Global existence and blow up of solutions for a class of coupled parabolic systems with logarithmic nonlinearity

1.   Introduction

In this paper, we consider the following initial-boundary value problem for a class of coupled parabolic systems with logarithmic nonlinearity.

where $(u_0, v_0)\in H^1_0(\Omega)\times H^1_0(\Omega)$, $T\in(0, +\infty)$, $\Omega\subset R^n (n\geq2)$ is a bounded domain with smooth boundary $\partial\Omega$ and $p$ satisfies the following assumptions:

Among the fields of mathematical physics, biosciences and engineering, problem (1.1) is one of the most important reaction-diffusion coupled systems with logarithmic nonlinearity. It can be used not only to predict the time evolution of various population density distributions, but also to describe the thermal propagation of a two-component combustible mixture ^[1,2,3]. In recent years, this kind of systematic research has attracted many mathematicians and has made remarkable progress ^[4,5,6,7,8]. In order to overcome the special difficulties brought by nonlinear terms, many new ideas and tools have been developed, which greatly enrich the theory of partial differential equations ^{[9,10,11,12,13,14]}.

In the past years, many authors made efforts to the investigation of the existence and blow up of solutions for such kinds of systems. Galaktionov et al. ^[15,16] investigated the following semilinear reaction-diffusion system

They proved the local and global existence of solutions for the initial boundary value problem of (1.3). Subsequently, Escobedo and Herrero ^[17] considered the initial boundary value problem of (1.3) for a bounded open domain on $R^n$ with smooth boundary. They obtained global solution under the condition $0 < pq\leq 1$, meanwhile global solution and blow up in finite time depending on sufficient small or large initial value and $pq > 1$. For more studies on problem (1.3) we refer the interested reader to ^[18,19,20] and references therein.

Recently, Xu et al. ^[21] considered the following nonlinear reaction-diffusion systems

When initial energy $J(u_0, v_0)\leq d$, by virtue of Galerkin method ^[22] and concave function method ^[23], global existence and finite time blow-up of the solutions for the problem (1.4) were obtained. When initial energy $J(u_0, v_0) > d$, they discussed global existence, finite time blow-up of solutions and tried to find out the corresponding initial data with arbitrarily high initial energy. What's more, by using comparison principle and the ideas in ^[24,25], they described the structures of the initial data and gave some sufficient conditions of the initial data which ensured the finite time blow up and global existence of the solutions, respectively.

Inspired by the above works, we aim to use the Galerkin method, logarithmic inequalities ^[26], and concave function method to prove the global existence, decay, finite time blow-up of solutions for problem (1.1) with initial energy $J(u_0, v_0)\leq d$. When high initial energy $J(u_0, v_0) > d$, by constructing two sets $\Phi_\alpha$ and $\Psi_\alpha$ defined as (5.1) and (5.2), we prove that the weak solution will blow up in finite or infinite time if the initial value belongs to $\Psi_\alpha$, while the weak solution will exist globally and tends to zero as time $t\rightarrow +\infty$ when the initial value belongs to $\Phi_\alpha$.

The organization of the remaining part of this paper is as follows. In Section 2, we introduce some preliminaries and lemmas of this paper. In Sections 3–5, we will give our main results and the corresponding proofs.

2.   Preliminary

Throughout this paper, we denote by $\|u\|_{\gamma}$ the norm of $L^{\gamma}(\Omega)$ for $1\leq \gamma \leq +\infty$ and by $\|u\|_{H_0^1(\Omega)}$ the norm of $H_0^{1}(\Omega)$. For $u \in L^\gamma(\Omega)$,

and for $u \in H_0^1(\Omega)$,

By virtue of Poincaré inequality, we know that $\|u\|^2_{H_{0}^{1}(\Omega)}$ and $\|\nabla u\|_{2}^{2}$ are equivalent norms to each other, i.e., there exist $C_1$ and $C_2$ such that

which is denoted by $\|u\|_{H_{0}^{1}(\Omega)}^{2} \simeq\|\nabla u\|_{2}^{2}$. In addition, we denoted by $(\cdot, \cdot)$ the inner product in $L^2(\Omega)$ and $c$ is an arbitrary positive number which may be different from line to line.

For $(u, v) \in H_0^1(\Omega)\times H^1_0(\Omega)$, we define the Nehari functional $I$ and energy functional $J$ as follows:

From (2.1) and (2.2), we have

Let

be the Nehari manifold. Furthermore, the potential well $W$ and its corresponding set $V$ are defined respectively by

where

is the depth of the potential well $W$.

To consider the weak solution with high energy level, we need to introduce some new notions.

and

Clearly, $\lambda_\alpha$ is nonincreasing, and $\Lambda_\alpha$ is nondecreasing with respect to $\alpha$, respectively.

Now, we give the definitions of the weak solution, maximal existence time and finite time blow up of the problem (1.1) as follows.

Definition 1. (Weak solution) We say that $(u, v)$ = $(u(x, t), v(x, t))\in L^\infty([0, T), H_0^1(\Omega)\times H_0^1(\Omega))$ with $(u_t, v_t)\in L^2([0, T), L^2(\Omega)\times L^2(\Omega))$ is a weak solution of problem (1.1) on $\Omega \times [0, T)$, if it satisfies the initial condition $u(x, 0) = u_0(x), v(x, 0) = v_0(x)$ in $H_0^1(\Omega)$,

and

for all $w_1, w_2\in H_0^1(\Omega)$ and $t\in(0, T)$. Moreover, for all $t\in (0, T)$, we have

Remark 1. For the global weak solution $(u(t), v(t)) = (u(x, t), v(x, t))$ of problem (1.1), we define the $\omega$-limit set of $(u_0, v_0)$ by

Definition 2. (Maximal existence time) Let $(u, v) = (u(x, t), v(x, t))$ be a weak solution of problem (1.1). We define the maximal existence time of $(u, v)$ as follows

(i) If $(u, v)$ exists for all $t\in[0, +\infty)$, then $T = +\infty$.

(ii) If there exists a $t_0\in(0, +\infty)$ such that $(u, v)$ exists for $0\leq t < t_0$, but it does not exist at $t = t_0$, then $T = t_0$.

Definition 3. (Finite time blow-up) Let $(u(t), v(t)) = (u(x, t), v(x, t))$ be a weak solution of problem (1.1). We say $(u(t), v(t))$ blows up in finite time if the maximal existence time $T$ is finite and

Lemma 1. Let $(u, v) \in H_0^1(\Omega)\times H^1_0(\Omega) \backslash \{(0, 0)\}$, then the following hold

(i) $\lim _{\lambda \rightarrow 0} J(\lambda u, \lambda v) = 0$, $\lim _{\lambda \rightarrow +\infty} J(\lambda u, \lambda v) = -\infty$.

(ii) There exists a unique $\lambda_* > 0$ such that $\left.\frac{d}{d \lambda} J(\lambda u, \lambda v)\right|_{\lambda = \lambda_{*}} = 0$.

(iii) $J(\lambda u, \lambda v)$ is increasing on $(0, \lambda_*)$, decreasing on $(\lambda_*, +\infty)$, and attains the maximum at $\lambda = \lambda_*$.

(iv) $I(\lambda u, \lambda v) > 0 \ \mathit{{for}} \ 0 < \lambda < \lambda_*$, $I(\lambda u, \lambda v) < 0 \ \mathit{{for}} \ \lambda_* < \lambda < +\infty$, and $I(\lambda_* u, \lambda_* v) = 0.$

Proof. (i) By definition of $J(u, v)$ and $\lambda > 0$, we have

Thus $\lim _{\lambda \rightarrow 0} J(\lambda u, \lambda v) = 0$, $\lim _{\lambda \rightarrow +\infty} J(\lambda u, \lambda v) = -\infty.$

(ii) Differentiating $J(\lambda u, \lambda v)$ with respect to $\lambda$, we get

Setting $g(\lambda) = \|u\|_{H_{0}^{1}(\Omega)}^{2}+\|v\|_{H_{0}^{1}(\Omega)}^{2}-2 \lambda^{2 p-2} \log \lambda^{2}\|u v\|_{p}^{p}-2 \lambda^{2 p-2} \int_{\Omega}|u v|^{p} \log (|u v|)dx$, we have $\lim_{\lambda \rightarrow 0} g(\lambda) = \|u\|_{H_{0}^{1}(\Omega)}^{2}+\|v\|_{H_{0}^{1}(\Omega)}^2 > 0, \ \lim_{\lambda \rightarrow +\infty} g(\lambda) = -\infty,$ and

Thus there exists a unique $\lambda_* > 0$ such that $g(\lambda_*) = 0$, i.e., $\left.\frac{d}{d \lambda} J(\lambda u, \lambda v)\right|_{\lambda = \lambda_{*}} = 0.$

(iii) It is easy to find that $J(\lambda u, \lambda v)$ is strictly increasing on $(0, \lambda_*]$, strictly decreasing on $(\lambda_*, +\infty)$ and taking the maximum at $\lambda = \lambda_*.$

(iv) Since

then the conclusion follows immediately.

Lemma 2. Assume (1.2) holds, let $(u, v) \in H_0^1(\Omega) \times H_0^1(\Omega)$ satisfy $I(u, v) < 0$, then

Proof. According to $I(u, v) < 0$ and Lemma 1, we have $(u, v)\neq(0, 0)$ and there exists a $\lambda_* \in (0, 1)$ such that $I(\lambda_*u, \lambda_*v) = 0$, i.e., $J(\lambda_*u, \lambda_*v)\geq d.$ For $\lambda > 0$, set

then

Hence $h(\lambda)$ is strictly increasing for $\lambda > 0$. Together with $\lambda_* \in (0, 1)$, it follows that $h(1) > h(\lambda_*)$. i.e.,

Then (2.6) follows immediately.

Lemma 3. Assume (1.2) holds, let $(u_0, v_0) \in H_0^1(\Omega) \times H_0^1(\Omega)$ and $(u(t), v(t)) = (u(x, t), v(x, t))$ be a weak solution of problem (1.1). If $J(u_0, v_0) < d$ and $I(u_0, v_0) < 0$, then $(u(t), v(t)) \in V$ for all $0\leq t \leq T$, where $T$ is the maximal existence time of ($u(t), v(t)$).

Proof. We will show that $(u(t), v(t))\in V$ for $0 \leq t\leq T$. Arguing by contradiction, suppose that $t_0 \in [0, T]$ be the smallest time for which $(u(t_0), v(t_0))\notin V$, then by the continuity of the $(u(t), v(t))$, we get $(u(t_0), v(t_0)) \in \partial V$. Hence, it follows that

or

If (2.7) is true, then $(u(t_0), v(t_0))\in \mathcal{N}$, $J(u(t_0), v(t_0)) > d$, which contradicts with (2.5). While if (2.8) is true, it also contradicts with (2.5). Consequently, we have $(u(t), v(t)) \in V$ for all $0\leq t \leq T.$

Lemma 4. Let (u, v) be a weak solution of problem (1.1). Then for all $t \in [0, T)$,

Proof. The proof of Lemma 4 directly follows by choosing $w_1 = u, w_2 = v$ in Definition 1.

3.   Low initial energy $J(u_0, v_0) < d$

In this section, we prove global existence and finite time blow up of solutions for problem (1.1) with the initial energy $J(u_0, v_0) < d$.

Theorem 1. Assume $(u_0, v_0)\in H_0^1(\Omega)\times H_0^1(\Omega)$ and (1.2) hold. If $J(u_0, v_0) < d$ and $I(u_0, v_0)\geq0$, then the problem (1.1) has a global solution $(u(t), v(t)) \in L^{\infty}((0, \infty); H_0^1(\Omega)\times H_0^1(\Omega))$ with $(u_t(t), v_t(t)) \in L^2((0, \infty); L^2(\Omega)\times L^2(\Omega))$ and $(u(t), v(t)) \in W$ for $0\leq t < \infty.$ Furthermore, if $I(u_0, v_0) > 0$, then there exists a $c > 0$ such that $\|u\|_{2}^{2}+\|v\|_{2}^{2}\leq (\|u_0\|_{2}^{2}+\|v_0\|_{2}^{2})e^{-2ct}$.

Proof. Since we know $J(u_0, v_0) < d$ and $I(u_0, v_0)\geq 0$, then it follows that

(i) If $0 < J(u_0, v_0) < d$ and $I(u_0, v_0)\geq 0$, then we have $I(u_0, v_0) > 0$. In fact, if $I(u_0, v_0) = 0$, then by the definition of $d$ in (2.4), we have $J(u_0, v_0)\geq d$, which is a contradiction.

(ii) If $J(u_0, v_0) = 0$ and $I(u_0, v_0)\geq 0$, then we obtain $(u_0, v_0) = (0, 0).$ In fact, if $(u_0, v_0)\neq (0, 0)$, then by the (2.3), we have $\frac{p-1}{2p}\Big(\|u_0\|_{H_0^1(\Omega)}^2+\|v_0\|_{H_0^1(\Omega)}^2\Big)+\frac{1}{p^2}\|u_0v_0\|_p^p < 0,$ which is also a contradiction.

(iii) If $J(u_0, v_0) < 0$ and $I(u_0, v_0)\geq 0$, then it is contradictive with (2.3).

From the discussions above, we consider the case $0 < J(u_0, v_0) < d$ and $I(u_0, v_0) > 0$. It is widely know that there is a basis $\{\omega_j(x)\}_{j = 1}^\infty$ of $H_0^1(\Omega)$ such that $\omega_j$ is an eigenfunction of the Laplacian operator corresponding to the eigenvalue $\lambda_ j$ and

Hence, we choose $\{\omega_j(x)\}_{j = 1}^\infty$ as the Galerkin basis for $-\Delta$ in $H_0^1(\Omega)$. Then we construct the Galerkin approximate solution $(u_m(x, t), v_m(x, t))$ of the problem (1.1),

which satisfy, for $j = 1, 2, \cdots, m$,

and

with initial condition $u_m(x, 0) = u_{0m}$, $v_m(x, 0) = v_{0m}$, where $u_{0m}$ and $v_{0m}$ are chosen in span $\{ \omega_1, \omega_2, \cdot \cdot \cdot, \omega_m \}$ so that

and

According to the standard ordinary differential equation theory, the system (3.1)–(3.4) admit a solution

where $T_0$ is the minimum of the existence time of $g_{jm}(t)$ and $h_{jm}(t)$ for each $m$. Thus $(u_m(x, t), v_m(x, t)) \in C^1\Big([0, T_0);H_0^1(\Omega)\times H_0^1(\Omega)\Big)$.

Next, multiplying (3.1) and (3.2) by $g'_{jm}(t)$ and $h'_{jm}(t)$, respectively, summing for $j$ from 1 to $m$, integrating with respect to $t$ from 0 to $t$ and adding these two equations, we get

From $J(u_0, v_0) < d$ and (3.3)–(3.4), we see that $J(u_{0m}, v_{0m}) < d$ for sufficiently large $m$. Then we get from (3.5) that

for sufficiently large $m$.

By (3.3) and (3.4) and $(u_0, v_0)\in W$, we know that $(u_{0m}, v_{0m})\in W$ for large enough $m$. Next, we prove $(u_m(x, t), v(x, t)) \in W$ for large enough $m$ and $0\leq t < T_{0}$. If it is false, then there exists $t_0 \in(0, T_0)$ such that $(u_m(x, t_0), v_m(x, t_0)) \in \partial W$, then $I(u_m(t_0), v_m(t_0)) = 0$ and $(u_m(t_0), v_m(t_0))\neq(0, 0)$, or $J(u_m(t_0), v_m(t_0)) = d$.

By (3.6), $J(u_m(t_0), v_m(t_0)) = d$ is not true. On the other hand, if $I(u_m(t_0), v_m(t_0)) = 0$ and $(u_m(t_0), v_m(t_0))\neq (0, 0)$, then by the definition of $d$, we have $J(u_m(t_0), v_m(t_0))\geq d$, which is also contradiction with (3.6). So $(u_m(x, t), v_m(x, t)) \in W$ for large enough $m$ and $0\leq t < T_{0}$.

From the fact $\Big(u_m(x, t), v_m(x, t)\Big) \in W$ for large enough $m$, (3.6) and

we obtain

for sufficiently large $m$, which gives

By (3.10), we know that $T_0 = +\infty$. Then by (3.8)–(3.10), there exist $u, v$ with theirs subsequences of $\{u_m\}_{j = 1}^{+\infty}$ and $\{v_m\}_{j = 1}^{+\infty}$, such that, as $m\rightarrow +\infty$,

Then it follows Aubin-lions compactness theorem ^[27] that

Clearly, this implies that

Furthermore, we get

On the other hand,

where $\sigma = \frac{(p+r)p}{p-1}$, $e = \frac{(p-1+r)p}{p-1}$, and since $|x^{q-1}\log x|\leq(e(q-1))^{-1}$ for $0 < x < 1$ while $x^{-\mu}\log x\leq (e\mu)^{-1}$ for $x\geq1, \mu > 0$. Choosing a positive real number $r$ so that $0 < 2e < 2^*$ and $0 < 2\sigma < 2^*$, and similar to the proof (3.17), we have

Hence, from (3.15)–(3.18) and Lion's Lemma(see ^[27], Lemma 1.3, p.12), we have

In view of (3.11)–(3.14) and (3.19), (3.20), for $j$ fixed, we can pass to the limit in (3.1) and (3.2) to get

and

for a.e., $t\in (0, +\infty)$. Since $\{\omega_j(x)\}_{j = 1}^\infty$ is the basis in $H_0^1(\Omega)$, we have

and

for any $w_1, w_2 \in H_0^1(\Omega)$ and a.e., $t \in (0, +\infty)$.

Fixing any $t \in (0, +\infty)$ and integrating (3.21) and (3.22) from 0 to $t$, we get

and

Similarly, integrating (3.1) and (3.2) from 0 to $t$, and passing to the limit, we get

and

From (3.23)–(3.26), we get $u(0) = u_0$ and $v(0) = v_0$.

According to (3.3), (3.4), (3.7), (3.11)–(3.14), (3.19), (3.20) and since the norm is weakly lower semicontinuous, we know that the energy inequality (2.5) holds. Then from Definition 2, $(u(t), v(t))$ is a global weak solution and $(u(t), v(t))\in W$.

Next, we will prove the algebraic decay of the global solution $u(x, t)$. Combining (2.3), (2.5) and $(u(t), v(t))\in W$, we have

As $I(u, v) < 0$, then there exists a $\lambda_* \in (0, 1)$ such that $I(\lambda_*u, \lambda_*v) = 0$. Furthermore, we get

It follows from (3.27) and (3.28) that

Due to the $I(\lambda_*u, \lambda_*v) = 0$, we have

i.e.,

Combining (3.29) with (3.30), we have

By the emdedding $H_0^1(\Omega)\hookrightarrow L^2(\Omega)$, we have

On the other hand, by Lemma 4, we know

Combining this equality with (3.31), we get

By Grönwall's inequality, we have

The proof of Theorem 1 is complete.

Theorem 2. Assume $(u_0, v_0) \in H_0^1(\Omega) \times H_0^1(\Omega)$ and (1.2) hold. If $J(u_0, v_0) < d$ and $I(u_0, v_0) < 0$, then the weak solution $(u(x, t), v(x, t))$ of the problem (1.1) blows up in finite time, i.e., there exists a $T > 0$ such that

Proof. Step 1: Blow-up in finite time

By contradiction, we suppose that $(u(t), v(t))$ is global weak solution of problem (1.1), then $T_{max} = +\infty$. Let

then

and

From (3.32) and energy inequality (2.5), it follows that

where the constant $c$ is from the Poincaré inequality $\|u\|_2^2\leq c\|\nabla u\|_2^2.$

Note that

then

Hence by (3.33) and (3.34) we know that

By Schwartz inequality, we have

It implies that

From Lemma 3 we have $I(u(t), v(t)) < 0$ for $0\leq t < +\infty$. Thus from Lemma 2 one has

Combing (3.36) and (2.5) we get

and

Hence for sufficiently large $t$, we have

Combining (3.35) with (3.37), we obtain

for sufficiently large $t$. Note that

It follows that there exists a finite time $T > 0$ such that $\lim _{t \rightarrow T} G^{-(p-1)}(t) = 0,$ i.e., $\lim _{t \rightarrow T} \int_{0}^{t}\|u\|_{2}^{2}+\|v\|_{2}^{2} d \tau = +\infty.$

Step 2: Upper bound estimation of the blow-up time.

We next give an upper bound estimate of $T$. Suppose $(u(t), v(t))$ be a solution of problem (1.1) with initial value $(u_0, v_0)$ satisfying $I(u_0, v_0) < 0$ and $J(u_0, v_0) < d$. By Step 1, the maximal existence time $T < \infty$. By Lemma 3, we get $(u(t), v(t))\in V$, $\forall t\in [0, T)$, i.e., $I(u(t), v(t)) < 0, t\in[0, T)$. For $T_1 \in(0, T)$, we define the auxiliary functional $M:[0, T_1]\rightarrow R$ which is defined by

with $\beta > 0$ and $\gamma > 0$ specified later. Through a direct calculation, we have

and

It follows from Lemma 2 and (2.5) that

From (3.39) and Hölder inequality, we have

According to the inequality

by setting $x = \|u_\tau\|_2$, $y = \|v_\tau\|_2$, $z = \|u\|_2$, $w = \|v\|_2$ in (3.41), we get

By the Hölder inequality, we get

From (3.38), (3.40) and (3.42), we have

Restricting $\beta$ to satisfy

we have

Define $y(t): = M^{1-p}(t)$ for $t\in[0, T_1]$, then by $M(t) > 0, M'(t) > 0$ we get

for all $t\in[0, T_1]$. It follows from $y''(t) < 0$ that

where

Combining (3.44) and the above inequalities, we can deduce

Then by the definition of $M(t)$ and above inequality we have

Hence, letting $T_1 \rightarrow T$, we get

For any $\beta$ satisfying (3.43), let $\gamma$ be large enough such that

then (3.45) lead to

Let

then

where $\Phi = \{(\beta, \gamma): \beta, \gamma \; {\rm { satisfy \; (3.43) \;and \;(3.46) \;respectively}\}}.$

Since

i.e., $\rho(\beta, \gamma)$ is decreasing with respect to $\beta$. Then we have

where

and

It is easy to get that $\rho_1(\gamma)$ achieves its minimum at

and

Thus, we have

The proof of Theorem 2 is complete.

4.   Critical initial energy $J(u_0, v_0)=d$

In this section, we prove global existence and blow up at finite time of solutions for problem (1.1) with the initial energy $J(u_0, v_0) = d$.

Theorem 3. Assume $(u_0, v_0)\in H_0^1(\Omega)\times H_0^1(\Omega)$ and (1.2) hold. If $J(u_0, v_0) = d$ and $I(u_0, v_0)\geq0$, then the problem (1.1) has a global solution $(u(t), v(t)) \in L^{\infty}(0, +\infty; H_0^1(\Omega)\times H_0^1(\Omega))$ with $(u_t(t), v_t(t)) \in L^2(0, +\infty; L^2(\Omega)\times L^2(\Omega))$ and $(u(t), v(t)) \in \overline{W} = W\cup \partial W$ for $0\leq t < \infty.$

Proof. Since $J(u_0, v_0) = d$, then $(u_0, v_0)\neq(0, 0)$. Let $\lambda_m = 1-\frac{1}{m}, \ (u_{0m}, v_{0m}) = \lambda_m(u_0, v_0), \ m = 1, 2, \cdot\cdot\cdot,$ and consider the following problem:

By $I(u_0, v_0)\geq0$ and Lemma 1, there exists a unique $\lambda^*\geq 1$ such that $I(\lambda^*u_0, \lambda^*v_0) = 0$. Due to the $\lambda_m < 1 < \lambda^*$, we get $I(\lambda_m u_0, \lambda_m v_0) > 0, J(\lambda_m u_0, \lambda_m v_0) < J(u_0, v_0) = d$. From Theorem 1, it follows that for each $m$ problem (4.1) admits a global solution $(u_m(t), v_m(t))\in L^{\infty}(0, +\infty; H_0^1(\Omega)\times H_0^1(\Omega))$ with $(u_{mt(t)}, v_{mt}(t)) \in L^2(0, +\infty; L^2(\Omega)\times L^2(\Omega))$ with the initial data

Furthermore, we have $(u_m(t), v_m(t)) \in W$ for $0\leq t < +\infty$,

and

From (4.2) and

we obtain

The remainder of the proof is similar to that in the proof of Theorem 1.

The proof of Theorem 3 is complete.

Theorem 4. Assume $(u_0, v_0) \in H_0^1(\Omega) \times H_0^1(\Omega)$ and (1.2) hold. If $J(u_0, v_0) = d$ and $I(u_0, v_0) < 0$, then the weak solution $(u(x, t), v(x, t))$ of the problem (1.1) blows up in finite time, i.e., there exists a $T > 0$ such that

Proof. By contradiction, we suppose that $(u(t), v(t))$ is a global weak solution of problem (1.1), then $T_{max} = +\infty$. Let

Taking into account to (3.35) still holds, combining the fact $J(u_0, v_0) = d$, we have

From continuities of $J(u, v)$ and $I(u, v)$ with respect to $t$, we know that there exists a sufficient small $t_1\in(0, +\infty)$ such that $J(u(t_1), v(t_1)) > 0$ and $I(u, v) < 0$ for $0 < t < t_1$. By $(u_t, u)+(v_t, v) = -I(u, v)$, we have $(u_t, u)+(v_t, v) > 0$ and $\|u_t\|_2^2+\|v_t\|_2^2 > 0$ for $t\in[0, t_1].$ From (2.5), we have $0 < J(u(t_1), v(t_1))\leq d-\int_0^{t_1} (\|u_t\|_2^2+\|v_t\|_2^2)dt < d$. Hence we take $t = t_1$ as the initial time, and obtain $(u(t_1), v(t_1))\in V.$ From Lemma 3 we have $I(u(t), v(t)) < 0$ for $t_1\leq t < +\infty$. Thus from Lemma 2 one has

Combing (4.4) and (2.5) we get

and

Hence for sufficiently large $t$, we have

Combining (4.3) with (4.5), we get

for sufficiently large $t$. Then similar to the proof of Theorem 2, i.e., there exists a finite time $T > 0$ such that $\lim _{t \rightarrow T} \int_{0}^{t}\|u\|_{2}^{2}+\|v\|_{2}^{2} d \tau = +\infty.$

The proof of Theorem 4 is complete.

5.   High initial energy $J(u_0, v_0) > d$

In this section, we investigate the conditions that ensure the global existence or blow up of solution for problem (1.1) with the initial energy $J(u_0, v_0) > d$.

Theorem 5. For any $\alpha \in (d, +\infty)$, the following conclusions hold.

(i) If $(u_0, v_0)\in \Phi_{\alpha}$, then the solution of the problem (1.1) exists globally and $(u(t), v(t))\rightarrow (0, 0)$, as $t\rightarrow \infty$;

(ii) If $(u_0, v_0)\in \Psi_{\alpha}$, then the solution of the problem (1.1) blows up in finite or infinite time, where

and

Proof. (i) Assume that $(u_0, v_0) \in \Phi_\alpha$, then by the definition of $\Phi_\alpha$ and the monotonicity property of $\lambda_\alpha$, we have $(u_0, v_0) \in \mathcal{N_+}$, $d < J(u_0, v_0)\leq\alpha$ and

We claim that $(u(t), v(t)) \in \mathcal N_+$. By contradiction, there exists a $t_0 \in (0, T)$ such that $(u(t), v(t)) \in \mathcal N_+$ for $t \in [0, t_0)$ and $(u(t_0), v(t_0))\in \mathcal N$. By Lemma 4, we have

From the definition of $\mathcal N_+$ and (5.4), we know that $\|u\|_{2}^{2}+\|v\|_{2}^{2}$ is strictly decreasing on $[0, t_0)$. On the other hand, by (2.5), we know that $J(u, v)$ is nonincreasing with respect to $t$. Therefore, we have

From (5.3), we get

By $(u(t_0), v(t_0)) \in \mathcal{N}$ and (5.3), we get $(u(t_0), v(t_0)) \in \mathcal{N}_{J(u_0, v_0)}$. According to the definition of $\lambda_{J(u_0, v_0)}$, we have

which contradicts with (5.5) and prove the claim. Hence, we have $(u(t), v(t)) \in \mathcal N_+$ for all $t\in [0, T)$ and $(u(t), v(t))\in J^{J(u_0, v_0)}$, i.e., $(u(t), v(t))\in J^{J(u_0, v_0)}\cap \mathcal N_+$ for all $t\in [0, T)$. From the definition of $\mathcal N_\alpha$, we have $(\|u\|_{H_0^1(\Omega)}^2+\|v\|_{H_0^1(\Omega)}^2) < \frac{2p}{p-2}J(u_0, v_0), \ \forall t \in [0, T)$, so $T = +\infty$. It indicates that $(u(t), v(t))$ is bounded uniformly in $H_0^1(\Omega)\times H_0^1(\Omega)$. Hence, $\omega$-limit set is not an empty set.

Next, for any $(\omega, \varphi) \in \omega(u_0, v_0)$, by the above discussions, we get

According to first inequality, it implies that $(\omega, \varphi)\in J^{J(u_0, v_0)}$. According to the second inequality and the definition of $\lambda_{J(u_0, v_0)}$, we know that $(\omega, \varphi) \notin \mathcal N_{J(u_0, v_0)}$. Since $\mathcal N_{J(u_0, v_0)} = \mathcal N \cap J^{J(u_0, v_0)}$, we obtain $(\omega, \varphi) \notin \mathcal N$. Hence, $\omega(u_0, v_0)\cap \mathcal N = \phi$. As $\mathcal N$ include the nontrivial solutions of the problem (1.1), we have $\omega(u_0, v_0) = {(0, 0)}$, i.e., $(u(t), v(t))\rightarrow (0, 0)$, as $t\rightarrow \infty$.

(ii) If $(u_0, v_0)\in \Psi_{\alpha}$, by the definition of $\Psi_{\alpha}$, it is clear that $(u_0, v_0) \in \mathcal N_-$ and $d < J(u_0, v_0)\leq \alpha$. Combing with the monotonicity of $\Lambda_\alpha$, we get

We claim that $(u(t), v(t)) \in \mathcal N_-$ for $t \in [0, T)$. By contradiction, if there exists a $t_1 \in (0, T)$ such that $(u(t), v(t)) \in \mathcal N_-$ for $t \in [0, t_1)$ and $(u(t_1), v(t_1))\in \mathcal N$. By Lemma 4, we have

Then by the definition of $\mathcal N_-$, we deduce that $\frac{1}{2}(\|u\|_{2}^{2}+\|v\|_{2}^{2})$ is strictly increasing on $[0, t_1)$. It along with (2.5) yields

By $(u(t_1), v(t_1)) \in \mathcal{N}$ and (5.6), we get $(u(t_1), v(t_1)) \in \mathcal{N}_{J(u_0, v_0)}$. Hence, it follows from the definition of $\Lambda_{J(u_0, v_0)}$ that

which is incompatible with (5.6), so we get $(u(t), v(t))\in J^{J(u_0, v_0)}\cap \mathcal N_-$ for all $t\in [0, T)$.

Next, we assume that $(u(t), v(t))$ exists globally, i.e., $T = +\infty$. For every $(\omega, \varphi)\in \omega(u_0, v_0)$, by the above discussions, we get

According to first inequality, this shows $(\omega, \varphi)\in J^{J(u_0, v_0)}$. According to the second inequality and the definition of $\Lambda_{J(u_0, v_0)}$, we know that $(\omega, \varphi) \notin \mathcal N_{J(u_0, v_0)}$. Since $\mathcal N_{J(u_0, v_0)} = \mathcal N \cap J^{J(u_0, v_0)}$, we obtain $(\omega, \varphi) \notin \mathcal N$. Hence, $\omega(u_0, v_0)\cap \mathcal N = \phi$. However, since $dist(0, \mathcal N_-) > 0$, we also have $(0, 0)\notin \omega(u_0, v_0)$. Thus, $\omega(u_0, v_0) = \emptyset$, it contraries to the assumption that $(u(t), v(t))$ is a global solution, then $T < \infty$.

The proof of Theorem 5 is complete.

Acknowledgments

This research was supported by Guizhou Provincial Science and Technology Projects (No.Qiankehe foundation-ZK[2021]YIBAN317), and by the project of Guizhou Minzu University under (No.GZMU[2019]YB04), and central leading local science and technology development special foundation for Sichuan province, P.R. China under (No.2021ZYD0020).

Conflict of interest

The authors declare there is no conflict of interest.