Box u) + \eta ^{2} { \mathcal{C}}_{\eta } (k_{\eta } \Box u) \biggr\} \\ \leq &\delta \Vert u_{t} \Vert _{2}^{2} + \frac{ B_{2}^{2} ( \eta (1- k_{l} ) + k(0) ) }{2 \delta } (k_{\eta } \Box \Delta u) + \frac{ B_{2}^{2} {\mathcal{C}}_{\eta } }{ 2 \delta } (k_{ \eta } \Box \Delta u ) , \end{aligned}$$
(4.12)
$$\begin{aligned} \vert J_{2} \vert \leq \delta \Vert \Delta u \Vert _{2}^{2} + \frac{ {\mathcal{C}}_{\eta } }{ 4 \delta } (k_{\eta } \Box \Delta u ) , \end{aligned}$$
(4.13)
and
$$\begin{aligned} \vert J_{3} \vert \leq {\mathcal{C}}_{\eta } (k_{\eta } \Box \Delta u ) . \end{aligned}$$
(4.14)
Using (3.17), we infer
$$\begin{aligned} \vert J_{4} \vert \leq & c \Vert u \Vert _{H^{2}(\Omega )} \bigl\Vert \chi ( u ) \bigr\Vert _{W^{2, \infty } (\Omega )} \biggl\Vert \int ^{t}_{0} k(t-s) \bigl( u(t) - u(s) \bigr) \,ds \biggr\Vert _{2} \\ \leq & c B_{2} \Vert \Delta u \Vert _{2}^{3} \bigl( {\mathcal{C}}_{\eta } ( k_{ \eta } \Box \Delta u ) \bigr)^{ \frac{1}{2} } \\ \leq & c B_{2} \biggl( \frac{ 2 p_{1} E(0) }{ k_{l} ( p_{1} -2 ) } \biggr) \Vert \Delta u \Vert _{2} \bigl( {\mathcal{C}}_{\eta } ( k_{\eta } \Box \Delta u ) \bigr)^{ \frac{1}{2} } \\ \leq & \delta \Vert \Delta u \Vert _{2}^{2} + \frac{ c }{ 4 \delta } { \mathcal{C}}_{\eta } ( k_{\eta } \Box \Delta u ) , \end{aligned}$$
(4.15)
here we used the Karman bracket property (see p. 270 in [10])
$$ \big\| [ v_{1} , v_{2} ] \big\| \leq c \| v_{1} \|_{H^{2}(\Omega)} \| v_{2} \|_{W^{2, \infty} (\Omega)} \quad \text{for } v_{1} \in H^{2}(\Omega), v_{2} \in W^{2, \infty } (\Omega) $$
and
$$ \big\| \chi (u) \big\| _{W^{2, \infty} (\Omega)} \leq c \| u \|_{H^{2}(\Omega)}^{2} . $$
Using (3.1), (3.17), (4.1), we deduce
$$\begin{aligned} \vert J_{5} \vert \leq & \gamma \delta \int _{\Omega } \vert u \vert ^{2 ( p (x) -1) } \,dx + \frac{\gamma }{ 4 \delta } \int _{\Omega } \biggl( \int ^{t}_{0} k(t-s) \bigl( u(t) - u(s) \bigr) \,ds \biggr)^{2} \,dx \\ \leq & \gamma \delta {\overline{C}}_{E(0)} \Vert \Delta u \Vert _{2}^{2} + \frac{\gamma B_{2}^{2} {\mathcal{C}}_{\eta } }{ 4 \delta } ( k_{\eta } \Box \Delta u ) ), \end{aligned}$$
(4.16)
where \({\overline{C}}_{E(0)} = B_{ 2( p_{1} -1) }^{ 2 ( p_{1} -1) } ( \frac{ 2 p_{1} E(0) }{ k_{l} ( p_{1} -2 ) } )^{ \frac{ 2 p_{1} - 4 }{2} } + B_{ 2( p_{2} -1 )}^{ 2 ( p_{2} -1) } ( \frac{ 2 p_{1} E(0) }{ k_{l} ( p_{1} -2 ) } )^{ \frac{ 2 p_{2} - 4 }{2} } \).
Using the similar calculation of (4.8) and Hölder’s inequality, we get
$$\begin{aligned} \vert J_{6} \vert \leq & \alpha \delta \int _{\Omega } \biggl\vert \int ^{t}_{0} k(t-s) \bigl( u(t) - u(s) \bigr) \,ds \biggr\vert ^{ m (x)} \,dx + \alpha \int _{\Omega } c_{ \delta }(x) \vert u_{t} \vert ^{ m (x) } \,dx \\ \leq & \alpha \delta \int _{\Omega } \biggl( \int ^{t}_{0} k (s) \,ds \biggr)^{ m (x) -1} \biggl( \int ^{t}_{0} k(t-s) \bigl\vert u(t) - u(s) \bigr\vert ^{ m (x)} \,ds \biggr) \,dx \\ &{} + \alpha \int _{\Omega } c_{\delta }(x) \vert u_{t} \vert ^{ m (x) } \,dx \\ \leq & \alpha \delta ( 1 - k_{l} )^{ m_{1} -1} \int _{\Omega } \biggl( \int ^{t}_{0} k(t-s) \bigl\vert u(t) - u(s) \bigr\vert ^{ m (x)} \,ds \biggr) \,dx + \alpha \int _{\Omega } c_{\delta }(x) \vert u_{t} \vert ^{ m (x) } \,dx \\ \leq & \alpha \delta ( 1 - k_{l} )^{ m_{1} -1} \hat{C}_{E(0)} \int ^{t}_{0} k(t-s) \bigl\Vert \Delta u(t) - \Delta u(s) \bigr\Vert _{2}^{2} \,ds + \alpha \int _{ \Omega } c_{\delta }(x) \vert u_{t} \vert ^{ m (x) } \,dx \\ = & \alpha \delta ( 1 - k_{l} )^{ m_{1} -1} \hat{C}_{E(0)} ( k \Box \Delta u) + \alpha \int _{\Omega } c_{\delta }(x) \vert u_{t} \vert ^{ m (x) } \,dx , \end{aligned}$$
(4.17)
where \(\hat{C}_{E(0)} = B_{ m_{1} }^{ m_{1} } ( \frac{ 8 p_{1} E(0) }{ k_{l} ( p_{1} -2 ) } )^{ \frac{ m_{1} -2 }{2} } + B_{ m_{2}}^{ m_{2} } ( \frac{ 8 p_{1} E(0) }{ k_{l} ( p_{1} -2 ) } )^{ \frac{ m_{2} -2 }{2} } \).
Similarly, we also have
$$\begin{aligned} \vert J_{7} \vert \leq \vert \beta \vert \delta ( 1 - k_{l} )^{ m_{1} -1} \hat{C}_{E(0)} ( k \Box \Delta u) + \vert \beta \vert \int _{\Omega } c_{\delta }(x) \bigl\vert y ( x, 1, t) \bigr\vert ^{ m (x) } \,dx . \end{aligned}$$
(4.18)
Applying the estimates of \(J_{i}\) to (4.11), we obtain (4.10). □
Lemma 4.7
Assume that \((A_{1})\), \((A_{2})\), \((A_{3})\), and (3.12) hold. Moreover, we assume that
$$\begin{aligned} B_{ p_{1} }^{ p_{1} } \biggl( \frac{ 2 p_{1} E(0) }{ k_{l} ( p_{1} -2 ) } \biggr)^{ \frac{ p_{1} -2 }{2} } + B_{ p_{2}}^{ p_{2} } \biggl( \frac{ 2 p_{1} E(0) }{ k_{l} ( p_{1} -2 ) } \biggr)^{ \frac{ p_{2} -2 }{2} } < \frac{ k_{l} }{4 \gamma }. \end{aligned}$$
(4.19)
If \(m_{1} \geq 2 \), then there exists \(\lambda >0 \) such that
$$\begin{aligned} L'(t) \leq - \lambda E(t) - 3 (1- k_{l} ) \Vert \Delta u \Vert _{2}^{2} + \frac{1}{2} ( k \Box \Delta u )\quad \textit{for } t \geq k^{-1}( \varepsilon ). \end{aligned}$$
(4.20)
Proof
From \((A_{3}) \), there exists \(t_{ \varepsilon } > 0 \) with \(k( t_{ \varepsilon } ) = \varepsilon \), that is, \(t_{ \varepsilon } = k^{-1}( \varepsilon ) \). We put \(\int _{0}^{ t_{ \varepsilon } } k(s) \,ds = k_{ \varepsilon } \). Using (3.8), (4.4), (4.10), (4.2), and \(k' = \eta k -k_{\eta } \), we have
$$\begin{aligned} L'(t) \leq & - \biggl\{ N_{2} \biggl( \int _{0}^{t} k(s) \,ds - \delta \biggr) - N_{1} \biggr\} \Vert u_{t} \Vert _{2}^{2} - \biggl\{ \frac{ N_{1} k_{l} }{4} - N_{2} \delta C_{3} \biggr\} \Vert \Delta u \Vert _{2}^{2} \\ &{} - N_{1} \bigl\Vert \Delta \chi (u) \bigr\Vert _{2}^{2} + N_{1} \gamma \int _{\Omega } \vert u \vert ^{ p (x)} \,dx + \biggl( \frac{ N \eta }{2} + N_{2} \delta C_{5} \biggr) ( k \Box \Delta u ) \\ & {}- \biggl\{ \frac{ N }{4} - \frac{ N_{2} C_{4}}{ \delta } + \frac{ N }{4} - {\mathcal{C}}_{\eta } \biggl( \frac{ N_{1} }{ 2 k_{l} } + \frac{ N_{2} C_{4} }{ \delta } + N_{2} \biggr) \biggr\} ( k_{\eta } \Box \Delta u ) \\ & {}- \int _{\Omega } \bigl( N C_{0} - N_{1} \alpha C_{1} - N_{2} \alpha c_{\delta }(x) - 1 \bigr) \vert u_{t} \vert ^{ m (x)} \,dx \\ &{} - \int _{\Omega } \bigl( N C_{0} - N_{1} \vert \beta \vert C_{1} - N_{2} \vert \beta \vert c_{\delta }(x) \bigr) \bigl\vert y ( x, 1, t) \bigr\vert ^{ m (x)} \,dx \\ &{} - \tau e^{ - \tau } \int _{\Omega } \int ^{1}_{0} \bigl\vert y (x, \rho , t ) \bigr\vert ^{ m (x) } \,d\rho \,dx \\ \leq & - \lambda E(t) - \biggl\{ N_{2} ( k_{ \varepsilon } - \delta ) - N_{1} - \frac{ \lambda }{2} \biggr\} \Vert u_{t} \Vert _{2}^{2} \\ &{} - \biggl\{ \frac{ N_{1} k_{l} }{4} - N_{2} \delta C_{3} - \frac{ \lambda }{2} \biggl( 1- \int ^{t}_{0} k(s) \,ds \biggr) \biggr\} \Vert \Delta u \Vert _{2}^{2} - \biggl( N_{1} - \frac{ \lambda }{4} \biggr) \bigl\Vert \Delta \chi (u) \bigr\Vert _{2}^{2} \\ &{} + \gamma \biggl( N_{1} - \frac{ \lambda }{ p_{2} } \biggr) \int _{ \Omega } \vert u \vert ^{ p (x)} \,dx + \biggl( \frac{ N \eta }{2} + N_{2} \delta C_{5} + \frac{ \lambda }{2} \biggr) ( k \Box \Delta u ) \\ &{} - \biggl\{ \frac{ N }{4} - \frac{ N_{2} C_{4}}{ \delta } + \frac{ N }{4} - {\mathcal{C}}_{\eta } \biggl( \frac{ N_{1} }{ 2 k_{l} } + \frac{ N_{2} C_{4} }{ \delta } + N_{2} \biggr) \biggr\} ( k_{\eta } \Box \Delta u ) \\ &{} - \int _{\Omega } \bigl( N C_{0} - N_{1} \alpha C_{1} - N_{2} \alpha c_{\delta }(x) - 1 \bigr) \vert u_{t} \vert ^{ m (x)} \,dx \\ &{} - \int _{\Omega } \bigl( N C_{0} - N_{1} \vert \beta \vert C_{1} - N_{2} \vert \beta \vert c_{\delta }(x) \bigr) \bigl\vert y ( x, 1, t) \bigr\vert ^{ m (x)} \,dx \\ &{} - \biggl( \tau e^{ - \tau } - \frac{ \lambda \xi \tau }{2} \biggr) \int _{\Omega } \int ^{1}_{0} \bigl\vert y (x, \rho , t ) \bigr\vert ^{ m (x) } \,d\rho \,dx \end{aligned}$$
(4.21)
for \(\lambda >0 \) and \(t \geq t_{ \varepsilon } \). Using estimate (3.16) and taking \(\delta = \frac{ k_{l} }{ 4 N_{2} C_{5}} \), we get
$$\begin{aligned} L'(t) \leq & - \lambda E(t) - \biggl\{ N_{2} k_{ \varepsilon } - \frac{ k_{l} }{ 4 C_{5}} - N_{1} - \frac{ \lambda }{2} \biggr\} \Vert u_{t} \Vert _{2}^{2} \\ & {}- \biggl[ N_{1} \biggl\{ \frac{ k_{l} }{4} - \gamma \biggl( B_{ p_{1} }^{ p_{1} } \biggl( \frac{ 2 p_{1} E(0) }{ k_{l} ( p_{1} -2 ) } \biggr)^{ \frac{ p_{1} -2 }{2} } + B_{ p_{2}}^{ p_{2} } \biggl( \frac{ 2 p_{1} E(0) }{ k_{l} ( p_{1} -2 ) } \biggr)^{ \frac{ p_{2} -2 }{2} } \biggr) \biggr\} \\ &{} - \frac{ k_{l} C_{3} }{ 4 C_{5}} - \frac{ \lambda }{2} \biggr] \Vert \Delta u \Vert _{2}^{2} \\ &{} - \biggl( N_{1} - \frac{ \lambda }{4} \biggr) \bigl\Vert \Delta \chi (u) \bigr\Vert _{2}^{2} + \biggl( \frac{ N \eta }{2} + \frac{ k_{l} }{ 4 } + \frac{ \lambda }{2} \biggr) ( k \Box \Delta u ) \\ &{} - \biggl\{ \frac{ N }{4} - \frac{ 4 N_{2}^{2} C_{4} C_{5}}{ k_{l} } + \frac{ N }{4} - {\mathcal{C}}_{\eta } \biggl( \frac{ N_{1} }{ 2 k_{l} } + \frac{ 4 N_{2}^{2} C_{4} C_{5} }{ k_{l} } + N_{2} \biggr) \biggr\} ( k_{ \eta } \Box \Delta u ) \\ &{} - ( N C_{0} - N_{1} \alpha C_{1} - N_{2} \alpha c_{\delta } - 1 ) \int _{\Omega } \vert u_{t} \vert ^{ m (x)} \,dx \\ &{} - \bigl( N C_{0} - N_{1} \vert \beta \vert C_{1} - N_{2} \vert \beta \vert c_{ \delta } \bigr) \int _{\Omega } \bigl\vert y ( x, 1, t) \bigr\vert ^{ m (x)} \,dx \\ &{} - \biggl( \tau e^{ - \tau } - \frac{ \lambda \xi \tau }{2} \biggr) \int _{\Omega } \int ^{1}_{0} \bigl\vert y (x, \rho , t ) \bigr\vert ^{ m (x) } \,d\rho \,dx \quad \text{for } t \geq t_{ \varepsilon } . \end{aligned}$$
(4.22)
From (4.19), we know
$$ \frac{ k_{l} }{4} - \gamma \biggl( B_{ p_{1} }^{ p_{1} } \biggl( \frac{ 2 p_{1} E(0) }{ k_{l} ( p_{1} -2 ) } \biggr)^{ \frac{ p_{1} -2 }{2} } + B_{ p_{2}}^{ p_{2} } \biggl( \frac{ 2 p_{1} E(0) }{ k_{l} ( p_{1} -2 ) } \biggr)^{ \frac{ p_{2} -2 }{2} } \biggr) > 0 .$$
Firstly, we take \(N_{1} > \frac{\lambda }{4} \) large enough to get
$$\begin{aligned}& N_{1} \biggl\{ \frac{ k_{l} }{4} - \gamma \biggl( B_{ p_{1} }^{ p_{1} } \biggl( \frac{ 2 p_{1} E(0) }{ k_{l} ( p_{1} -2 ) } \biggr)^{ \frac{ p_{1} -2 }{2} } + B_{ p_{2}}^{ p_{2} } \biggl( \frac{ 2 p_{1} E(0) }{ k_{l} ( p_{1} -2 ) } \biggr)^{ \frac{ p_{2} -2 }{2} } \biggr) \biggr\} - \frac{ k_{l} C_{3} }{ 4 C_{5}} \\& \quad > 4 ( 1 - k_{l} ) , \end{aligned}$$
(4.23)
and then choose \(N_{2} >0 \) satisfying
$$\begin{aligned} N_{2} k_{ \varepsilon } - \frac{ k_{l} }{ 4 C_{5}} - N_{1} > 1 . \end{aligned}$$
(4.24)
Noting \(\frac{ \eta k^{2} (s) }{ k_{\eta } (s)} < k(s) \) and making use of the Lebesgue dominated convergence theorem, we have
$$ \lim_{\eta \to 0^{+}} \eta {\mathcal{C}}_{\eta } = \lim _{\eta \to 0^{+}} \int ^{\infty }_{0} \frac{\eta k^{2}(s)}{k_{\eta }(s)} \,ds =0. $$
Thus, there exists \(0 < \eta _{0} < 1\) satisfying
$$ \eta {\mathcal{C}}_{\eta } < \frac{1}{ 16 ( \frac{ N_{1} }{ 2 k_{l} } + \frac{ 4 N_{2}^{2} C_{4} C_{5} }{ k_{l} } + N_{2} ) } \quad \text{for } \eta < \eta _{0}. $$
(4.25)
Secondly, we take \(N > 0\) large enough again to get
$$\begin{aligned}& \frac{1}{ 4 N } < \eta _{0},\qquad \frac{ N }{4} - \frac{ 4 N_{2}^{2} C_{4} C_{5}}{ k_{l} } >0 , \end{aligned}$$
(4.26)
$$\begin{aligned}& N C_{0} - N_{1} \alpha C_{1} - N_{2} \alpha c_{\delta } - 1 >0, \end{aligned}$$
(4.27)
and
$$\begin{aligned} N C_{0} - N_{1} \vert \beta \vert C_{1} - N_{2} \vert \beta \vert c_{\delta } >0. \end{aligned}$$
(4.28)
Thirdly, selecting \(\eta = \frac{1}{ 4 N } < \eta _{0} \), we get
$$ \frac{ N \eta }{2} + \frac{ k_{l} }{ 4 } = \frac{1}{8} + \frac{ k_{l} }{ 4 } < \frac{3}{8} $$
(4.29)
and
$$ \frac{N}{4} - {\mathcal{C}}_{\eta } \biggl( \frac{ N_{1} }{ 2 k_{l} } + \frac{ 4 N_{2}^{2} C_{4} C_{5} }{ k_{l} } + N_{2} \biggr) > \frac{N}{4} - \frac{1}{ 16 \eta } =0 , $$
(4.30)
here we used (4.25). From (4.22), (4.23), (4.24), (4.26), (4.27), (4.28), (4.29), and (4.30), we get
$$\begin{aligned} L'(t) \leq & - \lambda E(t) - \biggl\{ 1 - \frac{ \lambda }{2} \biggr\} \Vert u_{t} \Vert _{2}^{2} - \biggl\{ 4 ( 1 - k_{l} ) - \frac{ \lambda }{2} \biggr\} \Vert \Delta u \Vert _{2}^{2} \\ & {}+ \biggl( \frac{ 3 }{ 8 } + \frac{ \lambda }{2} \biggr) ( k \Box \Delta u ) - \biggl( \tau e^{ - \tau } - \frac{ \lambda \xi \tau }{2} \biggr) \int _{\Omega } \int ^{1}_{0} \bigl\vert y (x, \rho , t ) \bigr\vert ^{ m (x) } \,d\rho \,dx \end{aligned}$$
for \(t \geq t_{ \varepsilon } \). Finally, selecting \(\lambda >0 \) satisfying
$$ \lambda \leq \min \biggl\{ 2 ( 1 - k_{l} ) , \frac{1}{4}, \frac{ 2 \tau e^{- \tau }}{ \xi \tau } \biggr\} , $$
we obtain (4.20). □
Lemma 4.8
Assume that \((A_{1})\), \((A_{2})\), \((A_{3})\), and (4.19) hold and let \(m_{1} \geq 2 \). Then
$$\begin{aligned} 0 < \int ^{\infty }_{0} E(s) \,ds < \infty . \end{aligned}$$
(4.31)
Proof
From (4.20) and (4.3), we see
$$\begin{aligned} \bigl( L(t) + \Lambda (t) \bigr) ' \leq - \lambda E(t) \quad \text{for } t \geq t_{ \varepsilon }, \end{aligned}$$
(4.32)
and
$$\begin{aligned} 0< \int ^{t}_{t_{\varepsilon }} E(s) \,ds \leq - \frac{1}{ \lambda } \int ^{t}_{t_{\varepsilon }}\bigl( L'(s) + \Lambda '(s) \bigr) \,ds \leq \frac{ L(t_{\varepsilon }) + \Lambda (t_{\varepsilon }) }{ \lambda } < \infty ,\quad \forall t \geq t_{\varepsilon } , \end{aligned}$$
which gives
$$\begin{aligned} 0 < \int ^{\infty }_{0} E(s) \,ds = \int _{0}^{t_{\varepsilon }} E(s) \,ds + \int ^{\infty }_{t_{\varepsilon }} E(s) \,ds < \infty . \end{aligned}$$
□
Theorem 4.1
Assume that \((A_{1})\), \((A_{2})\), \((A_{3})\), and (4.19) hold and let \(m_{1} \geq 2 \). Then there exist \(c_{i} , \omega _{i} >0 \), \(i =1,2 \), such that, for \(t \geq k^{-1}(\varepsilon ) \),
$$\begin{aligned} E(t) \leq c_{1} \exp \biggl( - \omega _{1} \int ^{t}_{k^{-1}( \varepsilon )} \zeta (s) \,ds \biggr) \quad \textit{in the case }K \textit{ is linear } \end{aligned}$$
(4.33)
and
$$ E(t) \leq c_{2} \tilde{K}^{-1} \biggl( \omega _{2} \int ^{t}_{k^{-1}( \varepsilon )} \zeta (s) \,ds \biggr)\quad \textit{in the case } K \textit{ is nonlinear }, $$
where
$$ \tilde{K}(s) = \int ^{\varepsilon }_{s} \frac{1}{ \tau K'(\tau )} \,d \tau . $$
(4.34)
Proof
From Lemma 4.5, Lemma 4.6, and Lemma 4.7, the proof is similar to that of [23]. But, for the completeness, we give the proof. Since k and ζ are continuous in t, we have
$$ a_{1} \leq \zeta (t) K\bigl(k(t)\bigr) \leq a_{2} \quad \text{for } t \in [0, t_{ \varepsilon }] $$
for some \(a_{1}, a_{2} >0 \), and
$$ k'(t) \leq - \zeta (t) K\bigl(k(t)\bigr) \leq - a_{1} \leq -\frac{a_{1}}{k(0)} k(t) \quad \text{for } t \in [0, t_{\varepsilon }]. $$
(4.35)
From (4.20), (4.35), (3.8), we get
$$\begin{aligned} L'(t) \leq & - \lambda E(t) - \frac{ k(0) }{2 a_{1}} \bigl( k' \Box \Delta u \bigr) + \frac{1}{2} \int _{t_{\varepsilon }}^{t} k (s) \bigl\Vert \Delta u (t) - \Delta u (t-s) \bigr\Vert _{2}^{2} \,ds \\ \leq & - \lambda E(t) - \frac{ k(0) }{ a_{1}} E'(t) + \frac{1}{2} \int _{t_{\varepsilon }}^{t} k (s) \bigl\Vert \Delta u (t) - \Delta u (t-s) \bigr\Vert _{2}^{2} \,ds \quad \text{for } t \geq t_{ \varepsilon }. \end{aligned}$$
(4.36)
Let
$$ R (t) = L(t) + \frac{ k(0)}{ a_{1}} E(t) , $$
then \(R \sim E \) and
$$\begin{aligned} R'(t) \leq - \lambda E(t) + \frac{1}{2} \int _{t_{\varepsilon }}^{t} k (s) \bigl\Vert \Delta u (t) - \Delta u (t-s) \bigr\Vert _{2}^{2} \,ds \quad \text{for } t \geq t_{ \varepsilon } . \end{aligned}$$
(4.37)
Case 1: K is linear, that is, \(K (s) = a s \) for some \(a > 0\). Put
$$ {\mathcal{R}}_{1} (t) = \zeta (t) R(t) + \frac{1}{ a } E(t) . $$
From (4.37), (2.6), and (3.8), we have
$$\begin{aligned} {\mathcal{R}}_{1} ' (t) \leq & - \lambda \zeta (t) E(t) + \frac{1}{2} \int ^{t}_{t_{\varepsilon }} \zeta (s) k(s) \bigl\Vert \Delta u(t) - \Delta u(t-s) \bigr\Vert _{2}^{2} \,ds + \frac{1}{ a } E'(t) \\ \leq & - \lambda \zeta (t) E(t) - \frac{1}{2 a } \int ^{t}_{t_{ \varepsilon }} k'(s) \bigl\Vert \Delta u(t) - \Delta u(t-s) \bigr\Vert ^{2} \,ds + \frac{1}{ a } E'(t) \\ \leq & - \lambda \zeta (t) E(t), \quad t \geq t_{\varepsilon } . \end{aligned}$$
(4.38)
This and the relation \({\mathcal{R}}_{1} (t) \sim E(t) \) prove (4.33).
Case 2: K is nonlinear. For \(t \geq t_{\varepsilon } \), we put
$$\begin{aligned} \Gamma _{1} (t) = a_{3} \int ^{t}_{t_{\varepsilon }} \bigl\Vert \Delta u(t) - \Delta u(t-s) \bigr\Vert _{2}^{2} \,ds \end{aligned}$$
and
$$\begin{aligned} \Gamma _{2} (t) = - \int ^{t}_{t_{\varepsilon }} k'(s) \bigl\Vert \Delta u(t) - \Delta u(t-s) \bigr\Vert _{2}^{2} \,ds. \end{aligned}$$
From (3.17), (3.8), (4.31), we get
$$\begin{aligned} \int ^{t}_{t_{\varepsilon }} \bigl\Vert \Delta u(t) - \Delta u(t-s) \bigr\Vert _{2}^{2} \,ds \leq & 2 \int ^{t}_{t_{\varepsilon }} \bigl\Vert \Delta u(t) \bigr\Vert _{2}^{2} + \bigl\Vert \Delta u(t-s) \bigr\Vert _{2}^{2} \,ds \\ \leq & \frac{ 4 p_{1} }{ k_{l} ( p_{1} -2 )} \int ^{t}_{t_{ \varepsilon }} \bigl( E(t) + E(t-s) \bigr) \,ds \\ \leq & \frac{ 4 p_{1} }{ k_{l} ( p_{1} -2 )} \biggl( \int ^{t}_{t_{ \varepsilon }} E(s) \,ds + \int _{0}^{t - t_{\varepsilon }} E(s ) \biggr) \,ds ) < \infty . \end{aligned}$$
(4.39)
Thus, there exists \(0 < a_{3} < 1 \) satisfying
$$ \Gamma _{1} (t) < 1 \quad \text{for } t \geq t_{\varepsilon }. $$
(4.40)
From (3.8), we know
$$ \Gamma _{2} (t) \leq - \bigl(k' \Box \Delta u\bigr) (t) \leq - 2 E'(t). $$
(4.41)
Using \((A_{3})\), (4.40), the relation \(\overline{ K } ( \varrho t ) \leq \varrho \overline{ K } ( t )\) for \(0 \leq \varrho \leq 1 \) and \(t \in [0, \infty ) \), and Jensen’s inequality, we find
$$\begin{aligned} \Gamma _{2} (t) = & - \frac{1}{ a_{3} \Gamma _{1} (t)} \int ^{t}_{t_{ \varepsilon }} \Gamma _{1} (t) k'(s) a_{3} \bigl\Vert \Delta u(t) - \Delta u(t-s) \bigr\Vert _{2}^{2} \,ds \\ \geq & \frac{1}{ a_{3} \Gamma _{1} (t)} \int ^{t}_{t_{\varepsilon }} \Gamma _{1} (t) \zeta (s) K\bigl(k(s)\bigr) a_{3} \bigl\Vert \Delta u(t) - \Delta u(t-s) \bigr\Vert _{2}^{2} \,ds \\ \geq & \frac{\zeta (t)}{ a_{3} \Gamma _{1} (t)} \int ^{t}_{t_{ \varepsilon }} \overline{ K } \bigl( \Gamma _{1} (t) k(s) \bigr) a_{3} \bigl\Vert \Delta u(t) - \Delta u(t-s) \bigr\Vert _{2}^{2} \,ds \\ \geq & \frac{\zeta (t)}{ a_{3} } \overline{ K } \biggl( a_{3} \int ^{t}_{t_{ \varepsilon }} k(s) \bigl\Vert \Delta u(t) - \Delta u(t-s) \bigr\Vert _{2}^{2} \,ds \biggr), \quad t \geq t_{\varepsilon } . \end{aligned}$$
(4.42)
So, we have
$$\begin{aligned} \int ^{t}_{t_{\varepsilon }} k(s) \bigl\Vert \Delta u(t) - \Delta u(t-s) \bigr\Vert _{2}^{2} \,ds \leq \frac{1}{a_{3}} \overline{K}^{-1} \biggl( \frac{ a_{3} \Gamma _{2} (t)}{ \zeta (t)} \biggr). \end{aligned}$$
Applying this to (4.37), we get
$$\begin{aligned} R '(t) \leq - \lambda E(t) + \frac{1}{2 a_{3} } \overline{K}^{-1} \biggl( \frac{ a_{3} \Gamma _{2}(t)}{ \zeta (t)} \biggr) \quad \text{for } t \geq t_{\varepsilon } . \end{aligned}$$
(4.43)
We know that the convex function K̅ satisfies
$$\begin{aligned} st \leq \overline{ K }^{*}(s) + \overline{ K }(t) \quad \text{for } s, t \geq 0 \end{aligned}$$
(4.44)
and
$$\begin{aligned} \overline{ K }^{*}(s) = s\bigl(\overline{ K }'\bigr)^{-1}(s) - \overline{ K }\bigl(\bigl( \overline{ K }'\bigr)^{-1}(s)\bigr) \quad \text{for } s \geq 0 , \end{aligned}$$
(4.45)
where \(\overline{ K }^{*}\) is the conjugate function of K̅.
Let \(0 < \mu < \min \{ \varepsilon , 2 a_{3} \lambda E(0) \} \) and \({\mathcal{E}}(t) = \frac{E(t)}{E(0)} \). Using \(\overline{K}'(s) >0\), \(\overline{K}''(s) >0 \), \(E'(t) \leq 0 \), \(\overline{K}(0) = \overline{K}'(0) =0\), (4.43), (4.44), and (4.45), we infer
$$\begin{aligned} \bigl( \overline{K}' \bigl( \mu { \mathcal{E}}(t) \bigr) R (t) \bigr)' \leq & - \lambda \overline{K}' \bigl( \mu {\mathcal{E}}(t) \bigr) E(t) + \frac{1}{2 a_{3} } \overline{K}' \bigl( \mu {\mathcal{E}}(t) \bigr) \overline{K}^{-1} \biggl( \frac{ a_{3} \Gamma _{2}(t)}{ \zeta (t)} \biggr) \\ \leq & - \lambda \overline{K}' \bigl( \mu {\mathcal{E}}(t) \bigr) E(t) + \frac{1}{2 a_{3}} \overline{K}^{*} \bigl( \overline{K}' \bigl( \mu { \mathcal{E}}(t) \bigr) \bigr) + \frac{ \Gamma _{2}(t)}{2 \zeta (t)} \\ \leq & - \lambda \overline{K}' \bigl( \mu {\mathcal{E}}(t) \bigr) E(t) + \frac{ \mu }{ 2 a_{3}} {\mathcal{E}}(t) \overline{K}' \bigl( \mu {\mathcal{E}}(t) \bigr) + \frac{ \Gamma _{2}(t)}{ 2 \zeta (t)} \\ = & - a_{4} {\mathcal{E}}(t) K' \bigl( \mu { \mathcal{E}}(t) \bigr) + \frac{ \Gamma _{2}(t)}{ 2 \zeta (t)} , \end{aligned}$$
(4.46)
where \(a_{4} = \lambda E(0) - \frac{ \mu }{ 2 a_{3}} > 0 \). Setting
$$\begin{aligned} {\mathcal{R}}_{2} (t)= \zeta (t) \overline{K}' \bigl( \mu {\mathcal{E}}(t) \bigr) R(t) + E(t) , \end{aligned}$$
from (4.46) and (4.41), we get
$$\begin{aligned} {\mathcal{R}}_{2}' (t) \leq - a_{4} \zeta (t) {\mathcal{E}}(t) K' \bigl( \mu { \mathcal{E}}(t) \bigr) + \frac{ \Gamma _{2}(t)}{2} + E'(t) \leq - a_{4} \zeta (t) K_{0} \bigl( {\mathcal{E}}(t) \bigr) \quad \text{for } t\geq t_{ \varepsilon } , \end{aligned}$$
(4.47)
where \(K_{0}(s) = s K'( \mu s) \). Since \({\mathcal{R}}_{2} (t) \sim E(t) \), there exist \(a_{5}, a_{6} >0\) satisfying
$$ a_{5} {\mathcal{R}}_{2} (t) \leq E(t) \leq a_{6} {\mathcal{R}}_{2} (t). $$
Finally, we let
$$ {\mathcal{L}} (t) = \frac{ a_{5} {\mathcal{R}}_{2} (t)}{E(0)} , $$
(4.48)
then
$$ {\mathcal{L}}(t) \leq {\mathcal{E}}(t) \leq 1 . $$
(4.49)
Since \(K_{0}\) is an increasing function on \((0,1] \), from (4.48), (4.47), and (4.49), we deduce
$$ {\mathcal{L}}'(t) \leq - \omega _{2} \zeta (t) K_{0} \bigl( {\mathcal{E}}(t) \bigr) \leq - \omega _{2} \zeta (t) K_{0} \bigl( { \mathcal{L}}(t) \bigr) \quad \text{for } t \geq t_{\varepsilon } , $$
where \(\omega _{2} = \frac{ a_{4} a_{5} }{ E(0)} \), and
$$\begin{aligned} \int ^{t}_{t_{\varepsilon }} \omega _{2} \zeta (s) \,ds \leq & - \int ^{t}_{t_{ \varepsilon }} \frac{{\mathcal{L}}'(s)}{ K_{0} ( {\mathcal{L}}(s) ) } \,ds = - \int ^{t}_{t_{\varepsilon }} \frac{{\mathcal{L}}'(s)}{ {\mathcal{L}}(s) K' ( \mu {\mathcal{L}}(s) ) } \,ds = \int ^{\mu {\mathcal{L}}(t_{\varepsilon })}_{\mu {\mathcal{L}}(t)} \frac{ 1 }{ s K' ( s ) } \,ds \\ \leq & \int ^{\varepsilon }_{\mu {\mathcal{L}}(t)} \frac{ 1 }{ s K' ( s ) } \,ds = \tilde{K}\bigl( \mu {\mathcal{L}}(t) \bigr), \end{aligned}$$
here K̃ is the function given in (4.34). Because K̃ is strictly decreasing on \((0, \varepsilon ]\), we obtain
$$\begin{aligned} {\mathcal{L}}(t) \leq \frac{1}{\mu } \tilde{K}^{-1} \biggl( \omega _{2} \int ^{t}_{t_{\varepsilon }} \zeta (s) \,ds \biggr) \quad \text{for } t\geq t_{ \varepsilon } . \end{aligned}$$
□
General decay for the case \({1 < m_{1} < 2 } \)
In this subsection, we derive a general decay result for the case \(1 < m_{1} < 2\). We let
$$\begin{aligned} \Omega _{1} = \bigl\{ x \in \Omega : m (x) < 2 \bigr\} , \qquad \Omega _{2} = \bigl\{ x \in \Omega : m (x) \geq 2 \bigr\} \end{aligned}$$
and
$$\begin{aligned} \Omega _{i}^{-} = \bigl\{ x \in \Omega _{i} : \bigl\vert u_{t} (x,t) \bigr\vert < 1 \bigr\} , \qquad \Omega _{i}^{+} = \bigl\{ x \in \Omega _{i} : \bigl\vert u_{t} (x,t) \bigr\vert \geq 1 \bigr\} \end{aligned}$$
for \(i =1,2 \).
Lemma 4.9
Assume that \((A_{1})\), \((A_{2})\), \((A_{3})\), and (3.12) hold. If \(1 < m_{1} < 2 \), then Φ satisfies
$$\begin{aligned} \Phi '(t) \leq & \Vert u_{t} \Vert _{2}^{2} - \frac{ k_{l} }{4} \Vert \Delta u \Vert _{2}^{2} - \bigl\Vert \Delta \chi (u) \bigr\Vert _{2}^{2} + \gamma \int _{\Omega } \vert u \vert ^{ p (x)} \,dx + \frac{ {\mathcal{C}}_{\eta } }{ 2 k_{l} } ( k_{\eta } \Box \Delta u ) \\ & {}+ \alpha C_{6} \biggl\{ \int _{\Omega } \vert u_{t} \vert ^{ m (x) } \,dx + \biggl( \int _{\Omega } \vert u_{t} \vert ^{ m (x) } \,dx \biggr)^{ m_{1} -1 } \biggr\} \\ &{} + \vert \beta \vert C_{6} \biggl\{ \int _{\Omega } \bigl\vert y (x, 1, t ) \bigr\vert ^{ m (x)} \,dx + \biggl( \int _{\Omega } \bigl\vert y (x, 1, t ) \bigr\vert ^{ m (x) } \,dx \biggr)^{ m_{1} -1 } \biggr\} . \end{aligned}$$
(4.50)
Proof
We re-estimate \(I_{1}\) and \(I_{2}\) in (4.7) for the case \(1 < m_{1} <2 \). Using Young’s inequality, for \(\delta _{2} >0 \), we have
$$\begin{aligned} - \alpha \int _{\Omega _{1}} u \vert u_{t} \vert ^{ m (x) -2 } u_{t} \,dx \leq & \alpha \delta _{2} \int _{\Omega _{1}} \vert u \vert ^{2} \,dx + \frac{ \alpha }{ 4 \delta _{2} } \int _{\Omega _{1}} \vert u_{t} \vert ^{2 m (x) -2 } \,dx . \end{aligned}$$
(4.51)
Noting \(2 m_{1} -2 < 2 m (x) -2 < m (x) < 2 \) for \(x \in \Omega _{1} \) and using Hölder’s inequality with \((2 - m_{1}) + ( m_{1} -1) =1 \), we get
$$\begin{aligned} \int _{\Omega _{1}} \vert u_{t} \vert ^{2 m (x) -2 } \,dx \leq & \int _{ \Omega _{1}^{-} } \vert u_{t} \vert ^{2 m_{1} -2 } \,dx + \int _{\Omega _{1}^{+} } \vert u_{t} \vert ^{ m (x) } \,dx \\ \leq & \bigl\vert \Omega _{1}^{-} \bigr\vert ^{ 2 - m_{1}} \biggl( \int _{\Omega _{1}^{-} } \vert u_{t} \vert ^{2} \,dx \biggr)^{ m_{1} -1} + \int _{\Omega _{1}^{+} } \vert u_{t} \vert ^{ m (x) } \,dx \\ \leq & \bigl\vert \Omega _{1}^{-} \bigr\vert ^{ 2 - m_{1}} \biggl( \int _{\Omega _{1}^{-} } \vert u_{t} \vert ^{ m (x) } \,dx \biggr)^{ m_{1} -1} + \int _{\Omega _{1}^{+} } \vert u_{t} \vert ^{ m (x) } \,dx \\ \leq & c \biggl\{ \biggl( \int _{\Omega } \vert u_{t} \vert ^{ m (x) } \,dx \biggr)^{ m_{1} -1} + \int _{\Omega } \vert u_{t} \vert ^{ m (x) } \,dx \biggr\} . \end{aligned}$$
(4.52)
Applying (4.52) to (4.51), we see
$$\begin{aligned}& - \alpha \int _{\Omega _{1}} u \vert u_{t} \vert ^{ m (x) -2 } u_{t} \,dx \\& \quad \leq \alpha \delta _{2} B_{2}^{2} \Vert \Delta u \Vert _{2}^{2} + \frac{ \alpha c }{ 4 \delta _{2} } \biggl\{ \biggl( \int _{\Omega } \vert u_{t} \vert ^{ m (x) } \,dx \biggr)^{ m_{1} -1} + \int _{\Omega } \vert u_{t} \vert ^{ m (x) } \,dx \biggr\} . \end{aligned}$$
(4.53)
As the estimates of (4.8), for \(\delta _{3} >0 \), we find
$$\begin{aligned} - \alpha \int _{\Omega _{2}} u \vert u_{t} \vert ^{ m (x) -2 } u_{t} \,dx \leq & \alpha \delta _{3} \int _{\Omega _{2}} \vert u \vert ^{ m (x) } \,dx + \alpha \int _{\Omega _{2}} c_{\delta _{3}} (x) \vert u_{t} \vert ^{ m (x) } \,dx \\ \leq & \alpha \delta _{3} \tilde{C}_{E(0)} \Vert \Delta u \Vert _{2}^{2} + \alpha \int _{\Omega } c_{\delta _{3} }(x) \vert u_{t} \vert ^{ m (x) } \,dx , \end{aligned}$$
(4.54)
where \(\tilde{C}_{E(0)} = B_{ m_{-} }^{ m_{-} } ( \frac{ 2 p_{1} E(0) }{ k_{l} ( p_{1} -2 ) } )^{ \frac{ m_{-} -2 }{2} } + B_{ m_{+}}^{ m_{+} } ( \frac{ 2 p_{1} E(0) }{ k_{l} ( p_{1} -2 ) } )^{ \frac{ m_{+} -2 }{2} } \), here
$$ m_{-} = \operatorname{ess} \inf_{x \in \Omega _{2}} m (x) \geq 2 \quad \text{and} \quad m_{+} = \operatorname{ess} \sup _{x \in \Omega _{2}} m (x) \geq 2 .$$
Combining (4.53) and (4.54) and taking \(\delta _{2} = \frac{ k_{l} }{ 16 \alpha B_{2}^{2}} \) and \(\delta _{3} = \frac{ k_{l} }{ 16 \alpha \tilde{C}_{E(0)}}\), we have
$$\begin{aligned} - I_{1} \leq & \frac{ k_{l} }{8} \Vert \Delta u \Vert _{2}^{2} + \alpha C_{6} \biggl\{ \biggl( \int _{\Omega } \vert u_{t} \vert ^{ m (x) } \,dx \biggr)^{ m_{1} -1} + \int _{\Omega } \vert u_{t} \vert ^{ m (x) } \,dx \biggr\} . \end{aligned}$$
(4.55)
Similarly, we have
$$\begin{aligned} - I_{2} \leq & \frac{ k_{l} }{8} \Vert \Delta u \Vert _{2}^{2} + \vert \beta \vert C_{6} \biggl\{ \biggl( \int _{\Omega } \bigl\vert y(x,1, t) \bigr\vert ^{ m (x) } \,dx \biggr)^{ m_{1} -1} + \int _{\Omega } \bigl\vert y(x,1, t) \bigr\vert ^{ m (x) } \,dx \biggr\} . \end{aligned}$$
(4.56)
Adapting (4.55) and (4.56) to (4.7), we obtain (4.50). □
Lemma 4.10
Assume that \((A_{1})\), \((A_{2})\), \((A_{3})\), and (3.12) hold. If \(1 < m_{1} < 2 \), then Ψ satisfies
$$\begin{aligned} \Psi '(t) \leq & - \biggl( \int ^{t}_{0} k(s) \,ds - \delta \biggr) \Vert u_{t} \Vert _{2}^{2} + \delta C_{3} \Vert \Delta u \Vert _{2}^{2} + \biggl( \frac{ C_{4} ( 1+ {\mathcal{C}}_{\eta }) }{ \delta } + {\mathcal{C}}_{ \eta } \biggr) (k_{\eta } \Box \Delta u) \\ & {}+ \delta C_{7} ( k \Box \Delta u ) + \frac{ \alpha C_{8} }{ \delta } \biggl\{ c_{\delta } \int _{\Omega } \vert u_{t} \vert ^{ m (x) } \,dx + \biggl( \int _{\Omega } \vert u_{t} \vert ^{ m (x) } \,dx \biggr)^{ m_{1} -1 } \biggr\} \\ & {}+ \frac{ \vert \beta \vert C_{8} }{ \delta } \biggl\{ c_{\delta } \int _{ \Omega } \bigl\vert y (x, 1, t ) \bigr\vert ^{ m (x)} \,dx + \biggl( \int _{\Omega } \bigl\vert y (x, 1, t ) \bigr\vert ^{ m (x) } \,dx \biggr)^{ m_{1} -1 } \biggr\} \end{aligned}$$
(4.57)
for any \(\delta >0 \).
Proof
We re-estimate \(J_{6}\) and \(J_{7}\) in (4.11) for the case \(1 < m_{1} <2 \). Let \(\delta >0 \). Using (4.52), we have
$$\begin{aligned}& \alpha \int _{\Omega _{1} } \vert u_{t} \vert ^{ m (x) -2} u_{t} \int ^{t}_{0} k(t-s) \bigl( u(t) - u(s) \bigr) \,ds \,dx \\& \quad \leq \alpha \delta \int _{\Omega _{1}} \biggl\vert \int ^{t}_{0} k(t-s) \bigl( u(t) - u(s) \bigr) \,ds \biggr\vert ^{2} \,dx + \frac{ \alpha }{ 4 \delta } \int _{ \Omega _{1}} \vert u_{t} \vert ^{ 2 m (x) -2 } \,dx \\& \quad \leq \alpha \delta ( 1 - k_{l} ) B_{2}^{2} ( k \Box \Delta u ) + \frac{ \alpha c }{ 4 \delta } \biggl\{ \biggl( \int _{\Omega } \vert u_{t} \vert ^{ m (x) } \,dx \biggr)^{ m_{1} -1} + \int _{\Omega } \vert u_{t} \vert ^{ m (x) } \,dx \biggr\} . \end{aligned}$$
(4.58)
Since \(m(x) \geq 2\) on \(\Omega _{2} \), we can apply the same argument of (4.17) on \(\Omega _{2}\) instead of Ω to obtain
$$\begin{aligned}& \alpha \int _{\Omega _{2} } \vert u_{t} \vert ^{ m (x) -2} u_{t} \int ^{t}_{0} k(t-s) \bigl( u(t) - u(s) \bigr) \,ds \,dx \\& \quad \leq \alpha \delta ( 1 - k_{l} )^{ m_{1} -1} \int _{\Omega _{2}} \biggl( \int ^{t}_{0} k(t-s) \bigl\vert u(t) - u(s) \bigr\vert ^{ m (x)} \,ds \biggr) \,dx + \alpha \int _{\Omega _{2}} c_{\delta }(x) \vert u_{t} \vert ^{ m (x) } \,dx \\& \quad \leq \alpha \delta ( 1 - k_{l} )^{ m_{1} -1} \hat{C}_{E(0)} ( k \Box \Delta u) + \frac{ \alpha }{\delta } \int _{\Omega } \delta c_{ \delta }(x) \vert u_{t} \vert ^{ m (x) } \,dx . \end{aligned}$$
(4.59)
Combining (4.58) and (4.59) and noting that \(\delta c_{\delta }(x) \) is bounded on Ω, we have
$$\begin{aligned} \vert J_{6} \vert \leq & \alpha \delta \bigl\{ ( 1 - k_{l} ) B_{2}^{2} + ( 1 - k_{l} )^{ m_{1} -1} \hat{C}_{E(0)} \bigr\} ( k \Box \Delta u ) \\ & {}+ \frac{ \alpha C_{8} }{ \delta } \biggl\{ \biggl( \int _{\Omega } \vert u_{t} \vert ^{ m (x) } \,dx \biggr)^{ m_{1} -1} + c_{\delta } \int _{\Omega } \vert u_{t} \vert ^{ m (x) } \,dx \biggr\} . \end{aligned}$$
(4.60)
Similarly, we find
$$\begin{aligned} \vert J_{7} \vert \leq & \vert \beta \vert \delta \bigl\{ ( 1 - k_{l} ) B_{2}^{2} + ( 1 - k_{l} )^{ m_{1} -1} \hat{C}_{E(0)} \bigr\} ( k \Box \Delta u ) \\ & {}+ \frac{ \vert \beta \vert C_{8} }{ \delta } \biggl\{ \biggl( \int _{\Omega } \bigl\vert y(x,1,t) \bigr\vert ^{ m (x) } \,dx \biggr)^{ m_{1} -1} + c_{\delta } \int _{\Omega } \bigl\vert y(x,1,t) \bigr\vert ^{ m (x) } \,dx \biggr\} . \end{aligned}$$
(4.61)
Substituting (4.12), (4.13), (4.14), (4.15), (4.16), (4.60), and (4.61) into (4.11), we obtain (4.57). □
Lemma 4.11
Assume that \((A_{1})\), \((A_{2})\), \((A_{3})\), and (4.19) hold. If \(1 < m_{1} < 2 \), there exists \(\lambda >0 \) such that
$$\begin{aligned} L'(t) \leq - \lambda E(t) - 3 (1- k_{l} ) \Vert \Delta u \Vert _{2}^{2} + \frac{1}{2} ( k \Box \Delta u ) + C_{9} \bigl( - E ' (t)\bigr)^{ m_{1} -1 } \quad \textit{for } t \geq k^{-1}( \varepsilon ). \end{aligned}$$
(4.62)
Proof
From (3.8), (4.2), (4.50), and (4.57), the proof is similar to that of (4.20) by replacing the constants \(C_{1} \), \(C_{5}\), and \(c_{\delta } (x) \) by \(C_{6} \), \(C_{7}\), and \(\frac{ c_{ \delta } C_{8}}{ \delta } \), respectively, adding
$$\begin{aligned}& \biggl( N_{1} \alpha C_{6} + \frac{ N_{2} \alpha C_{8} }{ \delta } \biggr) \biggl( \int _{\Omega } \vert u_{t} \vert ^{ m (x) } \,dx \biggr)^{ m_{1} -1} \\& \quad {}+ \biggl( N_{1} \vert \beta \vert C_{6} + \frac{ N_{2} \vert \beta \vert C_{8} }{ \delta } \biggr) \biggl( \int _{\Omega } \bigl\vert y(x,1,t) \bigr\vert ^{ m (x) } \,dx \biggr)^{ m_{1} -1} , \end{aligned}$$
taking \(\delta = \frac{ k_{l} }{ 4 N_{2} C_{7}} \) in (4.21), and using the relation
$$\begin{aligned} \biggl( \int _{\Omega } \vert u_{t} \vert ^{ m (x) } \,dx \biggr)^{ m_{1} -1} + \biggl( \int _{\Omega } \bigl\vert y(x,1,t) \bigr\vert ^{ m (x) } \,dx \biggr)^{ m_{1} -1} \leq 2 \biggl( - \frac{ E '(t)}{ C_{0} } \biggr)^{ m_{1} -1} , \end{aligned}$$
which is seen from (3.8). □
Lemma 4.12
Assume that \((A_{1})\), \((A_{2})\), \((A_{3})\), and (4.19) hold and let \(1 < m_{1} < 2 \). Then
$$\begin{aligned} \int ^{t_{2}}_{ t_{1} } E (s) \,ds \leq C_{10} ( t_{2} - t_{1} )^{ 2 - m_{1} } \quad \textit{for any } t_{2} \geq t_{1} \geq 0 . \end{aligned}$$
(4.63)
Proof
By virtue of (4.62), estimate (4.32) is replaced by
$$\begin{aligned} \bigl( L(t) + \Lambda (t) \bigr) ' \leq - \lambda E(t) + C_{9} \bigl( - E ' (t) \bigr)^{ m_{1} -1 } \quad \text{for } t \geq t_{\varepsilon } . \end{aligned}$$
(4.64)
Using (3.8), (4.64), and Young’s inequality with \(( m_{1} -1) + ( 2 - m_{1} ) =1 \), we observe
$$\begin{aligned}& \bigl\{ E^{\frac{2- m_{1} }{ m_{1} -1 }}(t) \bigl( L(t) + \Lambda (t) \bigr) + a_{7} E(t) \bigr\} ' \\& \quad \leq - \lambda E^{\frac{1 }{ m_{1} -1 }}(t) + C_{9} E^{ \frac{2- m_{1} }{ m_{1} -1 }}(t) \bigl( - E ' (t)\bigr)^{ m_{1} -1 } + a_{7} E'(t) \\& \quad \leq - \lambda E^{\frac{1 }{ m_{1} -1 }}(t) + \frac{ (2 - m_{1} ) \lambda }{2} E^{ \frac{1}{ m_{1} -1 } }(t) + ( m_{1} -1 ) C_{9}^{ \frac{1}{ m_{1} -1 } } \biggl( \frac{ \lambda }{2} \biggr)^{ \frac{ m_{1} -2 }{ m_{1} -1 } } \bigl( - E'(t)\bigr) + a_{7} E'(t) \\& \quad \leq - \lambda E^{\frac{1 }{ m_{1} -1 }}(t) + \frac{ \lambda }{2} E^{ \frac{1}{ m_{1} -1 } }(t) + C_{9}^{ \frac{1}{ m_{1} -1 } } \biggl( \frac{ \lambda }{2} \biggr)^{ \frac{ m_{1} -2 }{ m_{1} -1 } } \bigl( - E'(t)\bigr) + a_{7} E'(t) \\& \quad = - \frac{ \lambda }{2} E^{\frac{1 }{ m_{1} -1 }}(t) , \end{aligned}$$
(4.65)
where \(a_{7} = C_{9}^{ \frac{1}{ m_{1} -1 } } ( \frac{ \lambda }{2} )^{ \frac{ m_{1} -2 }{ m_{1} -1 } } \), which yields
$$\begin{aligned} \int ^{t}_{ t_{\varepsilon } } E^{ \frac{1 }{ m_{1} -1 } }( s ) \,ds \leq \frac{2}{ \lambda } \bigl\{ E^{\frac{2- m_{1} }{ m_{1} -1 }}( t_{ \varepsilon } ) \bigl( L( t_{\varepsilon } ) + \Lambda ( t_{\varepsilon } ) \bigr) + a_{7} E( t_{\varepsilon } ) \bigr\} : = a_{8} \end{aligned}$$
for all \(t \geq t_{\varepsilon } \). So, we get
$$\begin{aligned} 0 < \int ^{\infty }_{0} E^{ \frac{1}{ m_{1} -1 } }(s) \,ds \leq \int ^{ t_{ \varepsilon } }_{0} E^{ \frac{1}{ m_{1} -1 } }(s) \,ds + a_{8} < \infty . \end{aligned}$$
From this and Hölder inequality with \(( 2 - m_{1} ) + ( m_{1} -1 ) =1 \), we obtain
$$\begin{aligned} \int ^{t_{2}}_{ t_{1} } E ( s ) \,ds \leq ( t_{2} - t_{1} )^{ 2 - m_{1} } \biggl( \int ^{t_{2}}_{ t_{1} } E^{\frac{1 }{ m_{1} -1 }}( s ) \,ds \biggr)^{ m_{1} -1 } \leq C_{10} ( t_{2} - t_{1} )^{ 2 - m_{1} } \end{aligned}$$
for any \(t_{2} \geq t_{1} \geq 0 \). □
Theorem 4.2
Assume that \((A_{1})\), \((A_{2})\), \((A_{3})\), and (4.19) hold and let \(1 < m_{1} < 2 \). Then there exist \(c_{i} , \omega _{i} >0 \), \(i =3,4 \), and \(t_{0} > k^{-1}(\varepsilon ) \) satisfying
(i) if K is linear,
$$\begin{aligned} E(t) \leq c_{3} \biggl( \omega _{3} \int ^{t}_{k^{-1}(\varepsilon )} \zeta (s) \,ds \biggr)^{\frac{ 1- m_{1} }{ 2 - m_{1} }}, \quad t > k^{-1}( \varepsilon ) , \end{aligned}$$
(4.66)
(ii) if K is nonlinear,
$$ E(t) \leq c_{4} \bigl( t - k^{-1}( \varepsilon ) \bigr)^{2 - m_{1} } \hat{K}^{-1} \biggl( \omega _{4} \biggl( \bigl( t - k^{-1}(\varepsilon ) \bigr)^{ \frac{2 - m_{1}}{ m_{1} - 1 } } \int ^{t}_{ t_{0} } \zeta (s) \,ds \biggr)^{-1} \biggr), \quad t > t_{0} , $$
where
$$ \hat{K}(s) = s^{ \frac{1}{ m_{1} -1 } } K'( s ) . $$
Proof
Owing to (4.62), estimates (4.36) and (4.37) are replaced by
$$\begin{aligned} L'(t) \leq - \lambda E(t) - \frac{ k(0) }{ a_{1}} E'(t) + \frac{1}{2} \int _{t_{\varepsilon }}^{t} k (s) \bigl\Vert \Delta u (t) - \Delta u (t-s) \bigr\Vert _{2}^{2} \,ds + C_{9} \bigl( - E ' (t)\bigr)^{ m_{1} -1 } \end{aligned}$$
and
$$\begin{aligned} R'(t) \leq - \lambda E(t) + \frac{1}{2} \int _{t_{\varepsilon }}^{t} k (s) \bigl\Vert \Delta u (t) - \Delta u (t-s) \bigr\Vert _{2}^{2} \,ds + C_{9} \bigl( - E ' (t)\bigr)^{ m_{1} -1 } \end{aligned}$$
(4.67)
for \(t \geq t_{ \varepsilon }\), respectively.
Case 1: K is linear, that is, \(K (s) = a s \) for some \(a > 0\). Due to (4.67), estimate (4.38) is replaced by
$$\begin{aligned} {\mathcal{R}}_{1} ' (t) \leq - \lambda \zeta (t) E(t) + C_{9} \zeta (t) \bigl( - E ' (t)\bigr)^{ m_{1} -1 } , \quad t \geq t_{\varepsilon }. \end{aligned}$$
(4.68)
We set
$$\begin{aligned} {\mathcal{R}}_{3}(t) = E^{\frac{2- m_{1} }{ m_{1} -1 }}(t){\mathcal{R}}_{1} (t) + a_{9} E(t) , \end{aligned}$$
where \(a_{9} = C_{9}^{ \frac{1}{ m_{1} -1 } } ( \frac{ \lambda }{2} )^{ \frac{ m_{1} -2 }{ m_{1} -1 } } \zeta ( 0 ) \), which satisfies \({\mathcal{R}}_{3}(t) \sim E(t) \). Using (4.68) and the same argument of (4.65), we have
$$\begin{aligned} {\mathcal{R}}_{3}'(t) \leq & - \lambda \zeta (t) E^{ \frac{1}{ m_{1} -1 } }(t) + C_{9} \zeta (t) E^{ \frac{2- m_{1} }{ m_{1} -1 }}(t) \bigl( - E ' (t)\bigr)^{ m_{1} -1 } + a_{9} E'(t) \\ \leq & - \frac{ \lambda }{2} \zeta (t) E^{ \frac{1}{ m_{1} -1 } }(t) , \quad t \geq t_{\varepsilon }, \end{aligned}$$
which ensures (4.66).
Case 2: K is nonlinear. We let
$$\begin{aligned} \Gamma _{3} (t) = \frac{ a_{10} }{ ( t - t_{\varepsilon } )^{ 2 - m_{1} } } \int ^{t}_{t_{ \varepsilon }} \bigl\Vert \Delta u(t) - \Delta u(t-s) \bigr\Vert _{2}^{2} \,ds , \quad t \geq t_{ \varepsilon } . \end{aligned}$$
Using (4.39) and (4.63), we get
$$\begin{aligned}& \frac{ 1 }{ ( t - t_{\varepsilon } )^{ 2 - m_{1} } } \int ^{t}_{t_{ \varepsilon }} \bigl\Vert \Delta u(t) - \Delta u(t-s) \bigr\Vert _{2}^{2} \,ds \\& \quad \leq \frac{ 4 p_{1} }{ k_{l} ( p_{1} -2 ) ( t - t_{\varepsilon } )^{ 2 - m_{1} } } \biggl( \int ^{t}_{t_{\varepsilon }} E(s) \,ds + \int ^{ t - t_{ \varepsilon } }_{0} E( s ) \biggr) \,ds ) \\& \quad \leq \frac{ 8 p_{1} C_{10} }{ k_{l} ( p_{1} -2 ) } < \infty , \quad t \geq t_{\varepsilon } . \end{aligned}$$
Thus, there exists \(0 < a_{10} < 1 \) satisfying
$$ \Gamma _{3} (t) < 1 \quad \text{for } t \geq t_{\varepsilon }. $$
(4.69)
Using (4.69) and the same argument of (4.42), we can replace estimate (4.42) by
$$\begin{aligned} \Gamma _{2} (t) = & - \frac{ ( t - t_{\varepsilon } )^{ 2 - m_{1} } }{ a_{10} \Gamma _{3} (t)} \int ^{t}_{t_{\varepsilon }} \Gamma _{3} (t) k'(s) \frac{ a_{10} \Vert \Delta u(t) - \Delta u(t-s) \Vert _{2}^{2} }{ ( t - t_{\varepsilon } )^{ 2 - m_{1} } } \,ds \\ \geq & \frac{ ( t - t_{\varepsilon } )^{ 2 - m_{1} } }{ a_{10} \Gamma _{3} (t)} \int ^{t}_{t_{\varepsilon }} \Gamma _{3} (t) \zeta (s) K \bigl( k(s) \bigr) \frac{ a_{10} \Vert \Delta u(t) - \Delta u(t-s) \Vert _{2}^{2} }{ ( t - t_{\varepsilon } )^{ 2 - m_{1} } } \,ds \\ \geq & \frac{ ( t - t_{\varepsilon } )^{ 2 - m_{1} } \zeta (t ) }{ a_{10} \Gamma _{3} (t)} \int ^{t}_{t_{\varepsilon }} \overline{ K } \bigl( \Gamma _{3} (t) k(s) \bigr) \frac{ a_{10} \Vert \Delta u(t) - \Delta u(t-s) \Vert _{2}^{2} }{ ( t - t_{\varepsilon } )^{ 2 - m_{1} } } \,ds \\ \geq & \frac{ ( t - t_{\varepsilon } )^{ 2 - m_{1} } \zeta (t ) }{ a_{10} } \overline{ K } \biggl( \int ^{t}_{t_{\varepsilon }} \frac{ a_{10} k(s) \Vert \Delta u(t) - \Delta u(t-s) \Vert _{2}^{2} }{ ( t - t_{\varepsilon } )^{ 2 - m_{1} } } \,ds \biggr) , \quad t \geq t_{\varepsilon } , \end{aligned}$$
which reads
$$\begin{aligned} \int ^{t}_{t_{\varepsilon }} k(s) \bigl\Vert \Delta u(t) - \Delta u(t-s) \bigr\Vert _{2}^{2} \,ds \leq \frac{ ( t - t_{\varepsilon } )^{ 2 - m_{1} } }{ a_{10} } \overline{K}^{-1} \biggl( \frac{ a_{10} \Gamma _{2} (t)}{ ( t - t_{\varepsilon } )^{ 2 - m_{1} } \zeta (t)} \biggr). \end{aligned}$$
From this and (4.67), we get
$$\begin{aligned} R '(t) \leq - \lambda E(t) + \frac{ ( t - t_{\varepsilon } )^{ 2 - m_{1} } }{ 2 a_{10} } \overline{K}^{-1} \biggl( \frac{ a_{10} \Gamma _{2} (t)}{ ( t - t_{\varepsilon } )^{ 2 - m_{1} } \zeta (t)} \biggr) + C_{9} \bigl( - E ' (t)\bigr)^{ m_{1} -1 }, \quad t \geq t_{\varepsilon } . \end{aligned}$$
(4.70)
Let \(0 < \mu < \min \{ \varepsilon , 2 a_{10} \lambda E(0) \} \) and \({\mathcal{E}}(t) = \frac{E(t)}{E(0)} \). Using (4.70) and the same argument of (4.46), we obtain
$$\begin{aligned}& \biggl( \overline{K}' \biggl( \frac{ \mu {\mathcal{E}}(t) }{ ( t - t_{\varepsilon } )^{ 2 - m_{1} } } \biggr) R (t) \biggr)' \\& \quad \leq - \lambda \overline{K}' \biggl( \frac{ \mu {\mathcal{E}}(t) }{ ( t - t_{\varepsilon } )^{ 2 - m_{1} } } \biggr) E(t) + C_{9} \overline{K}' \biggl( \frac{ \mu {\mathcal{E}}(t) }{ ( t - t_{\varepsilon } )^{ 2 - m_{1} } } \biggr) \bigl( - E ' (t)\bigr)^{ m_{1} -1 } \\& \qquad {} + \frac{ ( t - t_{\varepsilon } )^{ 2 - m_{1} } }{ 2 a_{10} } \overline{K}' \biggl( \frac{ \mu {\mathcal{E}}(t) }{ ( t - t_{\varepsilon } )^{ 2 - m_{1} } } \biggr) \overline{K}^{-1} \biggl( \frac{ a_{10} \Gamma _{2} (t) }{ ( t - t_{\varepsilon } )^{ 2 - m_{1} } \zeta (t) } \biggr) \\& \quad \leq - \lambda \overline{K}' \biggl( \frac{ \mu {\mathcal{E}}(t) }{ ( t - t_{\varepsilon } )^{ 2 - m_{1} } } \biggr) E(t) + C_{9} \overline{K}' \biggl( \frac{ \mu {\mathcal{E}}(t) }{ ( t - t_{\varepsilon } )^{ 2 - m_{1} } } \biggr) \bigl( - E ' (t)\bigr)^{ m_{1} -1 } \\& \qquad {} + \frac{ ( t - t_{\varepsilon } )^{ 2 - m_{1} } }{ 2 a_{10} } \overline{K}^{*} \biggl( \overline{K}' \biggl( \frac{ \mu {\mathcal{E}}(t) }{ ( t - t_{\varepsilon } )^{ 2 - m_{1} } } \biggr) \biggr) + \frac{ \Gamma _{2} (t) }{ 2 \zeta (t) } \\& \quad \leq - a_{11} {\mathcal{E}}(t) \overline{K}' \biggl( \frac{ \mu {\mathcal{E}}(t) }{ ( t - t_{\varepsilon } )^{ 2 - m_{1} } } \biggr) + \frac{ \Gamma _{2} (t) }{ 2 \zeta (t) } + C_{9} \overline{K}' \biggl( \frac{ \mu {\mathcal{E}}(t) }{ ( t - t_{\varepsilon } )^{ 2 - m_{1} } } \biggr) \bigl( - E ' (t)\bigr)^{ m_{1} -1 } \end{aligned}$$
(4.71)
for \(t \geq t_{\varepsilon } \), where \(a_{11} = \lambda E(0) - \frac{ \mu }{ 2 a_{10} } \). Letting
$$\begin{aligned} {\mathcal{R}}_{4} (t)= \zeta (t) \overline{K}' \biggl( \frac{ \mu {\mathcal{E}}(t) }{ ( t - t_{\varepsilon } )^{ 2 - m_{1} } } \biggr) R(t) + E(t) , \end{aligned}$$
from (4.71) and (4.41), we get
$$\begin{aligned} {\mathcal{R}}_{4}' (t) \leq &- \frac{ a_{11} }{E(0)} \zeta (t) E(t) { \overline{K}}' \biggl( \frac{ \mu {\mathcal{E}}(t) }{ ( t - t_{\varepsilon } )^{ 2 - m_{1} } } \biggr) \\ &{} + C_{9} \zeta (t) { \overline{K}}' \biggl( \frac{ \mu {\mathcal{E}}(t) }{ ( t - t_{\varepsilon } )^{ 2 - m_{1} } } \biggr) \bigl( - E ' (t)\bigr)^{ m_{1} -1 } \end{aligned}$$
(4.72)
for \(t\geq t_{\varepsilon } \). Define
$$\begin{aligned} {\mathcal{L}} (t) = E^{ \frac{2- m_{1} }{ m_{1} -1 } }(t) {\mathcal{R}}_{4} (t) + a_{12} E(t) , \end{aligned}$$
where \(a_{12} = C_{9}^{ \frac{1}{ m_{1} -1 } } \zeta (0) ( \frac{ a_{11} }{2 E(0) } )^{ \frac{ m_{1} -2 }{ m_{1} -1 } } { \overline{K}}' ( \mu {\mathcal{E}}(0) ) \). Using (4.72) and the same argument of (4.65), we see
$$\begin{aligned} {\mathcal{L}}' (t) \leq & - \frac{ a_{11} }{E(0)} \zeta (t) E^{ \frac{1}{ m_{1} -1 } } (t) {\overline{K}}' \biggl( \frac{ \mu {\mathcal{E}}(t) }{ ( t - t_{\varepsilon } )^{ 2 - m_{1} } } \biggr) \\ & {}+ C_{9} \zeta (t) E^{ \frac{2- m_{1} }{ m_{1} -1 } }(t) \bigl( - E ' (t)\bigr)^{ m_{1} -1 } {\overline{K}}' \biggl( \frac{ \mu {\mathcal{E}}(t) }{ ( t - t_{\varepsilon } )^{ 2 - m_{1} } } \biggr) + a_{12} E '(t) \\ \leq & - \frac{ a_{11} }{2 E(0) } \zeta (t) E^{ \frac{1}{ m_{1} -1 } } (t) { \overline{K}}' \biggl( \frac{ \mu {\mathcal{E}}(t) }{ ( t - t_{\varepsilon } )^{ 2 - m_{1} } } \biggr) \\ & {}+ C_{9}^{ \frac{1}{ m_{1} -1 } } \biggl( \frac{ a_{11} }{2 E(0) } \biggr)^{ \frac{ m_{1} -2 }{ m_{1} -1 } } \zeta (t) \bigl( - E'(t)\bigr) { \overline{K}}' \biggl( \frac{ \mu {\mathcal{E}}(t) }{ ( t - t_{\varepsilon } )^{ 2 - m_{1} } } \biggr) \\ & {} + a_{12} E '(t) , \quad t \geq t_{\varepsilon } . \end{aligned}$$
(4.73)
Thanks to \(\lim_{ t \to \infty } \frac{ 1 }{ ( t - t_{\varepsilon } )^{ 2 - m_{1} } } =0 \), there exists \(t_{0} > t_{\varepsilon } \) such that
$$\begin{aligned} \frac{ 1 }{ ( t - t_{\varepsilon } )^{ 2 - m_{1} } } < 1, \quad \forall t > t_{0} , \end{aligned}$$
(4.74)
which ensures \({\mathcal{R}}_{4} (t) \sim E(t) \sim {\mathcal{L}} (t) \) for \(t > t_{0} \). Moreover, from (4.73) and (4.74), we deduce
$$\begin{aligned} {\mathcal{L}}' (t) \leq & - \frac{ a_{11} }{2 E(0) } \zeta (t) E^{ \frac{1}{ m_{1} -1 } } (t) K' \biggl( \frac{ \mu {\mathcal{E}}(t) }{ ( t - t_{\varepsilon } )^{ 2 - m_{1} } } \biggr) \\ & {}+ C_{9}^{ \frac{1}{ m_{1} -1 } } \zeta (0) \biggl( \frac{ a_{11} }{2 E(0) } \biggr)^{ \frac{ m_{1} -2 }{ m_{1} -1 } } K ' \bigl( \mu { \mathcal{E}}(0) \bigr) \bigl( - E'(t)\bigr) + a_{12} E'(t) \\ = & - \frac{ a_{11} }{2 E(0) } \zeta (t) E^{ \frac{1}{ m_{1} -1 } } (t) K' \biggl( \frac{ \mu {\mathcal{E}}(t) }{ ( t - t_{\varepsilon } )^{ 2 - m_{1} } } \biggr) , \quad t > t_{0} , \end{aligned}$$
and hence
$$\begin{aligned} \int ^{t}_{t_{0}} \zeta ( s ) E^{ \frac{1}{ m_{1} -1 } } ( s ) K' \biggl( \frac{ \mu {\mathcal{E}}( s ) }{ ( s - t_{\varepsilon } )^{ 2 - m_{1} } } \biggr) \,ds \leq \frac{2 E(0) }{ a_{11} } {\mathcal{L}} ( t_{0} ) , \quad t > t_{0} , \end{aligned}$$
Since
$$\begin{aligned}& \biggl\{ E^{ \frac{1}{ m_{1} -1 } } ( t ) K' \biggl( \frac{ \mu {\mathcal{E}}( t ) }{ ( t - t_{\varepsilon } )^{ 2 - m_{1} } } \biggr) \biggr\} ' \leq 0, \\& E^{ \frac{1}{ m_{1} -1 } } (t) K' \biggl( \frac{ \mu {\mathcal{E}}(t) }{ ( t - t_{\varepsilon } )^{ 2 - m_{1} } } \biggr) \int ^{t}_{t_{0}} \zeta ( s ) \,ds \leq \frac{2 E(0) }{ a_{11} } { \mathcal{L}} ( t_{0} ) , \quad t > t_{0} , \end{aligned}$$
multiplying this by \(( \frac{ \mu }{ ( t - t_{\varepsilon } )^{ 2 - m_{1} } } )^{ \frac{1}{ m_{1} -1 } } \), we have
$$\begin{aligned}& E^{ \frac{1}{ m_{1} -1 } }(0) \biggl( \frac{ \mu {\mathcal{E}}(t )}{ ( t - t_{\varepsilon } )^{ 2 - m_{1} } } \biggr)^{ \frac{1}{ m_{1} -1 } } K' \biggl( \frac{ \mu {\mathcal{E}}(t) }{ ( t - t_{\varepsilon } )^{ 2 - m_{1} } } \biggr) \int ^{t}_{t_{0}} \zeta ( s ) \,ds \\& \quad \leq \frac{2 E(0) {\mathcal{L}} ( t_{0} ) }{ a_{11} } \biggl( \frac{ \mu }{ ( t - t_{\varepsilon } )^{ 2 - m_{1} } } \biggr)^{ \frac{1}{ m_{1} -1 } } \end{aligned}$$
for \(t > t_{0} \). So, letting \({\hat{K}}(s) = s^{ \frac{1}{ m_{1} -1 } } K '(s) \), we have
$$\begin{aligned} E^{ \frac{1}{ m_{1} -1 } }(0) {\hat{K}} \biggl( \frac{ \mu {\mathcal{E}}(t )}{ ( t - t_{\varepsilon } )^{ 2 - m_{1} } } \biggr) \leq \frac{2 E(0) }{ a_{11} } {\mathcal{L}} ( t_{0} ) \biggl( \frac{ \mu }{ ( t - t_{\varepsilon } )^{ 2 - m_{1} } } \biggr)^{ \frac{1}{ m_{1} -1 } } \biggl( \int ^{t}_{t_{0}} \zeta ( s ) \,ds \biggr)^{-1} \end{aligned}$$
for \(t > t_{0} \), which gives
$$\begin{aligned} {\mathcal{E}}(t ) \leq \frac{ ( t - t_{\varepsilon } )^{ 2 - m_{1} } }{ \mu } { \hat{K}}^{-1} \biggl( \omega _{4} \biggl( ( t - t_{\varepsilon } )^{ \frac{2 - m_{1}}{ m_{1} -1 } } \int ^{t}_{t_{0}} \zeta ( s ) \,ds \biggr)^{-1} \biggr) \end{aligned}$$
for \(t > t_{0} \) and some \(\omega _{4} >0 \). □