Some remarks on Blow-up of solutions to Nakao's problem

TL;DR

This paper investigates the blow-up behavior of solutions to Nakao's problem using nonlinear ordinary differential inequalities to gain insights into the conditions leading to solution blow-up.

Contribution

It introduces a method based on nonlinear ODE inequalities to analyze the blow-up phenomena in Nakao's problem, providing a new perspective.

Findings

Identifies conditions under which solutions blow up.

Provides a framework for analyzing blow-up using differential inequalities.

Offers insights into the nature of solution singularities.

Abstract

In this note we try to understand the blow-up of solutions to Nakao's problem by using nonlinear ordinary differential inequalities.

Equations

$$\{\begin{array}{rlrl}& u_{tt}-\Delta u+u_t=|v{|}^p,& & x\in {\mathbb{R}}^n,t>0,\\ & v_{tt}-\Delta v=|u{|}^q,& & x\in {\mathbb{R}}^n,t>0,\\ & (u,u_t,v,v_t)(0,x)=(u_0,u_1,v_0,v_1)(x),& & x\in {\mathbb{R}}^n,\end{array}$$

$${\alpha }_{\text{N,W}}:=\max \left\{\frac{q/2+1}{pq-1}+\frac{1}{2};\frac{q+1}{pq-1};\frac{p+1}{pq-1}\right\}\ge \frac{n}{2},$$

$$\{\begin{array}{rlrl}& u_{tt}-\Delta u=|v{|}^p,& & x\in {\mathbb{R}}^n,t>0,\\ & v_{tt}-\Delta v=|u{|}^q,& & x\in {\mathbb{R}}^n,t>0,\\ & (u,u_t,v,v_t)(0,x)=(u_0,u_1,v_0,v_1)(x),& & x\in {\mathbb{R}}^n,\end{array}$$

$${\alpha }_{\text{W}}:=\max \left\{\frac{p+2+q^{-1}}{pq-1};\frac{q+2+p^{-1}}{pq-1}\right\}=\frac{n-1}{2}.$$

$$\{\begin{array}{rlrl}& u_{tt}-\Delta u+u_t=|v{|}^p,& & x\in {\mathbb{R}}^n,t>0,\\ & v_{tt}-\Delta v+v_t=|u{|}^q,& & x\in {\mathbb{R}}^n,t>0,\\ & (u,u_t,v,v_t)(0,x)=(u_0,u_1,v_0,v_1)(x),& & x\in {\mathbb{R}}^n,\end{array}$$

$${\alpha }_{\text{DW}}:=\max \left\{\frac{p+1}{pq-1},\frac{q+1}{pq-1}\right\}=\frac{n}{2}.$$

$$\displaystyle(u,v)\in\big{(}\mathcal{C}\big{(}[0,T],H^{1}(\mathbb{R}^{n})\big{)}\cap\mathcal{C}^{1}\big{(}[0,T],L^{2}(\mathbb{R}^{n})\big{)}\big{)}^{2}$$

$$\displaystyle(u,v)\in\big{(}\mathcal{C}\big{(}[0,T],H^{1}(\mathbb{R}^{n})\big{)}\cap\mathcal{C}^{1}\big{(}[0,T],L^{2}(\mathbb{R}^{n})\big{)}\big{)}\times\big{(}\mathcal{C}\big{(}[0,T],L^{\frac{2(n+1)}{n-1}}(\mathbb{R}^{n})\big{)}\big{)}$$

$$\displaystyle(u,v)\in\big{(}\mathcal{C}\big{(}[0,T],H^{1}(\mathbb{R}^{n})\big{)}\cap\mathcal{C}^{1}\big{(}[0,T],L^{2}(\mathbb{R}^{n})\big{)}\big{)}^{2}$$

$${\int }_{{\mathbb{R}}^n}u({\tilde{t}}_0,x)dx\text{mbox}and{\int }_{{\mathbb{R}}^n}v({\tilde{t}}_0,x)dx$$

$$\alpha =\max \left\{\frac{q+1}{pq-1};\frac{2+2p^{-1}}{pq-1}\right\}\ge \frac{n-1}{2},$$

$$F_1(t)={\int }_{{\mathbb{R}}^n}u(t,x)dx\text{and}{F}_2(t)={\int }_{{\mathbb{R}}^n}v(t,x)dx$$

$$F_1({\tilde{t}}_0)={\int }_{{\mathbb{R}}^n}u({\tilde{t}}_0,x)dx\text{mbox}and{F}_2({\tilde{t}}_0)={\int }_{{\mathbb{R}}^n}v({\tilde{t}}_0,x)dx$$

$$F_1(t):$$

$$F_2(t):$$

$${\int }_{{\mathbb{R}}^n}{u}_j(x)dx>0\text{and}{\int }_{{\mathbb{R}}^n}{v}_j(x)dx>0\text{for}j=0,1.$$

$$F_1(t)$$

$$\frac{d{F}_1}{dt}(t)+F_1(t)$$

$$\frac{d^2{F}_1}{d{t}^2}(t)+\frac{d{F}_1}{dt}(t)$$

$$F_2(t)$$

$$\frac{d^2{F}_2}{d{t}^2}(t)$$

$${\beta }_2+{\alpha }_{2}q$$

$${\alpha }_2+{\beta }_{2}p$$

$$F_1(t)$$

$$\frac{d{F}_1}{dt}(t)+F_1(t)$$

$$\frac{d^2{F}_1}{d{t}^2}(t)+\frac{d{F}_1}{dt}(t)$$

$$F_2(t)$$

$$\frac{d^2{F}_2}{d{t}^2}(t)$$

$$\text{supp }u(t,\cdot ),v(t,\cdot )\subset \{|x|\le t+R\}\text{ for any }t>0.$$

$$\frac{d^2{F}_1}{d{t}^2}(t)+\frac{d{F}_1}{dt}(t)$$

$$\frac{d^2{F}_2}{d{t}^2}(t)$$

$$\displaystyle\Big{|}\int_{\mathbb{R}^{n}}v(t,x)\,dx\Big{|}^{p}$$

$$\displaystyle\Big{|}\int_{\mathbb{R}^{n}}u(t,x)\,dx\Big{|}^{q}$$

$$\frac{d^2{F}_1}{d{t}^2}(t)+\frac{d{F}_1}{d{t}^2}(t)$$

$$\frac{d^2{F}_2}{d{t}^2}(t)$$

$${\psi }_1(t,x):=e^{-\frac{\sqrt{5}-1}{2}t}\varphi (x)\text{and}{\psi }_2(t,x):=e^{-t}\varphi (x),$$

$$\varphi (x):={\int }_{{\mathbb{S}}^{n-1}}{e}^{x\cdot \omega }d{\sigma }_{\omega }\sim |x{|}^{-\frac{n-1}{2}}{e}^{|x|}\text{as}|x|\to \infty ,$$

Full text

1. Introduction

The goal of this note is to understand nonexistence results for global (in time) solutions to the Cauchy problem for the weakly coupled system of semilinear damped wave equation and semilinear wave equation, that is, blow-up of solutions to the following weakly coupled system:

[TABLE]

where $n\geq 1$ and $p,q>1$.

The problem of critical curve for the weakly coupled system (1.1) was proposed by Professor Misuhiro Nakao, Emeritus of Kyushu University (see [14, 18]). Here “critical curve” means that the threshold condition of a pair of exponents $(p,q)$ for global (in time) existence of small data Sobolev solutions and blow-up of classes of local (in time) Sobolev solutions. Recently, by applying the test function method (e. g. [10, 20]) the author of [18] proved that if the condition

[TABLE]

holds for spatial dimensions $n\geq 1$, then local (in time) Sobolev solutions $(u,v)$ to (1.1) in general blow up in finite time. However, in the $p-q$ plane for a pair of exponents $(p,q)$ the curve $\alpha_{\text{N,W}}=\frac{n}{2}$ is optimal only when $n=1$ because the condition $\alpha_{\text{N,W}}\geq\frac{1}{2}$ means that every local (in time) Sobolev solution blows up for $1<p,q<\infty$. When $n\geq 2$, the condition (1.2) seems not to be optimal. Thus, if a pair of exponents $(p,q)$ does not satisfy the condition (1.2), the questions of global (in time) existence or nonexistence of Sobolev solutions to Nakao’s problem (1.1) are still open.

We sketch now some historical background to (1.1). Since Nakao’s problem (1.1) is related to a weakly coupled system of semilinear damped wave equation and semilinear wave equation, we recall some results for weakly coupled systems for semilinear wave equations and semilinear damped wave equations, respectively, in the following.

On one hand, the following weakly coupled system of semilinear wave equations

[TABLE]

for $n\geq 1$ with $p,q>1$, has been widely studied in recent years. The papers [2, 3, 4, 1, 8, 7, 6, 9] investigated that the critical curve for (1.3) is described by the condition

[TABLE]

In other words, if $\alpha_{\text{W}}<\frac{n-1}{2}$, then there exists a unique global (in time) Sobolev solution for small data; else if $\alpha_{\text{w}}\geq\frac{n-1}{2}$, in general, local (in time) Sobolev solutions blow up in finite time.

On the other hand, let us consider the weakly coupled system of semilinear classical damped wave equations

[TABLE]

for $n\geq 1$ with $p,q>1$. The critical curve for (1.5) is described by the condition

[TABLE]

which has been investigated by the authors of [17, 11, 12, 13].

From the above results of critical curves for (1.3) and (1.5), we may expect that the critical curve for (1.1) is between (1.4) and (1.6). However, we should underline that the critical curve for (1.1) is not a simple combination of (1.4) and (1.6) because the critical curve to (1.1) seems to be influenced by varying degrees between semilinear wave equation and semilinear damped wave equation.

Let us explain the difficulties to derive blow-up of solutions. To obtain blow-up of solutions to (1.3) when $\alpha_{\text{W}}\geq\frac{n-1}{2}$, the authors of [3, 4, 2, 9] mainly applied generalized Kato’s lemmas to a system of nonlinear ordinary differential inequalities and constructed some contradictions. However, the authors of [13, 17] derive blow-up of solutions to (1.5) when $\alpha_{\text{DW}}\geq\frac{n}{2}$ by applying the test function method. So, it is interesting to see what method is suitable for us to derive blow-up of solution to (1.1). Although [18] has applied the test function method to (1.1), the result seems to be not optimal for $n\geq 2$. Furthermore, it is not trivial but interesting to see how do semilinear damped wave equation and semilinear wave equation affect each other. In this note we only try to understand what happen if we apply directly generalized Kato’s lemma.

The rest of this note is organized as follows. In Subsection 2.1 we show our main results, including local (in time) existence of solutions and blow-up of solutions to (1.1). In Subsection 2.2 we introduce the overview of our approach, especially, the system of nonlinear ordinary differential inequalities in Lemmas 2.1 and 2.2. In Sections 3 and 4 we prove Lemmas 2.1 and 2.2, respectively. In Section 5 we prove our main results by using the tools developed in the previous sections.

2. Main results and overview of our approach

2.1. Main results

First of all, let us introduce a result of local (in time) existence of Sobolev solutions to the Cauchy problem (1.1). The proof of the following theorem is quite standard (see for example [16, 15, 5]). Therefore, we will only sketch the proof in Subsection 5.1.

Theorem 2.1**.**

Let us assume (u_{0},u_{1};v_{0},v_{1})\in\big{(}H^{1}(\mathbb{R}^{n})\times L^{2}(\mathbb{R}^{n})\big{)}^{2} having compact support, that is, $\text{supp }u_{0},u_{1},v_{0},v_{1}\subset\{|x|\leq R\}$ with a positive constant $R$.

• •

If $1<p,q<\infty$ for $n=1,2$, then there exists a positive $T$ and a uniquely determined local (in time) energy solution $(u,v)$ such that

[TABLE]

with $\text{supp }u(t,\cdot),v(t,\cdot)\subset\{|x|\leq t+R\}$.

• •

If $1<p\leq\frac{n+3}{n-1}$ and $1<q\leq\frac{n}{n-2}$ for $n\geq 3$, then there exists a positive $T$ and a uniquely determined local (in time) Sobolev solution $(u,v)$ such that

[TABLE]

with $\text{supp }u(t,\cdot),v(t,\cdot)\subset\{|x|\leq t+R\}$.

Remark 2.1**.**

In the cases $n\geq 3$ we still can prove local (in time) existence of energy solution

[TABLE]

with $\text{supp }u(t,\cdot),v(t,\cdot)\subset\{|x|\leq t+R\}$, if $1<p,q\leq\frac{n}{n-2}$. One can see the result in the recent paper [18]. Nevertheless, we observe that $\frac{n+3}{n-1}>\frac{n}{n-2}$ for all $n\geq 3$. For this reason, the consideration of v\in\mathcal{C}\big{(}[0,T],L^{\frac{2(n+1)}{n-1}}(\mathbb{R}^{n})\big{)} with $\text{supp }v(t,\cdot)\subset\{|x|\leq t+R\}$ allows us to get a larger admissible range of the exponent $p$.

Now we state the blow-up results for (1.1). The proof will be shown in Subsection 5.2.

Theorem 2.2**.**

Let (u_{0},u_{1};v_{0},v_{1})\in\big{(}\mathcal{C}^{2}(\mathbb{R}^{n})\times\mathcal{C}^{1}(\mathbb{R}^{n})\big{)}^{2} be nonnegative (but not vanishing) have their support. Assume (u,v)\in\big{(}\mathcal{C}^{2}([0,T)\times\mathbb{R}^{n})\big{)}^{2} is the maximal, with respect to time interval, classical solution to the Cauchy problem (1.1). Moreover, we assume that there exists a large time $\tilde{t}_{0}\in(0,T)$ such that

[TABLE]

are suitably large (but not infinity). If the exponents satisfy

• •

$1<p,q<\infty$* for $n=1$,*

• •

$1<p,q<\frac{2n}{n-1}$* for $n=2,3$,*

• •

$1<p\leq\frac{n+3}{n-1}$, $1<q\leq\frac{n}{n-2}$ for $n\geq 4$,

and the following condition:

[TABLE]

then the local (in time) classical solution $(u,v)$ blows up in finite time, that is, $T<\infty$.

Remark 2.2**.**

Under the assumptions of exponents $p,q$ and the hypothesis for initial data in Theorem 2.2, according to Theorem 2.1 we know that there exists a unique local (in time) classical solution to Nakao’s problem (1.1).

Remark 2.3**.**

We remark that if we assume suitably large data in Theorem 2.2, then from the proof of Lemma 2.1 we may assert that the time-dependent functions

[TABLE]

are strictly increasing functions for $t\geq 0$. For this reason, the assumption for suitably large values of

[TABLE]

are trivially satisfied.

Remark 2.4**.**

For the one dimensional Nakao’s problem, the condition in Theorem 2.2 is always valid and there is no restriction to the exponents of the power nonlinearities.

Remark 2.5**.**

Actually, the assumption $1<p,q<\frac{2n}{n-1}$ for $n\geq 2$ in Theorem 2.2 is a usual assumption when we use the method of proving blow-up for nonlinear ordinary differential inequalities. One may observe this assumption when we prove blow-up of solutions to the single semilinear wave equation (see [15, 19]) and weakly coupled systems of semilinear wave equations (see [3, 9]). However, due to the symmetry of their models and the proof of the condition $\alpha_{\text{W}}\geq\frac{n-1}{2}$, the condition $1<p,q<\frac{2n}{n-1}$ is trivially valid.

Remark 2.6**.**

The authors find that the restriction for $F_{1}(\tilde{t}_{0})$ and $F_{2}(\tilde{t}_{0})$ is too strong. We will improve the result in the future.

2.2. Overview of our approach

Throughout this note we will consider the time-dependent functions

[TABLE]

where $(u,v)$ is a classical solution of (1.1).

Since we require compactly supported data, by the property of finite speed of propagation and Theorem 2.1 it follows that also $(u,v)(t,\cdot)$ is compactly supported with respect to the spatial variables as long as the solution exists with respect to $t$. Thus, if we prove that $F_{1}(t),F_{2}(t)$ blow up in finite time, then $(u,v)$ blows up in finite time as well.

In Section 3 we will prove the next lemma by employing the idea of [19].

Lemma 2.1**.**

Let us consider the Cauchy problem (1.1) with compactly supported initial data with $\text{supp }u_{0},u_{1},v_{0},v_{1}\subset\{|x|\leq R\}$ for a positive constant $R$. Moreover, we assume

[TABLE]

Moreover, we assume $1<p,q<\infty$ for $n=1$, and $1<p,q<\frac{2n}{n-1}$ for $n\geq 2$. Then, the functions $F_{1}(t)$, $F_{2}(t)$ satisfy the following system of second-order nonlinear differential inequalities:

[TABLE]

with $C_{3}>0$ defined in (3.8), respectively, for large time $t\geq T_{0}$.

Remark 2.7**.**

To investigate the lower bounds estimate of $F_{1}(t)$ in Lemma 2.1, we need to assume $p<\frac{2n}{n-1}$ when $n\geq 2$. Similarly, to get the lower bounds estimate of $F_{2}(t)$ in Lemma 2.1, we need to assume $q<\frac{2n}{n-1}$ when $n\geq 2$.

Then, in Section 4 we may prove blow-up of the functions $F_{1}(t),F_{2}(t)$ in finite time by using the following generalized Kato’s lemma.

Lemma 2.2**.**

Let $p,q>1$, $\alpha_{1},\beta_{1}>0$, $\alpha_{2},\beta_{2},\beta_{3}\geq 0$. Moreover, we assume

[TABLE]

Suppose that $F_{1}=F_{1}(t)$ and $F_{2}=F_{2}(t)$ belong to $\mathcal{C}^{2}[0,T]$ and satisfy the following system of second-order nonlinear differential inequalities for $t\geq T_{0}$:

[TABLE]

where all $k_{0},k_{1},k_{2},k_{3},k_{4}$ are positive constants. Moreover, we assume that there exists a large $\widetilde{T}_{0}$ such that $\widetilde{T}_{0}\in(T_{0},T)$ such that F_{1}\big{(}\widetilde{T}_{0}\big{)} and F_{2}\big{(}\widetilde{T}_{0}\big{)} are suitably large (but not infinity). Then, the functions $F_{1}(t),F_{2}(t)$ blow up in finite time, that is, $T<\infty$.

3. Proof of Lemma 2.1

3.1. Derivation of a system of nonlinear ordinary differential inequalities

Let $(u,v)$ be the local (in time) classical solutions to the Cauchy problem (1.1). We define the functions $F_{1}=F_{1}(t)$ and $F_{2}=F_{2}(t)$ as in (2.2) and (2.3), respectively. From the property of finite speed of propagation and our assumption on the initial data, we have

[TABLE]

By applying the divergence theorem and the compact support property of solutions, we have

[TABLE]

Using Hölder’s inequality leads to

[TABLE]

In other words, we obtain

[TABLE]

Here and in the following we use $C$ as a universal positive constant. In order to get lower bounds for the functions $F_{1}=F_{1}(t)$ and $F_{2}=F_{2}(t)$, we now introduce the functions $\psi_{1}=\psi_{1}(t,x)$ and $\psi_{2}=\psi_{2}(t,x)$ as follows:

[TABLE]

where

[TABLE]

and $\mathbb{S}^{n-1}$ is the $n-1$ dimensional sphere. By the compactness of the unit sphere $\mathbb{S}^{n-1}$ we know that the function $\phi\in\mathcal{C}^{\infty}(\mathbb{R}^{n})$ satisfies

[TABLE]

Actually, the fact that $\phi(x)>0$ for $x\in\mathbb{R}^{n}$ comes from the monotonicity of the exponential function. Moreover, from direct calculations, we may assert that

[TABLE]

We apply reverse Hölder’s inequality to obtain

[TABLE]

In the following steps we will estimate the time-dependent functions $J_{1}=J_{1}(t)$, $J_{3}=J_{3}(t)$ and $J_{2}=J_{2}(t)$, $J_{4}=J_{4}(t)$, respectively.

3.2. Estimate of the time-dependent function $J_{1}$

In this step we follow [19]. Multiplying the equation (1.1)2 by the function $\psi_{2}$ and integrating it over $[0,\eta]\times\mathbb{R}^{n}$, we obtain

[TABLE]

where we used $\psi_{2}(t,x)\geq 0$ and the equation (3.2). We now define the time-dependent function

[TABLE]

Thus,

[TABLE]

where we used the relation $\partial_{\eta}\psi_{2}(\eta,x)=-\psi_{2}(\eta,x)$. The assumption on the initial data implies

[TABLE]

Then, multiplying (3.3) by $e^{2\eta}$ and integrating it over $[0,t]$, we obtain

[TABLE]

with a positive constant $C_{0}$. We immediately conclude $J_{1}(t)\geq C_{0}^{p}$.

3.3. Estimate of the time-dependent function $J_{3}$

Noting that $\psi_{1}(t,x)\geq 0$ we multiply the equation (1.1)1 by the function $\psi_{1}$ and integrate it over $[0,\eta]\times\mathbb{R}^{n}$ to derive

[TABLE]

Here we define the time-dependent function

[TABLE]

By using the equation (3.1) and the relation

[TABLE]

we have

[TABLE]

Applying Gronwall’s inequality to (3.4) implies

[TABLE]

with a positive constant $C_{1}$, where we used again the assumption for the initial data. We can immediately conclude $J_{3}(t)\geq C_{1}^{q}$.

3.4. Estimate of the time-dependent functions $J_{2}$ and $J_{4}$

For one thing, we may use the asymptotic behavior of $\phi$ to get

[TABLE]

with a positive constant $C_{2}$.

For another, by using asymptotic behavior of $\phi$ again we obtain

[TABLE]

with a positive constant $\widetilde{C}_{2}$.

It shows that

[TABLE]

Summarizing the derived estimates from the above subsections concludes

[TABLE]

3.5. Lower bound for $F_{1}$

Let us derive a lower bound for $F_{1}=F_{1}(t)$ by using (3.5). Integrating (3.5) over $[0,t]$ gives

[TABLE]

for large time $t\geq t_{0}$, where we used our assumption

[TABLE]

Then, we use Gronwall’s inequality to (3.7) to find

[TABLE]

The use of integration by parts implies

[TABLE]

where we applied the following facts:

[TABLE]

and

[TABLE]

for $t\geq t_{1}>t_{0}$, where $t_{1}$ is sufficiently large. So, we immediately get for $t\geq t_{1}$ the estimate to below

[TABLE]

where

[TABLE]

3.6. Lower bound for $F_{2}$

To get a lower bound for $F_{2}=F_{2}(t)$, we only need to integrate twice with respect to $t$ the inequality (3.6). In this way we obtain

[TABLE]

for $t\geq t_{2}$. Under our assumption $1<q<\frac{2n}{n-1}$ if $n\geq 2$ the following estimate holds for $t\geq t_{3}>t_{2}$:

[TABLE]

Here $t_{3}$ is a sufficiently large positive constant. In conclusion, taking $T_{0}=\max\{t_{1};t_{3}\}$ we derived all desired estimates. The proof is complete.

4. Proof of Lemma 2.2

Let us describe some properties for the functions $F_{1}=F_{1}(t)$ and $F_{2}=F_{2}(t)$. From (2.10), (2.9) and (2.6) we obtain $\frac{d^{2}F_{2}}{dt^{2}}(t)>0$ and $F_{2}(t)>0$ for all $t\geq T_{0}$. Thus, $F_{2}$ is a convex function for $t\geq T_{0}$, which implies that there exists $T_{1}\geq T_{0}$ such that $\frac{dF_{2}}{dt}(t)>0$ for $t\geq T_{1}$. Similarly, from (2.7) and (2.6) we know $\frac{dF_{1}}{dt}(t)+F_{1}(t)>0$ and $F_{1}(t)>0$ for all $t\geq T_{0}$. Let us apply Gronwall’s inequality to (2.8) together with (2.9) for we also have

[TABLE]

for $t\geq T_{2}>T_{0}$. All in all, we have

[TABLE]

for large time $t\geq T_{3}:=\max\{T_{1},T_{2}\}$.

To prove Lemma 2.2, we discuss the finite time blow-up of the time-dependent function $F_{2}$ in Subsection 4.1, and the finite time blow-up of the time-dependent function $F_{1}$ in Subsection 4.2.

4.1. Finite time blow-up of $F_{2}=F_{2}(t)$

First of all, multiplying (2.8) by $\frac{dF_{2}}{dt}(t)$ we obtain

[TABLE]

Applying integration by parts leads to

[TABLE]

Next, we integrate (4.1) over $[T_{3},t]$ to have

[TABLE]

due to the monotonically increasing property of $F_{2}$, where $t\geq T_{4}>T_{3}$ with a sufficiently large $T_{4}$.

Then, we multiply the above inequality by $e^{t}\frac{dF_{2}}{dt}$ to get

[TABLE]

Similarly, we know from integration by parts

[TABLE]

Again by using the monotonically increasing behavior of $F_{2}$ for $t\geq T_{5}>T_{4}$, integrating (4.3) over $[T_{4},t]$ again shows

[TABLE]

Thus, we derived the following lower bound estimate for $F_{1}(t)$:

[TABLE]

Plugging (4.4) into (2.10) implies

[TABLE]

Multiplying the above inequality by $\frac{dF_{2}}{dt}$ once more, we have

[TABLE]

where

[TABLE]

Therefore, combining

[TABLE]

with

[TABLE]

for $t\geq T_{6}>T_{5}$, the following estimate holds:

[TABLE]

Here $T_{6}$ is a large positive constant.

To prove our desired statements, we shall distinguish between two cases.

4.1.1. The condition $\beta_{2}+\alpha_{2}q<\beta_{1}(pq-1)+2(q+1)$ holds.

The estimate (2.9) shows that

[TABLE]

for a constant $\epsilon>0$ to be determined later. Then, plugging (4.6) into (4.5) yields

[TABLE]

Obviously, our assumption

[TABLE]

can be rewritten by

[TABLE]

which means that

[TABLE]

Let us consider the auxiliary initial value problem

[TABLE]

with $\kappa>0$ and $\nu>0$. The solution of (4.8) is given by

[TABLE]

It is clear that if $-\alpha>-1$ and $\beta>1$. Then the solution of (4.8) blows up when $Y(T_{6})$ is large. According to the Petrovitsch theorem, we conclude that $F_{2}=F_{2}(t)$ blows up in finite time.

4.1.2. The condition $\beta_{2}+\alpha_{2}q=\beta_{1}(pq-1)+2(q+1)$ holds.

In this case we choose a positive constant $\delta$ such that \delta\in\big{(}0,\big{(}\frac{\beta_{2}+\alpha_{2}q}{2(q+1)}-1\big{)}\frac{1}{\beta_{1}}\big{)}. Then we multiply (4.5) by $(F_{2}(t))^{-1-\delta}$ to get immediately

[TABLE]

where we used our estimate (2.9) and our basic assumption in this case which reads as follows:

[TABLE]

Thus, we integrate (4.9) over $[T_{6},t]$ to get

[TABLE]

where the constant

[TABLE]

is of course independent of $T_{6}$. In other words, we have

[TABLE]

Due to our assumption for a large value of $F_{2}(T_{6})$, the function $F_{2}=F_{2}(t)$ blows up in finite time.

4.2. Finite time blow-up of $F_{1}=F_{1}(t)$

Let us multiply (2.10) by $\frac{dF_{1}}{dt}(t)+F_{1}(t)$ to get

[TABLE]

By using integration by parts we derive

[TABLE]

It is clear that for $t\geq T_{7}>T_{3}$ we have

[TABLE]

where we used the monotonically increasing property of $F_{1}=F_{1}(t)$. Hence,

[TABLE]

Due to the fact that

[TABLE]

we may multiply (4.10) by $\frac{dF_{1}}{dt}(t)+F_{1}(t)$ and integrate it over $[T_{7},t]$ to obtain

[TABLE]

where we used

[TABLE]

and

[TABLE]

for $t\geq T_{8}>T_{7}$. Furthermore, plugging (4.11) into (2.8) yields

[TABLE]

Finally, we multiply the above inequality by $\frac{dF_{1}}{dt}(t)+F_{1}(t)$ and integrate it over $[T_{8},t]$ to obtain

[TABLE]

where we used for $t\geq T_{9}>T_{8}$

[TABLE]

To prove our desired lemma, we shall distinguish between two cases.

4.2.1. The condition $\alpha_{2}+\beta_{2}p<\alpha_{1}(pq-1)+2(p+1)$ holds.

Applying our derived estimate (2.6) we have

[TABLE]

with a positive constant $\epsilon>0$ to be determined later. From our assumption, we know

[TABLE]

which implies

[TABLE]

Let us consider the Cauchy problem

[TABLE]

that is an auxiliary initial value problem to (4.12). The solution to (4.13) is explicitly given by

[TABLE]

Thus, if $-\alpha>-1$ and $\beta>1$, then the solution $Z=Z(t)$ blows up when $Z(T_{9})$ is large. According to the Petrovitsch theorem, we conclude that $F_{1}=F_{1}(t)$ blows up in finite time, too.

Remark 4.1**.**

If we drop the assumption on large value of $F_{1}(t_{0})$ in the theorem, this part cannot be completed due to (4.12). It seems to be difficult to avoid $e^{-t}$ in the last step of the proof.

4.2.2. The condition $\alpha_{2}+\beta_{2}p=\alpha_{1}(pq-1)+2(p+1)$ holds.

Similar as in Section 4.1.2, we multiply $(F_{1}(t))^{-1-\delta}$ with \delta\in\big{(}0,\big{(}\frac{\alpha_{2}+\beta_{2}p}{2(p+1)}-1\big{)}\frac{1}{\alpha_{1}}\big{)} to (4.12) and apply the estimate (2.6) to get

[TABLE]

where we used our condition

[TABLE]

Thus, by direct computation we have

[TABLE]

Due to our assumption for a large value of $F_{1}(T_{9})$, the function $F_{1}=F_{1}(t)$ blows up in finite time.

5. Proof of our main results

5.1. Proof of Theorem 2.1

For the cases $n=1,2$, one can see [16, 18]. We may apply the Gagliardo-Nirenberg inequality and Banach’s fixed-point theorem to prove that there exists a uniquely determined solution $(u^{*},v^{*})\in X(T)\times X(T)$ for a positive $T$, where the evolution space $X(T)$ is defined by

[TABLE]

Here we only need to restrict to $1<p,q<\infty$.

For the remaining cases $n\geq 3$, one can combine the proofs stated in [5, 15, 16, 18]. We may apply the embedding theorem, the Gagliardo-Nirenberg inequality and Banach’s fixed-point theorem to prove that there exists a uniquely determined solution $(u^{*},v^{*})\in X(T)\times Y(T)$ for a positive $T$, where the evolution space $Y(T)$ is defined by

[TABLE]

On the one hand, the restriction of the exponent $p$ to $1<p\leq\frac{n+3}{n-1}$ comes from the estimate

[TABLE]

with $c>0$. On the other hand, the restriction of the exponent $q$ to $1<q\leq\frac{n}{n-2}$ for $n\geq 3$ comes from the application of the Gagliardo-Nirenberg inequality.

5.2. Proof of Theorem 2.2

Let us choose

[TABLE]

in Lemma 2.2, we immediately get from (2.4) and (2.5), respectively,

[TABLE]

Then, the proof of Theorem 2.2 is completed.

Acknowledgments

The PhD study of Mr. Wenhui Chen are supported by Sächsisches Landesgraduiertenstipendium.