On decay and blow-up of solutions for a system of viscoelastic equations with weak damping and source terms

In this article, we investigate a system of two viscoelastic equations with Dirichlet boundary conditions. Under some suitable assumptions on the function \(g_{i}(\cdot )\), \(f_{i}(\cdot ,\cdot )\) (\(i=1,2\)) and the initial data, we obtain general and optimal decay results. Moreover, for certain initial data, we establish a finite time blow-up result. This work generalizes and improves earlier results in the literature. The conditions of the relaxation functions \(g_{1}(t)\) and \(g_{2}(t)\) in our work are weak and seldom appear in previous literature, which is an important breakthrough.

In this paper, we investigate the following initial-boundary problem:

where Ω is a bounded domain of \(R^{n}\) (\(n\geq 1\)) with a smooth boundary ∂Ω, u and v represent the transverse displacements of wave. The functions \(g_{1}\) and \(g_{2}\) denote the kernel of the memory term, and the nonlinear functions \(f_{1}\) and \(f_{2}\) will be specified later. Problem (1.1) describes the propagation of some material possessing a capacity of storage and dissipation of mechanical energy. Models of this type arise in the theory of viscoelasticity and physics.

Firstly, we present some results related to viscoelastic equations. The single viscoelastic equation of the form

which has been extensively studied by many authors, and related results concerning existence, decay, and blow-up have been recently established (see [6, 11]). Here, we understand \(\triangle u_{tt}\), \(\vert u _{t} \vert ^{m-2}u_{t}\), \(\int _{0}^{t}g(t-\tau )\bigtriangleup u(\tau )d \tau \), and \(\vert u \vert ^{p-2}u\) to be the dispersion term, weak damping term, viscoelasticity dissipative term, and source term, respectively. This type of problem usually appears as a model in nonlinear viscoelasticity.

As \(a=b=0\) in the presence of the strong damping term, Cavalcanti et al. [1] dealt with the equation

where Ω is a bounded domain with smooth boundary ∂Ω in \(R^{n}\) (\(n\geq 1\)) and \(\rho >0\). They established a global existence result when the constant \(\gamma \geq 0\) and an exponential decay result for the case \(\gamma >0\). Later, this result was improved by Messaoudi and Tatar [8] to a situation where a source term is presented. By using perturbation techniques, they established a global existence and an exponential decay result.

When \(a=b=1\) in the absence of the dispersion term, the following problem

in a bounded domain and \(m>2\), \(p>2\) was studied by Song [10]. The author proved the nonexistence of global solutions of problem (1.4) with bounded positive initial energy.

The case of \(\rho =0\) in the absence of the viscoelasticity term problem (1.2) has been discussed by many authors. For example, Chen and Liu [2] studied the following equation:

subject to the same boundary and initial conditions as problem (1.4). Under some suitable conditions on the initial data, they proved the global existence of solutions in both cases which are polynomial and exponential decay in the energy space respectively.

In the case of \(a=b=1\), \(m=2\), in the presence of the strong damping term and without the dispersion term, the following viscoelastic equation

was studied by Li and He [4]. By using some properties of the convex functions, they obtained a general decay rate result. Moreover, they established a finite time blow-up result for solutions with negative initial energy and positive initial energy.

For a coupled system, the following system of viscoelastic equations

was considered by Mustafa [9]. Under some suitable conditions, they proved the well-posedness and obtained a generalized stability result.

In the work [5], Liu considered the following problem:

where \(\gamma _{1},\gamma _{2}>0\) and \(0<\rho \leq \frac{2}{n-2}\) if \(n\geq 3\) or \(\rho >0\) if \(n=1,2\). Under the following assumptions on the relaxation functions:

where \(\zeta _{1}\) and \(\zeta _{2}\) are positive constants. By exploiting the perturbed energy method, a uniform decay of the energy result was established.

Recently, Houari et al. [3] investigated the following system:

where Ω is an open bounded domain of \(R^{n}\) with a smooth boundary ∂Ω, m, \(r\geq 1\). Under some suitable conditions, they obtained a general decay, which depends on the relaxation functions.

As for a single wave equation, in the absence of the source term, the damping term assures global existence. On the other hand, without the damping term, the source term may cause finite time blow-up of solution. Hence, it is valuable to study the viscoelastic equation with damping and source terms.

Our aim in this work is to establish the global existence, decay, and blow-up result of solutions to problem (1.1). By adopting and modifying the methods used in [9], we establish the general and optimal decay and blow-up results, while we should overcome the additional difficulty caused by the changes of the conditions of the relaxation functions \(g_{1}(t)\) and \(g_{2}(t)\). In [9], the relaxation functions \(g_{i}(t)\) (\(i=1,2\)) satisfy \(g_{i}'(t)\leq - \zeta _{i}g_{i}(t)\) for all \(t\geq 0\), where \(\zeta _{i}(t)\) are positive nonincreasing functions. In this paper, the conditions have been replaced by \(g_{i}'(t)\leq -\zeta _{i}(t)g_{i}^{\gamma }(t)\), \(\gamma \in [1, 3/2)\). As far as we know, the conditions of the relaxation functions \(g_{i}(t)\) in our work seldom appear in previous literature, which is an important breakthrough. Our main novel contribution is an extension and improvement of the previous result from [9].

This paper is organized as follows. In Sect. 2, we give material needed for our work. In Sect. 3, we prove the global existence. In Sect. 4, we present some technical lemmas needed in the proof of our result. Section 5 is devoted to the general decay result. In Sect. 6, we carry out the proof of finite time blow-up result.

In this section, we present some material needed for our work. First, we make the following assumptions.

1. (G1)

   \(g_{i}: \mathbb{R_{+}}\rightarrow \mathbb{R_{+}}\) (for \(i=1,2\)) are \(C^{1}\) nonincreasing functions satisfying

   $$ 1- \int _{0}^{\infty }g_{1}(\tau )\,d\tau =l>0,\quad\quad 1- \int _{0}^{\infty }g_{2}(\tau )\,d\tau =k>0, \quad g_{i}(0)>0. $$

   (2.1)
2. (G2)

   There exist two nonincreasing differentiable functions \(\zeta _{1}, \zeta _{2}:\mathbb{R_{+}}\rightarrow \mathbb{R_{+}}\), with \(\zeta _{1}(0), \zeta _{2}(0)>0\) and satisfying

   $$ g_{i}'(t)\leq -\zeta _{i}(t)g_{i}^{\gamma }(t), \quad t\geq 0, 1\leq \gamma < \frac{3}{2}, \text{for }i=1,2. $$

   (2.2)
3. (G3)

   For the functions \(f_{1}\) and \(f_{2}\), we note that

   $$\begin{aligned}& f_{1}(u,v)= \vert u+v \vert ^{2(p+1)}(u+v)+ \vert u \vert ^{p}u \vert v \vert ^{p+2}, \end{aligned}$$

   (2.3)
   $$\begin{aligned}& f_{2}(u,v)= \vert u+v \vert ^{2(p+1)}(u+v)+ \vert u \vert ^{p+2}v \vert v \vert ^{p}, \end{aligned}$$

   (2.4)

   where p satisfies

   $$ p>-1 \quad \text{if } n=1,2 \quad \text{and} \quad -1< p\leq \frac{3-n}{n-2} \quad \text{if } n\geq 3. $$

   (2.5)

It is easy to verify that

where

Remark 2.1

There are many functions \(g_{i}(t)\) and \(\zeta _{i}(t)\) (for \(i=1,2\)) satisfying (G1) and (G2). An example of such functions is

Definition 2.2

A pair of functions \((u,v)\) defined on \([0,T]\) is called a weak solution of system (1.1) if \(u,v\in C_{w}([0,T],H_{0}^{1}(\varOmega ))\), \(u _{t},v_{t}\in C_{w}([0,T],L^{2}(\varOmega ))\), \((u(x,0),v(x,0))=(u_{0},v _{0})\in H_{0}^{1}(\varOmega )\times H_{0}^{1}(\varOmega )\), \((u_{t}(x,0),v _{t}(x,0))=(u_{1},v_{1})\in H_{0}^{1}(\varOmega )\times H_{0}^{1}(\varOmega )\) and \((u,v)\) satisfies

for a.e. \(t\in [0,T]\) and all test functions \(\phi ,\psi \in H_{0} ^{1}(\varOmega )\). Here, \(C_{w}([0,T],X)\) denotes the space of weakly continuous functions from \([0,T]\) into Banach space X.

Proposition 2.3

Let \((u_{0},v_{0})\in H_{0}^{1}(\varOmega )\times H_{0}^{1}(\varOmega )\) and \((u_{1},v_{1})\in L^{2}(\varOmega )\times L^{2}(\varOmega )\). Assume that (G1)–(G3) hold. Then there exists a unique local weak solution \((u,v)\) of system (1.1) defined in \([0,T_{m}]\) for some \(T_{m}>0\) small enough.

In this section, we prove the global existence of the solution to system (1.1).

First, we define

and the energy function associated with system (1.1)

where

Lemma 3.1

Let (G1)–(G3) hold and \((u,v)\) be the solution of (1.1), then we have

Proof

Multiplying the first equation in system (1.1) by \(u_{t}\) and integrating over Ω gives

Then

For the last term on the left-hand side of (3.8), we get

By combining (3.4), (3.8), and (3.9), we deduce

Similarly, multiplying the second equation in (1.1) by \(v_{t}\) and integrating over Ω yields

Finally, by adding (3.10) to (3.11), (3.6) is established. □

The following lemma is important to prove the global existence of solution.

Lemma 3.2

([3])

Assume that (2.5) holds. Then there exists \(\eta >0\) such that, for any \((u,v)\in H_{0}^{1}(\varOmega )\times H_{0}^{1}(\varOmega )\), we obtain

Proof

Exploiting Minkowski’s inequality, we have

From Hölder’s inequality and Young’s inequality, we derive

Then, combining (3.13), (3.14) and the embedding \(H_{0}^{1}( \varOmega )\hookrightarrow L^{2(p+2)}(\varOmega )\) leads to (3.12). □

Lemma 3.3

Let \((u_{0},v_{0})\in H_{0}^{1}(\varOmega )\times H_{0}^{1}(\varOmega )\) and \((u_{1},v_{1})\in L^{2}(\varOmega )\times L^{2}(\varOmega )\). Assume that (G1)–(G3) hold. If

then \(I(t)>0\) for \(t\in [0,T_{m}]\).

Proof

Since \(I(0)>0\), then by the continuity of \(I(t)\), there exists \(T^{*}< T_{m}\) such that \(I(t)>0\), \(\forall t\in [0,T^{*}]\). By using (3.1) and (3.2), we have

From (2.1), (3.1), (3.2), and (3.6), we infer that

It follows from (2.7), (3.12), and (3.15) that

Combining (3.15) and (3.18), we deduce that

Therefore

By repeating this procedure, \(T^{*}\) is extended to \(T_{m}\). □

Lemma 3.4

Assume that (G1)–(G3) hold. If \((u_{0},v_{0})\in H_{0} ^{1}(\varOmega )\times H_{0}^{1}(\varOmega )\) and \((u_{1},v_{1})\in L^{2}( \varOmega )\times L^{2}(\varOmega )\) and satisfy (3.15), then the solution of system (1.1) is bounded and global in time.

Proof

From Lemma 3.3, (3.6), and (3.17), we see that

therefore

which implies that the solution of system (1.1) is global and bounded. □

In this section, we present some lemmas needed for the proof of our result.

Lemma 4.1

([3])

There exist two constants \(a_{0}>0\) and \(a_{1}>0\) such that

Proof

The right-hand side of (4.1) is trivial. If \(u=v=0\), for the left-hand side of (4.1), the result is also trivial. If, without loss of generality, \(v\neq 0\), then either \(\vert u \vert \leq \vert v \vert \) or \(\vert u \vert > \vert v \vert \).

If \(\vert u \vert \leq \vert v \vert \), we have

Now we consider the continuous function

then we obtain that \(\min j(\omega )\geq 0\). If \(\min j(\omega )=0\), then there exists \(\omega _{0}\in [-1,1]\) such that

This infers that \(\vert 1+\omega _{0} \vert = \vert \omega _{0} \vert =0\), which is impossible. Hence \(\min j(\omega )=2a_{0}>0\). Thus

It follows from (4.5) that

and then

If \(\vert u \vert > \vert v \vert \), we deduce that

Hence, this gives the desired result. □

Lemma 4.2

([3])

There exist two positive constants \(\lambda _{1}\) and \(\lambda _{2}\) such that

Proof

From (2.3), we easily get

By (4.10) and Young’s inequality, with \(q=\frac{2p+3}{p+1}\) and \(q'=\frac{2p+3}{p+2}\), we have

therefore

Then, by using Poincare’s inequality and (2.5), we observe that

In the same way, we have

 □

Lemma 4.3

(Jensen inequality)

Suppose that G is a concave function on \([a,b]\), \(f:\varOmega \rightarrow [a,b]\) and h are integrable functions on Ω, with \(h(x) \geq 0\), and \(\int _{\varOmega }h(x)\,dx=r>0\), then

For the special case \(G(y)=y^{\frac{1}{\gamma }}\), \(y\geq 0\), \(\gamma >1\), we obtain

Lemma 4.4

Assume that \(g_{i}\) satisfies (G1) and (G2) for \(i=1,2\), then

Proof

From (G1) and (G2), we see that

Integrating (4.18) over \((0,+\infty )\) and using the fact that \(0\leq \theta <2-\gamma \), we get

 □

Lemma 4.5

([7])

If (G1)–(G3) hold, \(u\in L^{\infty }(0,T,H_{0}^{1}(\varOmega ))\), for \(0<\theta <1\), we obtain

By taking \(\theta =\frac{1}{2}\), we have

Proof

For \(q>1\), we derive

Applying Hölder’s inequality, we deduce

By taking \(q=\frac{\gamma -1+\theta }{\gamma -1}\), we arrive at

Then, by taking \(\theta =\frac{1}{2}\) in (4.24), (4.21) is established. □

Similarly,

and

Lemma 4.6

([7])

Suppose that \(g_{1}\) satisfies (G1) and (G2), then we have

Proof

Multiplying both sides of (4.21) by \(\zeta _{1}(t)\) and by using (G2), (3.6), and (4.17) gives

 □

Likewise,

In this section, we establish three related lemmas before proving our result.

Lemma 5.1

If (G1)–(G3) and (3.15) hold, the functional \(\phi (t)\) defined by

satisfies, along solutions of (1.1),

Proof

A differentiation of \(\phi (t)\) with respect to t, it follows from system (1.1) that

For the first term on the right-hand side of (5.2), by using Young’s inequality and the fact that \(\int _{0}^{t}g_{1}(s)\,ds\leq \int _{0}^{\infty }g_{1}(s)\,ds =1-l\), for \(\eta =\frac{l}{1-l}>0\), we get

Similar calculations also yield, for \(\eta _{1}=\frac{k}{1-k}>0\),

Applying Young’s inequality and Poincare’s inequality, for some \(\beta _{1}>0\), we obtain

Likewise, for some \(\beta _{2}>0\), we have

Inserting (5.3)–(5.6) into (5.2) yields

Now, we pick \(\beta _{1}, \beta _{2}>0\) small enough such that

Finally, a combination of (5.7) and (5.8) gives (5.1). □

Lemma 5.2

Suppose that (G1)–(G3) and (3.15) hold. The functional \(\psi _{1}(t)\) defined by

satisfies, along solutions of (1.1),

where \(\alpha _{2}=\lambda _{1} (\frac{2(p+2)}{p+1}E(0) )^{2p+2}\), \(\lambda _{1}\) is the constant in Lemma 4.2.

Proof

Taking the derivative of \(\psi _{1}(t)\) with respect to t and using system (1.1) gives

For the first term on the right-hand side of (5.10), by exploiting (G1), Young’s inequality, and Cauchy–Schwarz inequality, for any \(\delta >0\), we get

As for the second term in (5.10), recall that \((a+b)^{2}\leq 2(a ^{2}+b^{2})\), for \(\eta _{2}=1\), we obtain

The third term can be handled by

For the forth term, it follows from (4.9) that

where \(\alpha _{2}=\lambda _{1} (\frac{2(p+2)}{p+1}E(0) )^{2p+2}\).

The fifth term on the right-hand side of (5.10) can be estimated as

Taking into account estimates (5.11)–(5.13), estimate (5.9) is established. □

Similar computations also yield the following.

Lemma 5.3

Suppose that (G1)–(G3) and (3.15) hold. The functional \(\psi _{2}(t)\) defined by

satisfies, along solutions of system (1.1),

where \(\alpha _{3}=\lambda _{2} (\frac{2(p+2)}{p+1}E(0) )^{2p+2}\), \(\lambda _{2}\) is the constant in Lemma 4.2.

Now, we define the functional

where \(\epsilon _{1}\) and \(\epsilon _{2}\) are positive constants, \(\phi (t)\) is given in Lemma 5.1 and \(\psi (t):=\psi _{1}(t)+\psi _{2}(t)\).

Lemma 5.4

([5])

Let \((u,v)\) be the solution of system (1.1) and assume that (3.15) holds. Then there exist constant \(\epsilon >0\) small enough and \(M>0\) large enough such that the following relation

holds for two positive constants \(\beta _{1}\) and \(\beta _{2}\).

Theorem 5.5

Assume (G1)–(G3) and (3.15) hold. Let \((u_{0},v_{0}) \in H_{0}^{1}(\varOmega )\times H_{0}^{1}(\varOmega )\) and \((u_{1},v_{1}) \in L^{2}(\varOmega )\times L^{2}(\varOmega )\). Then, for each \(t_{0}>0\), there exist positive constants K, k, \(k_{1}\), \(k_{2}\) such that the solution of system (1.1) satisfies, for all \(t\geq t_{0}\),

Furthermore, if

then

where \(\zeta (t)=\min \{\zeta _{1}(t),\zeta _{2}(t) \}\).

Proof

From (G1), we know \(g_{1}\) and \(g_{2}\) are positive, then for any \(t\geq t_{0}>0\), we have

Taking derivative of (5.17) with respect to t and using (3.6), (5.1), and (5.16) yields

At this point, we pick \(\delta >0\) small enough such that

and

As long as δ is fixed, the choice of any two positive constants \(\epsilon _{1}\) and \(\epsilon _{2}\) satisfying

and

will make

Hence, there exist two positive constants m and C such that

Case 1. \(\gamma =1\):

Let \(\zeta (t)=\min \{\zeta _{1}(t),\zeta _{2}(t) \}\), since \(\zeta _{1}(t)\) and \(\zeta _{2}(t)\) are nonincreasing differentiable functions, then we get that \(\zeta (t)\) is nonincreasing.

When \(\gamma =1\), from (2.2), we easily get

Multiplying both sides of (5.24) by \(\zeta (t)\) and applying (3.6), (4.27), (4.29), and (5.25) yields

Setting \(F(t)=\zeta (t)W(t)+CE(t)\), then clearly \(F\sim E\). Recalling that \(W\sim E\geq 0\) by (3.21) and (5.18), \(\zeta '(t) \leq 0\) by (G2), we get that \(\zeta '(t)W(t)\leq 0\), then together with (5.26), we have, for some \(k>0\),

Integrating (5.27) over \([t_{0},t]\) gives

Therefore, by using the fact that \(F(t)\sim E(t)\), we derive

Case 2. \(1<\gamma <\frac{3}{2}\):

Multiplying both sides of (5.24) by \(\zeta (t)\), using (4.27) and (4.29) leads to

Multiplying (5.30) by \(\zeta ^{\alpha }(t)E^{\alpha }(t)\), with \(\alpha =2\gamma -2\), we get

Exploiting Young’s inequality with \(q=1+\alpha \) and \(q'=\frac{1+ \alpha }{\alpha }\) gives

We pick \(\epsilon <\frac{m}{C}\) and recall that \(\zeta '(t)\leq 0\), \(\zeta (t)>0\) by (G2), \(E'(t)\leq 0\) by (3.6), and \(W\thicksim E\geq 0\) by (3.21) and (5.18), then together with (5.32) we have, for some \(c_{1}>0\),

Next, we take \(F(t)=\zeta ^{\alpha +1}WE^{\alpha }+CE\), which is clearly equivalent to \(E(t)\), then there exists \(a_{0}>0\) such that

Integrating (5.34) over \((t_{0},t)\) and recalling that \(F(t)\thicksim E(t)\) and \(\alpha =2\gamma -2\), we obtain

From (5.21) and (5.35), we infer that

Setting \(\lambda _{1}(t)=\int _{0}^{t} \Vert \nabla u(t)-\nabla u(t-\tau ) \Vert _{2} ^{2}\,d\tau \), by using (3.3), we deduce

Similarly, let \(\lambda _{2}(t)=\int _{0}^{t} \Vert \nabla v(t)-\nabla v(t- \tau ) \Vert _{2}^{2}\,d\tau \), we have \(\lambda _{2}(t)<+\infty \).

From (5.24) and recalling that \(\zeta (t)=\min \{\zeta _{1}(t), \zeta _{2}(t) \}\), \(\zeta _{1}(t)\) and \(\zeta _{2}(t)\) are nonincreasing differentiable functions, we arrive at

Exploiting Jensen’s inequality, for the second term on the right-hand side of (5.38), with \(G(y)=y^{\frac{1}{\gamma }}\), \(y>0\), \(f(\tau )= \zeta ^{\gamma }(\tau )g^{\gamma }(\tau )\) and \(h(\tau )= \Vert \nabla u(t)- \nabla u(t-\tau ) \Vert _{2}^{2}\), where we assume that \(\lambda _{1}(t), \lambda _{2}(t)>0\), otherwise we get \(\Vert \nabla u(t)-\nabla u(t-\tau ) \Vert = \Vert \nabla v(t)-\nabla v(t-\tau ) \Vert =0\), then by using (5.24) we deduce

Since \(\zeta _{1}(t)\) and \(\zeta _{2}(t)\) are nonincreasing, then for some \(C_{1}>0\), estimate (5.38) becomes

Multiplying both sides of (5.39) by \(\zeta ^{\alpha }(t)E^{\alpha }(t)\), with \(\alpha =\gamma -1\), we deduce

Applying Young’s inequality, with \(q=1+\alpha \), and \(q'=\frac{1+ \alpha }{\alpha }\) leads to

Then, by taking \(\sigma <\frac{m}{C_{1}}\) and recalling that \(\zeta '(t)\leq 0\), \(\zeta (t)>0\) by (G2), \(E'(t)\leq 0\) by (3.6), and \(W(t)\thicksim E(t)\geq 0\) by (3.21) and (5.18), together with (5.41), we get, for some \(C_{2}>0\),

which implies

Let \(L=\zeta ^{\alpha +1}E^{\alpha }W+CE\), then clearly \(L\thicksim E\), we obtain, for some \(C_{3}>0\),

Integrating (5.44) over \((t_{0},t)\) and recalling that \(L\thicksim E\) and \(\alpha =\gamma -1\) yields

This completes the proof. □

In this section, we carry out the proof of the finite time blow-up result.

Theorem 6.1

If (G1) and (G3) hold and the initial energy \(E(0)<0\). Assume that \(g_{i}\) satisfies

Then the solution of system (1.1) blows up in finite time.

Proof

First, we define

where \(E(t)\) is defined in (3.3). By (G1) and (3.6), we find that

Noting the assumption \(E(0)<0\), from (4.1), (6.2), and (6.3), we get

From (4.1), we see that

Let

where ϵ is a positive constant to be chosen later and

Taking the derivative of \(L(t)\) and using system (1.1) gives

It follows from Young’s inequality that

Similarly,

A combination of (6.8)–(6.10) leads to

Let \(0< r\leq \min \{l, k \}\), then by the expression of \(E(t)\) and \(H(t)\), we obtain

Inserting (6.12) into (6.11), we arrive at

By using (G1), (6.1), and \(r=\min \{l, k \}\), we infer that

For the last two terms of (6.13), by using Hölder’s inequality and (6.3)–(6.5) yields

where \(K_{1}=c_{0}^{\frac{-1}{2(p+2)}} \vert \varOmega \vert ^{\frac{p+1}{2(p+2)}}\).

By (6.7), we know \(\frac{1}{2(p+2)}-\frac{1}{2}+\alpha <0\), it follows from Young’s inequality and (6.3) that

Likewise,

We pick

then it follows from (6.13)–(6.18) that

where

Now, we choose \(0<\epsilon <1\) small enough such that

A combination of (6.14), (6.19), and (6.21) yields

where C is a positive constant. It is clear that \(L(t)\) is increasing on \([0,T)\) and

In the case of \(\int _{\varOmega }u_{0}u_{1}\,dx+\epsilon \int _{\varOmega }v _{0}v_{1}\,dx\geq 0\), no further restriction on ϵ is needed. For the case \(\int _{\varOmega }u_{0}u_{1}\,dx+\epsilon \int _{\varOmega }v_{0}v _{1}\,dx<0\), we assume that

Therefore, in either case, we have

Now we prove that \(L(t)\) satisfies the following inequality:

where C is a positive constant, \(1<\frac{1}{1-\alpha }<2\) assumed by (6.7). At this point, we distinguish two cases.

Case 1. \(\int _{\varOmega }uu_{t}\,dx+\int _{\varOmega }vv_{t}\,dx\leq 0\) for some \(t\in [0,T)\). For such t, we obtain

which together with (6.22), then (6.26) follows for all such t.

Case 2. \(\int _{\varOmega }uu_{t}\,dx+\int _{\varOmega }vv_{t}\,dx\geq 0\) for all \(t\in [0,T)\). Since \(0<\alpha <\frac{1}{2}\), \(1<\frac{1}{1-\alpha }<2\), and \(0<\epsilon <1\), then we deduce

Exploiting Hölder’s inequality and Young’s inequality, we see that

From (6.7), we get that \(\frac{1}{(1-2\alpha )(p+1)}-1<0\). Then, by (6.3) and (6.5), we have

Similarly,

By combining (6.28)–(6.31), we deduce that

It follows from (6.22) and (6.32) that

which shows (6.26) follows for Case 2.

Integrating (6.26) over \((0,t)\) yields

This shows that \(G(t)\) blows up in finite time

Therefore, the solution of system (1.1) blows up in finite time. □

Acknowledgements

The authors are highly grateful for the referees’ valuable suggestions which improved this work a lot.

Availability of data and materials

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

Funding

Not applicable.

Affiliations

1. School of Mathematics and Statistics, Wuhan University, Wuhan, China

   • Luofei He

Contributions

The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.

Corresponding author

Correspondence to Luofei He.

Competing interests

The authors declare to have no competing interests.

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