In this section, we establish several lemmas needed for the proof of our main result.
Lemma 3.1
There exist two positive constants
\(c_{1}\)
and
\(c_{2}\)
such that
$$\begin{aligned} \int _{0}^{L} x \bigl\vert f_{i}(u,v) \bigr\vert ^{2} \,dx \leq c_{i} \bigl(\ell _{1} \Vert u_{x} \Vert ^{2}_{L^{2}_{x}}+\ell _{2} \Vert v_{x} \Vert ^{2}_{L^{2}_{x}} \bigr)^{2r+3},\quad i=1,2. \end{aligned}$$
(19)
Proof
We prove inequality (19) for \(f_{1}\) and the same result holds for \(f_{2}\). It is clear that
$$\begin{aligned} \bigl\vert {f_{1}(u,v)} \bigr\vert &\le {C \bigl({ \vert {u+v} \vert }^{2r+3}+{ \vert {u} \vert }^{r+1} { \vert {v} \vert }^{r+2} \bigr)} \\ & \le C \bigl({ \vert u \vert }^{2r+3}+{ \vert v \vert }^{2r+3}+{ \vert {u} \vert }^{r+1} { \vert {v} \vert }^{r+2} \bigr). \end{aligned}$$
(20)
From (20) and Young’s inequality, with
$$\begin{aligned} q=\frac{2r+3}{r+1},\qquad q^{\prime }=\frac{2r+3}{r+2}, \end{aligned}$$
we get
$$\begin{aligned} { \vert {u} \vert }^{r+1}{ \vert {v} \vert }^{r+2}\le {c_{1} { \vert {u} \vert }^{2r+3} + c_{2}{ \vert {v} \vert }^{2r+3}}, \end{aligned}$$
hence
$$\begin{aligned} \bigl\vert f_{1}(u,v) \bigr\vert \le {C \bigl[{ \vert {u} \vert }^{2r+3} + { \vert {v} \vert }^{2r+3} \bigr]}. \end{aligned}$$
Consequently, by using (7), (12), (13) and the embedding \(V_{0} \hookrightarrow L^{2(2r+3)}\), we obtain
$$\begin{aligned} \int _{0}^{L}x{ \bigl\vert {f_{1}(u,v)} \bigr\vert }^{2}\,dx &\le C \bigl( \Vert u \Vert ^{2(2r+3)}_{L_{x}^{2(2r+3)}}+ \Vert v \Vert ^{2(2r+3)}_{L_{x}^{2(2r+3)}} \bigr) \\ &\le {c_{1}{\bigl(\ell _{1} { \Vert {u_{x}} \Vert }_{L^{2}_{x}}^{2}+\ell _{2}{ \Vert { v_{x}} \Vert }_{L^{2}_{x}}^{2} \bigr)}^{2r+3}}. \end{aligned}$$
This completes the proof of Lemma 3.1. □
Lemma 3.2
([22])
There exist positive constants d and \(t_{0}\) such that, for any \(t\in [0,t_{0}]\), we have
$$\begin{aligned} k_{i}^{\prime }(t)\le -d k_{i}(t),\quad i=1,2. \end{aligned}$$
(21)
Lemma 3.3
If \((A1)\) holds. Then, for any \(w\in V_{0}\), \(0<\alpha <1\) and \(i=1,2\), we have
$$\begin{aligned} \int _{0}^{L}x \biggl( \int _{0}^{t} k_{i}(t-s) \bigl(w(t)-w(s) \bigr)\,ds \biggr)^{2}\,dx \leq C_{\alpha,i}(h_{i}\circ w) (t), \end{aligned}$$
(22)
where \(C_{\alpha,i}:=\int _{0}^{\infty }\frac{k_{i}^{2}(s)}{\alpha k_{i}(s)-k_{i}'(s)}\,ds\) and \(h_{i}(t):=\alpha k_{i}(t)-k_{i}'(t)\).
Proof
The proof of this lemma goes similar to the one in [22]. □
Lemma 3.4
Under the assumptions \((A1)\) and \((A2)\), the functional
$$\begin{aligned} \Phi (t):= \int _{0}^{L}xuu_{t} \,dx+ \int _{0}^{L}xvv_{t} \,dx, \end{aligned}$$
satisfies, along with the solution of system (1), the estimate
$$\begin{aligned} \Phi ^{\prime }(t) \le{}& \Vert u_{t} \Vert ^{2}_{L^{2}_{x}}+ \Vert v_{t} \Vert ^{2}_{L^{2}_{x}} - \frac{\ell _{1}}{2} \Vert u_{x} \Vert ^{2}_{L^{2}_{x}}- \frac{\ell _{2}}{2} \Vert v_{x} \Vert ^{2}_{L^{2}_{x}} \\ &{} +C_{\alpha,1}(h_{1}\circ u_{x}) (t)+C_{\alpha,2}(h_{2}\circ v_{x}) (t)+ \int _{0}^{L}xF(u,v)\,dx. \end{aligned}$$
(23)
Proof
Direct differentiation, using (1), yields
$$\begin{aligned} \Phi ^{\prime }(t)={}& \int _{0}^{L}xu^{2}_{t}\,dx+ \biggl(1- \int _{0}^{t}k_{1}(s)\,ds \biggr) \int _{0}^{L}xu_{x}^{2}\,dx \\ &{} + \int _{0}^{L}xu_{x} \int _{0}^{t}k_{1}(t-s) \bigl(u_{x}(s)-u_{x}(t)\bigr)\,ds\,dx \\ &{} + \int _{0}^{L}xv^{2}_{t}\,dx+ \biggl(1- \int _{0}^{t}k_{1}(s)\,ds \biggr) \int _{0}^{L}xv_{t}^{2}\,dx \\ &{} + \int _{0}^{L}xv_{x} \int _{0}^{t}k_{2}(t-s) \bigl(v_{x}(s)-v_{x}(t)\bigr)\,ds\,dx \\ &{} + \int _{0}^{L}x \bigl(uf_{1}(u,v)+vf_{2}(u,v) \bigr)\,dx. \end{aligned}$$
(24)
Using Young’s inequality, we obtain, for any \(\delta _{1}, \delta _{2}\in (0,1)\),
$$\begin{aligned} \Phi ^{\prime }(t)\le{}& \int _{0}^{L}xu^{2}_{t}\,dx- \ell _{1} \int _{0}^{L}xu_{x}^{2}\,dx+ \frac{\delta _{1}}{2} \int _{0}^{L}xu_{x}^{2}\,dx \\ &{} +\frac{1}{2\delta _{1}} \int _{0}^{L}x \biggl( \int _{0}^{t}k_{1}(t-s) \bigl(u_{x}(s)-u_{x}(t)\bigr)\,ds \biggr)^{2}\,dx \\ &{} + \int _{0}^{L}xv^{2}_{t}\,dx- \ell _{2} \int _{0}^{L}xv_{x}^{2}\,dx+ \frac{\delta _{2}}{2} \int _{0}^{L}xv_{x}^{2}\,dx \\ &{} +\frac{1}{2\delta _{2}} \int _{0}^{L}x \biggl( \int _{0}^{t}k_{2}(t-s) \bigl(v_{x}(s)-v_{x}(t)\bigr)\,ds \biggr)^{2}\,dx \\ &{} + \int _{0}^{L}x F(u,v)\,dx. \end{aligned}$$
(25)
Taking \(\delta _{1}=\ell _{1}\) and \(\delta _{2}=\ell _{2}\) and using Lemma 3.3, we have
$$\begin{aligned} \Phi ^{\prime }(t)\le{}& \int _{0}^{L}xu^{2}_{t}\,dx- \frac{\ell _{1}}{2} \int _{0}^{L}xu_{x}^{2} \,dx+cC_{\alpha,1}(h_{1}\circ u_{x}) (t) \\ &{} + \int _{0}^{L}xv^{2}_{t}\,dx- \frac{\ell _{1}}{2} \int _{0}^{L}xv_{x}^{2} \,dx+cC_{ \alpha,2}(h_{2}\circ v_{x}) (t)+ \int _{0}^{L}x F(u,v)\,dx. \end{aligned}$$
(26)
□
Let us introduce the functionals
$$\begin{aligned} \chi _{1}(t):=- \int _{0}^{L}x u_{t} \int _{0}^{t}k_{1}(t-s) \bigl(u(t)-u(s) \bigr)\,ds\,dx \end{aligned}$$
and
$$\begin{aligned} \chi _{2}(t):=- \int _{0}^{L} xv_{t} \int _{0}^{t}k_{2}(t-s) \bigl(v(t)-v(s) \bigr)\,ds\,dx. \end{aligned}$$
Lemma 3.5
Assume that \((A1)\) and \((A2)\) hold. Then the functional
$$\begin{aligned} \chi (t):=\chi _{1}(t)+\chi _{2}(t) \end{aligned}$$
satisfies, along with the solution of (1), the following estimate:
$$\begin{aligned} \chi '(t)\le{}& {-} \biggl( \int _{0}^{t}k_{1}(s)\,ds-\delta \biggr) \Vert u_{t} \Vert ^{2}_{L^{2}_{x}}+c\delta \Vert u_{x} \Vert ^{2}_{L^{2}_{x}}+\frac{c}{\delta }(C_{\alpha,1}+1) (h_{1} \circ u_{x}) (t) \\ &{} - \biggl( \int _{0}^{t}k_{2}(s)\,ds-\delta \biggr) \Vert v_{t} \Vert ^{2}_{L^{2}_{x}}+c\delta \Vert v_{x} \Vert ^{2}_{L^{2}_{x}}+ \frac{c}{\delta }(C_{\alpha,2}+1) (h_{2}\circ v_{x}) (t), \end{aligned}$$
(27)
where \(0<\delta <1\).
Proof
Direct differentiation, using (1), gives
$$\begin{aligned} \chi _{1}'(t)={}&{-} \biggl( \int _{0}^{t}k_{1}(s)\,ds \biggr) \int _{0}^{L}xu_{t}^{2} \\ &{} + \biggl(1- \int _{0}^{t}k_{1}(s)\,ds \biggr) \int _{0}^{L}xu_{x}(t) \int _{0}^{t}k_{1}(t-s) \bigl(u_{x}(t)-u_{x}(s)\bigr)\,ds\,dx \\ &{} + \int _{0}^{L}x \biggl( \int _{0}^{t}k_{1}(t-s) \bigl(u_{x}(t)-u_{x}(s)\bigr)\,ds \biggr)^{2}\,dx \\ &{} - \int _{0}^{L}x f_{1}(u,v) \int _{0}^{t}k_{1}(t-s) \bigl(u(t)-u(s) \bigr)\,ds\,dx \\ &{} - \int _{0}^{L}x u_{t} \int _{0}^{t}k_{1}'(t-s) \bigl(u(t)-u(s)\bigr)\,ds\,dx. \end{aligned}$$
(28)
Using Young’s inequality and Lemma 3.3, we get, for any \(0<\delta <1\), the following:
$$\begin{aligned} &\biggl(1- \int _{0}^{t}k_{1}(s)\,ds \biggr) \int _{0}^{L}xu_{x}(t) \int _{0}^{t}k_{1}(t-s) \bigl(u_{x}(t)-u_{x}(s)\bigr)\,ds\,dx \\ &\qquad{} + \int _{0}^{L}x \biggl( \int _{0}^{t}k_{1}(t-s) \bigl\vert u_{x}(t)-u_{x}(s) \bigr\vert \,ds \biggr)^{2} \,dx \\ &\quad \leq \delta \int _{0}^{L}xu_{x}^{2}+ \frac{c}{\delta } \int _{0}^{L}x \biggl( \int _{0}^{t}k_{1}(t-s) \bigl\vert u_{x}(t)-u_{x}(s) \bigr\vert \,ds \biggr)^{2} \,dx \\ &\quad \leq \delta \int _{0}^{L}xu_{x}^{2}+ \frac{c}{\delta }C_{\alpha,1}(h_{1} \circ u_{x}) (t). \end{aligned}$$
(29)
Using Young’s inequality, (18), (19) and (22), we have
$$\begin{aligned} & \int _{0}^{L} xf_{1}(u,v) \int _{0}^{t}k_{1}(t-s) \bigl(u(t)-u(s) \bigr)\,ds\,dx \\ &\quad \le \delta \biggl( \int _{0}^{L} x{ \bigl\vert f_{1}(u,v) \bigr\vert }^{2} \,dx \biggr)+ \frac{1}{4\delta } \int _{0}^{L}x \biggl( \int _{0}^{t}k_{1}(t-s) \bigl(u(t)-u(s) \bigr)\,ds \biggr)^{2} \,dx \\ &\quad \le {c_{1} \delta {\bigl(\ell _{1} \Vert u_{x} \Vert _{L_{x}^{2}}^{2}+\ell _{2} \Vert v_{x} \Vert _{L_{x}^{2}}^{2} \bigr)}^{2r+3}}+\frac{c}{\delta }C_{\alpha,1}(h_{1} \circ u_{x}) (t) \\ &\quad \le c_{1}\delta { \biggl(\frac{2(r+2)}{r+1}E(0) \biggr)}^{2r+1}\bigl(\ell _{1} \Vert u_{x} \Vert _{L_{x}^{2}}^{2}+\ell _{2} \Vert v_{x} \Vert _{L_{x}^{2}}^{2}\bigr)+ \frac{c}{\delta }C_{\alpha,1}(h_{1} \circ u_{x}) (t) \\ &\quad \leq c\delta \Vert u_{x} \Vert _{L_{x}^{2}}^{2}+c \delta \Vert v_{x} \Vert _{L_{x}^{2}}^{2}+ \frac{c}{\delta }C_{\alpha,1}(h_{1}\circ u_{x}) (t). \end{aligned}$$
(30)
Also, by applying Young’s inequality and Lemma 3.3, we obtain, for any \(0<\delta <1\),
$$\begin{aligned} &- \int _{0}^{L} xu_{t} \int _{0}^{t}k_{1}'(t-s) \bigl(u(t)-u(s)\bigr)\,ds\,dx \\ &\quad = \int _{0}^{L} x u_{t} \int _{0}^{t} h_{1}(t-s) \bigl(u(t)-u(s) \bigr)\,ds\,dx- \int _{0}^{L} x u_{t} \int _{0}^{t}\alpha k_{1}(t-s) \bigl(u(t)-u(s)\bigr)\,ds\,dx \\ &\quad \leq \delta \Vert u_{t} \Vert _{L_{x}^{2}}^{2}+ \frac{1}{2\delta } \biggl( \int _{0}^{t}h_{1}(s)\,ds \biggr) (h_{1}\circ u) (t)+\frac{c}{\delta }C_{\alpha,1}(h_{1} \circ u) (t) \\ &\quad \leq \delta \Vert u_{t} \Vert _{L_{x}^{2}}^{2}+ \frac{c}{\delta }(C_{\alpha,1}+1) (h_{1} \circ u_{x}) (t). \end{aligned}$$
(31)
Similarly, we have
$$\begin{aligned} - \int _{0}^{L} xv_{t} \int _{0}^{t}k_{2}'(t-s) \bigl(v(t)-v(s)\bigr)\,ds\,dx\leq \delta \Vert v_{t} \Vert _{L_{x}^{2}}^{2}+\frac{c}{\delta }(C_{\alpha,2}+1) (h_{2} \circ v_{x}) (t). \end{aligned}$$
(32)
A combination of all the above estimates gives
$$\begin{aligned} \chi _{1}'(t)\le - \biggl( \int _{0}^{t}k_{1}(s)\,ds-\delta \biggr) \Vert u_{t} \Vert ^{2}_{L^{2}_{x}}+c\delta \Vert u_{x} \Vert ^{2}_{L^{2}_{x}}+\frac{c}{\delta }(C_{\alpha,1}+1) (h_{1} \circ u_{x}) (t). \end{aligned}$$
(33)
Repeating the same calculations with \(\chi _{2}\), we obtain
$$\begin{aligned} \chi _{2}'(t)\le - \biggl( \int _{0}^{t}k_{2}(s)\,ds-\delta \biggr) \Vert v_{t} \Vert ^{2}_{L^{2}_{x}}+c\delta \Vert v_{x} \Vert ^{2}_{L^{2}_{x}}+\frac{c}{\delta }(C_{\alpha,2}+1) (h_{2} \circ v_{x}) (t). \end{aligned}$$
(34)
Therefore, (33) and (34) imply (27), which completes the proof of Lemma 3.5. □
Lemma 3.6
Assume that \((A1)\) and \((A2)\) hold. Then the functionals \(J_{1}\) and \(J_{2}\) defined by
$$\begin{aligned} J_{1}(t):= \int _{0}^{L}x \int _{0}^{t}K_{1}(t-s) \bigl\vert u_{x}(s) \bigr\vert ^{2}\,ds\,dx \end{aligned}$$
and
$$\begin{aligned} J_{2}(t):= \int _{0}^{L}x \int _{0}^{t}K_{2}(t-s) \bigl\vert v_{x}(s) \bigr\vert ^{2}\,ds\,dx \end{aligned}$$
satisfy, along with the solution of (1), the estimates
$$\begin{aligned} & J_{1}'(t)\leq 3(1-\ell ) \Vert u_{x} \Vert _{L_{x}^{2}}^{2}-\frac{1}{2}(k_{1} \circ u_{x}) (t), \end{aligned}$$
(35)
$$\begin{aligned} & J_{2}'(t)\leq 3(1-\ell ) \Vert v_{x} \Vert _{L_{x}^{2}}^{2}-\frac{1}{2}(k_{2} \circ v_{x}) (t), \end{aligned}$$
(36)
where \(K_{i}(t):=\int _{t}^{\infty }k_{i}(s)\,ds\) (for \(i=1,2\)) and \(\ell =\min \{\ell _{1},\ell _{2}\}\).
Proof
We will prove inequality (35) and the same proof also holds for (36). By Young’s inequality and the fact that \(K_{1}^{\prime }(t)=-k_{1}(t)\), we see that
$$\begin{aligned} J_{1}^{\prime }(t)={}&K_{1}(0) \int _{0}^{L}x \bigl\vert u_{x}(t) \bigr\vert ^{2}\,dx- \int _{0}^{L}x \int _{0}^{t}k_{1}(t-s) \bigl\vert u_{x}(s) \bigr\vert ^{2} \,dx \\ ={}&{-} \int _{0}^{L}x \int _{0}^{t}k_{1}(t-s) \bigl\vert u_{x}(s)- u_{x}(t) \bigr\vert ^{2} \,ds \,dx \\ &{} -2 \int _{0}^{L} xu_{x}(t). \int _{0}^{t}k_{1}(t-s) \bigl( u_{x}(s)- u_{x}(t)\bigr)\,ds\,dx+K_{1}(t) \int _{0}^{L}x \bigl\vert u_{x}(t) \bigr\vert ^{2} \,dx. \end{aligned}$$
Now,
$$\begin{aligned} &-2 \int _{0}^{L}x u_{x}(t). \int _{0}^{t}k_{1}(t-s) \bigl( u_{x}(s)-u_{x}(t)\bigr)\,ds\,dx \\ &\quad \le 2(1-\ell _{1}) \int _{0}^{L}x \bigl\vert u_{x}(t) \bigr\vert ^{2} \,dx+ \frac{\int _{0}^{t}k_{1}(s)\,ds}{2(1-\ell _{1})} \int _{0}^{L}x \int _{0}^{t}k_{1}(t-s) \bigl\vert u_{x}(s)-u_{x}(t) \bigr\vert ^{2} \,ds\,dx. \end{aligned}$$
Using the facts that \(K_{1}(0)=1-\ell _{1}\) and \(\int _{0}^{t}k_{1}(s)\,ds \le 1-\ell _{1}\), (35) is established. □
Lemma 3.7
The functional L defined by
$$\begin{aligned} L(t):=NE(t)+N_{1} \phi (t)+N_{2} \chi (t) \end{aligned}$$
satisfies, for a suitable choice of \(N,N_{1},N_{2}\ge 1\),
$$\begin{aligned} L(t)\sim E(t) \end{aligned}$$
(37)
and the estimate
$$\begin{aligned} L'(t)\leq {}&{-}4(1-\ell ) \bigl( \Vert u_{x} \Vert _{L_{x}^{2}}^{2}+ \Vert v_{x} \Vert ^{2}_{L_{x}^{2}} \bigr)- \bigl( \Vert u_{t} \Vert ^{2}_{L_{x}^{2}}+ \Vert v_{t} \Vert ^{2}_{L_{x}^{2}} \bigr) \\ &{} +c \int _{0}^{L} xF(u,v)\,dx+\frac{1}{4} \bigl[(k_{1}\circ u_{x}) (t)+(k_{2} \circ v_{x}) (t) \bigr], \quad\forall t\geq t_{0}, \end{aligned}$$
(38)
where \(t_{0}\) is introduced in Lemma 3.2and \(\ell =\min \{\ell _{1},\ell _{2}\}\).
Proof
It is not difficult to prove that \(L(t)\sim E(t)\). To establish (38), we choose \(\delta =\frac{\ell }{4cN_{2}}\) where \(\ell =\min \{\ell _{1},\ell _{2}\}\). We set \(C_{\alpha }=\max \{C_{\alpha,1},C_{\alpha,2}\}\) and \(k_{0}=\min \lbrace \int _{0}^{t_{0}}k_{1}(s)\,ds,\int _{0}^{t_{0}}k_{2}(s)\,ds \rbrace >0\). Now using (23) and (28) and recalling the fact that \(k_{i}'=\alpha k_{i}-h_{i}\), we obtain, for any \(t\geq t_{0}\),
$$\begin{aligned} L'(t)\leq{}&{ -}\frac{\ell }{4}(2N_{1}-1) \bigl( \Vert u_{x} \Vert ^{2}_{L_{x}^{2}}+ \Vert v_{x} \Vert ^{2}_{L_{x}^{2}} \bigr)- \biggl(k_{0}N_{2}-\frac{\ell }{4c}-N_{1} \biggr) \bigl( \Vert u_{t} \Vert ^{2}_{L_{x}^{2}}+ \Vert v_{t} \Vert ^{2}_{L_{x}^{2}} \bigr) \\ &{} -N_{1} \int _{0}^{L}x F(u,v)\,dx+\frac{\alpha }{2}N \bigl[(k_{1}\circ u_{x}) (t)+(k_{2} \circ v_{x}) (t) \bigr] \\ &{} - \biggl[\frac{1}{2}N-\frac{4c^{2}}{\ell }N^{2}_{2}-C_{\alpha } \biggl( \frac{4c^{2}}{\ell }N_{2}^{2}+cN_{1} \biggr) \biggr] \bigl[(h_{1}\circ u_{x}) (t)+(h_{2} \circ v_{x}) (t) \bigr]. \end{aligned}$$
First, we choose \(N_{1}\) so large such that \(\frac{\ell }{4}(2N_{1}-1)>4(1-\ell )\).
Then we select \(N_{2}\) large enough so that \(k_{0} N_{2}-\frac{\ell }{4c}-N_{1}>1\). Now, one can use the Lebesgue dominated convergence theorem with the fact that \(\frac{\alpha k_{i}^{2}(s)}{\alpha k_{i}(s)-k_{i}'(s)}< k_{i}(s)\), for \(i=1,2\), to prove that
$$\begin{aligned} \lim_{\alpha \rightarrow 0^{+}}\alpha C_{\alpha }=0. \end{aligned}$$
Therefore, there exists \(\alpha _{0}\in (0,1)\) such that if \(\alpha <\alpha _{0}\), then, we get \(\alpha C_{\alpha }< \frac{1}{8 [\frac{4c^{2}}{\ell }N^{2}_{2}+cN_{1} ]}\). Then, by letting \(\alpha =\frac{1}{2N}<\alpha _{0}\), we get \(\frac{1}{4}N-\frac{4c^{2}}{\ell }N^{2}_{2}>0\). This leads to
$$\begin{aligned} \frac{1}{2}N-\frac{4c^{2}}{\ell }N^{2}_{2}-C_{\alpha } \biggl[ \frac{4c^{2}}{\ell }N_{2}^{2}+cN_{1} \biggr]>\frac{1}{4}N- \frac{4c^{2}}{\ell }N^{2}_{2}>0. \end{aligned}$$
Then, (38) is established. □