On a singular parabolic $p$-Laplacian equation with logarithmic nonlinearity

1.   Introduction

In this paper, we are concerned with the following initial-boundary problem:

where $\Omega \subset {\mathbb{R}^N}({N \geq N_{\Omega}})$ is a bounded domain with smooth boundary $\partial \Omega$, ${\Delta _p}u = {\rm {div}}({\left| {\nabla u} \right|^{p - 2}}\nabla u)$ with ${u_0} \in W_0^{1, p}(\Omega)$, $x = \left({{x_1}, {x_2}, \ldots, {x_N}} \right) \in {\mathbb{R}^N}$ with $\left| x \right| = \sqrt {x_1^2 + x_2^2 + \cdots + x_N^2}$, and the parameters satisfy

As is well known, according to the law of conservation, many diffusion processes with reactions can be described by the following equation (see ^[1]):

where $u(x, t)$ stands for the mass concentration in chemical reaction processes or temperature in heat conduction, at position $x$ in the diffusion medium and time $t$. The function $D$ is called the diffusion coefficient or the thermal diffusivity, the term $\nabla \cdot (D\nabla u)$ represents the rate of change due to diffusion, and $f(x, t, u, \nabla u)$ is the rate of change due to reaction.

In the past few years, many researchers had focused on Equation (1.2). For more details, one can refer to ^[2,3,4,5,6]. For the source $f(x, t, u, \nabla u) = u^q$, there has already been much discussion. For example, for $D = \left|x\right|^2$, in 2004, Tan ^[7] considered the existence and asymptotic estimates of global solutions as well as finite time blow-up of local solutions based on the classical Hardy inequality ^[8]. Han ^[9] considered the blow-up properties of solutions to the following non-Newton filtration equation with special a medium void:

A new criterion for the solutions to blow up in finite time was established by using the Hardy inequality. Moreover, the upper and lower bounds for the blow-up time were also estimated. The results solved an open problem proposed by Liu ^[10] in 2016.

When the source $f(u)$ is a logarithmic nonlinearity, Deng and Zhou ^[11] investigated the following semilinear heat equation with singular potential and logarithmic nonlinearity

under an appropriate initial-boundary value condition. They did make full use of the logarithmic Sobolev inequality in ^[12,13] to handle the difficulty caused by the logarithmic nonlinear term $u \ln {\left|u\right|}$. Taking the combination of a family of potential wells, the existence of global solutions and infinite time blow-up solutions were obtained.

Liu and Fang ^[14] considered a fourth-order singular parabolic equation involving logarithmic nonlinearity and $p$-biharmonic operator

and they established the local solvability by the technique of cut-off combining with the methods of Faedo-Galerkin approximation and multiplier. Meantime, by virtue of the family of potential wells, they used the technique of modified differential inequality and the improved logarithmic Sobolev inequality to obtain the global solvability and the infinite and finite time blow-up phenomena, and derived the upper bound of blow-up time as well as the estimate of the blow-up rate. Furthermore, the results of blow-up with arbitrary initial energy and extinction phenomena were presented.

Motivated by these works, in this paper, we consider the Problem (1.1) with the presence of nonlinear diffusion $\triangle _p u : = {\rm div}(\left|\nabla u\right|^{p-2}\nabla u)$ and logarithmic nonlinearity $\left|u\right|^{q-2}u \ln \left|u\right|$. To the best our knowledge, this is the first work in the literature that takes into account a singular parabolic $p$-Laplacian equation with logarithmic nonlinearity.

The rest of this paper is organized as follows. In Section 2, we introduce some symbols and definitions. In Section 3, we prove the local existence and uniqueness theorem. In Section 4, we prove the global existence and asymptotic behavior theorems of solutions. In Section 5, the blow-up phenomena of solutions are discussed. Finally, the extinction phenomenon of the solution is given in Section 6.

2.   Preliminaries

In this section, we introduce some notations and lemmas that will be used throughout the paper. In what follows, we denote by $\left\|\cdot\right\|_r(r\geq 1)$ the norm in $L^r(\Omega)$ and by $(\cdot, \cdot)$ the $L^2(\Omega)$ inner product. When $p > 1, \; p\neq2,$ we use $W^{1, p}_0(\Omega)$ to denote the Sobolev space such that both $u$ and $\nabla u$ belong to $L^p(\Omega)$ for any $u\in W^{1, p}_0(\Omega)$, denote by $W^{-1, p^{\prime}}(\Omega)$ its dual space, and by $\langle\cdot, \cdot\rangle$ the duality pairing between them. We will equip $W_{0}^{1, p}(\Omega)$ with the norm $\left\|u\right\|_{W_0^{1, p}} = \left\|\nabla u\right\|_{p}$, which is equivalent to the full one due to the Poincarés inequality. We use $\lambda_1 > 0$ to denote the first eigenvalue of $-\triangle$ in $\Omega$ under the homogeneous Dirichlet boundary condition. We also use notation $X_0$ to denote $W_0^{1, p}(\Omega) \backslash \left\{0\right\}$.

Due to the presence of the inverse coefficient ${\left|x\right|^{-s}}$, it is worth emphasizing the difference between the two cases when $0 \in \Omega$ and $0 \notin \Omega$.

If $0 \in \Omega$, then ${\left|x\right|^{-s}}$ develops a singularity. This necessitates the use of the Hardy-Sobolev inequality, which is valid for $N_{\Omega} \geq 3$, in the proofs of our main results.

On the other hand, if $0 \notin \Omega$, then there is no singularity and (1.1) can be regarded as a slight extension. In this case, our results are valid for all $N \in \left\{ {1, 2, 3, \ldots } \right\}$. To deal with these two cases simultaneously, we employ the notation

First, for Problem (1.1), we introduce the potential energy functional

and the Nehari functional

By a direct computation,

By $I(u)$ and $J(u)$, we define the potential well:

and the Nehari manifold

The depth of the potential well is defined as

The solution $u(x, t)$ to Problem (1.1) is considered in weak sense as follows. Sometimes $u(x, t)$ will be simply written as $u(t)$ if no confusion arises.

Lemma 2.1. ^[15] Let $\mu$ be a positive number. Then we have the following inequalities:

Lemma 2.2. ^[15,16] Assume that $q < \frac{Np}{N-p}$, i.e., $q < \infty$ for $N \leq p$ and $r\leq q < \frac{Np}{N-p}$ for $N > p$ and $r \geq 1$. Then for any $u\in W_0^{1, p}(\Omega)$, it holds that

where $\theta \in (0, 1)$ is determined by $\theta = \left({\frac{1}{r} - \frac{1}{q}} \right){\left({\frac{1}{N} - \frac{1}{p} + \frac{1}{r}} \right)^{ - 1}}$ and the constant $C_G > 0$ depends on $N, p, q$, and $r$.

Remark 2.1. From $p > \frac{2N}{N+2}$, we deduce

Then by the Sobolev inequality, we have $W_0^{1, p}(\Omega)\hookrightarrow L^{q+a}(\Omega)$ for $p > 1$ and $\forall\; a\geq0$.

Lemma 2.3. Let $u(t) \in X_0$ and $p, q$ satisfy $\max\left\{\frac{2N}{N+2}, 1\right\} < p \leq q < p\left(1 +\frac{2}{N}\right).$ We have the following statements:

$(i)$ If $0 < \left\| u\right\|_p \leq r$, then $I(u) \geq 0$;

$(ii)$ If $I(u) < 0$, then $\left\| u\right\|_p > r$;

$(iii)$ If $I(u) = 0$, then $\left\|u\right\|_p = 0$ or $\left\|u\right\|_p \geq r$,

where

Proof. (ⅰ) A direct computation yields

Then, by the definition of $I(u)$, we have

where $B_*$ is the imbedding constant for $W_0^{1, p}(\Omega)\hookrightarrow L^{q+\alpha}(\Omega)$. If $0 < \left\|\nabla u\right\|_p \leq r$, this implies that $\left\| {\nabla u} \right\|_p^{q + \alpha - p} \le \frac{1}{{B_*^{q + \alpha}}}$. Therefore, we gain $I(u) \geq 0$ by (2.5).

(ⅱ) From (2.5) and $I(u) < 0$, we can see that

which means that

(ⅲ) If ${I}\left(u \right) = 0$, then from (2.5) we attain

The prove is complete. □

Next, in Lemma 2.4, we describe some basic properties of the fiber mapping $J(\lambda u)$ that can be verified directly.

Lemma 2.4. ^[17] Assume that $u\in X_0$, then

$(i)$ $\lim_{\lambda\rightarrow 0^{+}}J(\lambda u) = 0$, $\lim_{\lambda\rightarrow +\infty}J(\lambda u) = -\infty$.

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

$(iii)$ $J(\lambda u)$ is increasing on $0 < \lambda < \lambda^{\ast}$, decreasing on $\lambda^{\ast} < \lambda < +\infty$, and attains the maximum at $\lambda = \lambda^{\ast}$.

$(iv)$ $I(\lambda u) > 0$ for $0 < \lambda < \lambda^{\ast}$, $I\left(\lambda u\right) < 0$ for $\lambda^{\ast} < \lambda < +\infty$, and $I(\lambda^{\ast}u) = 0$.

Lemma 2.5. ^[15,18] (Logarithmic Sobolev Inequality). Let $q > 1$, $\mu > 0$, and $u \in W_0^{1, q}\left(\mathbb{R}^N\right) \backslash\left\{0\right\}$. Then we have

where

Remark 2.2. If $u \in W_0^{1, q} (\Omega)\backslash\left\{0\right\}$, then by defining $u(x) = 0$ for $x \in \mathbb{R}^N \backslash\Omega$, we derive

for any real number $\mu > 0$.

Lemma 2.6. ^[14,19] (Hardy-Sobolev inequality). Let $R^N = R^k \times R^{N-k}, \; 2 \leq k \leq N$ and $x = (y, z) \in R^N = R^k \times R^{N-k}$. For given $n, \; \beta$ satisfying $1 < p < N$, $0 \leq \beta \leq p$, and $\beta < k$, let $m(\beta, N, p) = \frac{p(N -\beta)}{(N - p)}$. Then there exists a positive constant $C_H$ depending on $\beta, N, p$, and $k$ such that for any $u \in W_0^{1, p}(R^N)$, it holds that

Remark 2.3. (i) When $m = p = \beta$, this inequality is the classical Hardy inequality. (ii) If $m = 2, \beta = s$ in Lemma $2.4$, we have $p = \frac{2N}{N-s+2} > 2$, and then Lemma 2.6 becomes

Lemma 2.7. ^[15,20] Let $f:\mathbb{R}^+\rightarrow \mathbb{R}^+$ be a nonincreasing function and $\sigma$ be a positive constant such that

Then we have

$(i)$ $f(t) \le f(0){e^{1 - \omega t}}$, for all $t \ge 0$, whenever $\sigma = 0$.

$(i)$ $f(t) \le f(0){\left({\frac{{1 + \sigma }}{{1 + \omega \sigma t}}} \right)^{\frac{1}{\sigma }}}$, for all $t \ge 0$, whenever $\sigma > 0$.

The following is the concavity lemma.

Lemma 2.8. ^[21,22,23] Suppose that a positive, twice-differentiable function $\Psi (t)$ satisfies the inequality

where $\theta > 0$. If $\Psi(0) > 0$ and $\Psi^{\prime}(0) > 0$, then $\Psi(t)\rightarrow \infty$ as

Lemma 2.9. ^[24] Suppose that $0 < l < r \leq 1$ and $\epsilon_1, \epsilon_2 \geq 0$ are positive constants. If nonnegative and absolutely continuous function $h(t)$ satisfies

then we have

and

where ${\epsilon_0} = {\epsilon_1} - {\epsilon_2}{h^{r - l}}\left(0 \right)$ and ${T_0} = \frac{{{h^{1 - l}}\left(0 \right)}}{{{\epsilon_0}\left({1 - l} \right)}}$.

Definition 2.1. (Weak Solution). A function $u : = u(x, t) \in L^{\infty}(0, T; X_0)$ with $\left|x\right|^{-\frac{s}{2}}u_t\in L^2(0, T; L^2(\Omega))$ is called a weak solution of Problem (1.1) on $\Omega \times [0, T)$ if $u(x, 0) = u_0(x)$ in $X_0$ and

for any $v \in W_0^{1, p}(\Omega)$. Moreover,

Definition 2.2. (Maximal Existence Time). Let $u(x, t)$ be a weak solution of Problem (1.1), we define the maximal existence time $T_{max}$ as follows:

$(i)$ If $T_{\rm max} = +\infty$, we say that the solution $u(t)$ is global;

$(ii)$ If $T_{\rm max} < +\infty$, we say that the solution $u(t)$ blows up in finite time and $T_{\rm max}$ is the blow-up time.

Definition 2.3. (Finite Time Blow-Up). Let $u(x, t)$ be a weak solution of Problem (1.1), then $u(x, t)$ is called the finite time blow-up if the maximal existence time $T_{max} < +\infty$ and

Definition 2.4. (Infinite Time Blow-Up). Let $u(x, t)$ be a weak solution of Problem (1.1), then $u(x, t)$ is called the infinite blow-up if

3.   Local existence

In this section, we state the local existence and uniqueness of weak solutions to Problem (1.1).

Theorem 3.1. Let $u_0 \in X_0$, and $p, q$ satisfy $\max\left\{\frac{2N}{N+2}, 1\right\} < p \leq q < p\left(1 +\frac{2}{N}\right).$ Then there exist a $T > 0$ and a unique weak solution $u(x, t)\in L^{\infty}(0, T; X_0)$ of Problem (1.1) with $\left|x\right|^{-\frac{s}{2}}u_t \in L^2(0, T;L^2(\Omega))$ satisfying $u(0) = u_0$. Moreover, $u(x, t)$ satisfies the energy equality

Proof. We divide the proof of Theorem 3.1 into 5 steps.

Step 1. Approximate problem

In order to deal with the singular potential, we introduce the cut-off function

We denote the solutions corresponding to ${\rho _n}$ of Problem (1.1) as $u_n$,

We noticed that $u_{n0} \in C_0^{\infty}(\Omega)$, and then $u_{n0} \to {u_0}\left(x \right)\; in\; W_0^{1, p}(\Omega)$. Let $\left\{ {{\omega _j}} \right\}_{j = 1}^\infty$ be a system of basis in $W_{0}^{1, p}(\Omega)$ which is normalized orthogonal in $L^2(\Omega)$ and construct the approximate solution

We solve the problem

and

as $k \rightarrow +\infty, n\rightarrow +\infty$. Hence $\left\{a^k_{nj}\right\}_{j = 1}^k$ is determined by the following Cauchy problem:

where

Therefore, the standard theory of ordinary differential equations yields that there exists a $T > 0$ such that $a^k_{nj}(t)\in C^1([0, T])$. As a consequence, $u^k_n \in C^1([0, T], W_0^{1, p}(\Omega))$.

Step 2: Priori estimates

We discuss the following two cases:

Case 1: $\max\left\{\frac{2N}{N+2}, 1\right\} < p \leq q$ and $2\leq q < p\left(1 +\frac{2}{N}\right)$

Multiply (3.2) by $a^k_{nj}(t)$, sum for $j = 1, \cdots, k$, and recall $u^k_n(x, t)$ to find

Integrating over $(0, t)$ on both sides of (3.4), we get,

Set

Combining the above equalities, and we have

From Lemma 2.1, we get

Then, by Lemma 2.2 and Young's inequality, (3.7) becomes

where $\varepsilon \in (0, 1)$, and $\theta = \left(\frac{1}{2}- \frac{1}{q + \mu}\right)\left(\frac{1}{N}-\frac{1}{p}+\frac{1}{2}\right)^{-1} = \frac{(q+\mu-2)Np}{(q+\mu)(2p-2N+Np)}$. We note that since $0 < \mu < p(1 +\frac{2}{N}) - q$, $\theta(q + \mu) < p$ holds. Let

then $\alpha > 1$ since $\max\{1, \frac{2N}{N+2}\} < p \leq q$, $2\leq q < p\left(1+\frac{2}{N}\right)$. Besides, since $\Omega$ is a bounded domain in $\mathbb{R}^N$, it leads to

where $C(\Omega)$ is related to $\Omega$.

Thus, from (3.5), (3.6), (3.8), and (3.9), we get

and

Therefore,

where $1 - {\left({e\mu } \right)^{ - 1}}{C_G}\varepsilon > 0,$ ${C_1} = \frac{{{S^k_n}\left(0 \right)}}{{1 - {{\left({e\mu } \right)}^{ - 1}}{C_G}\varepsilon }}$, and ${C_2} = \frac{{{{\left({e\mu } \right)}^{ - 1}}{C_G}C\left(\varepsilon \right)C(\Omega)}}{{1 - {{\left({e\mu } \right)}^{ - 1}}{C_G}\varepsilon }}$. From the Gronwall-Bellman-Bihari inequality, we obtain

and

where ${C_3}$ is a constant which is dependent on $T$.

Multiplying (3.2) by $\left[a_{nj}^k(t)\right]_t$, summing on $j = 1, 2, \cdots, k$, and then integrating on $(0, t)$, we know that

By the continuity of the functional $J$ and (3.3), there exists a constant $C > 0$ satisfying

Applying (2.1), (3.5), (3.8), (3.11), (3.12), and (3.13), we obtain

where ${C_4} = \frac{{2{C_G}C\left(\varepsilon \right){C(\Omega)}}}{{e\mu }}{\left({{C_3}} \right)^\alpha }$, for all $n, k \in N^+$.

Case 2: $\max\{1, \frac{2N}{N+2}\} < p \leq q < 2$

Combining $\ln \left| {u\left(x \right)} \right| < \frac{{{{\left| {u\left(x \right)} \right|}^a }}}{a }$ a.e. $x\in\Omega, \forall\; a > 0$ and (3.5), and taking $a = 2 - q$, we obtain

Together with (3.9), it can become

Then by means of Gronwall's inequality, we have

and

where $C_5 = S_n^k(0)e^{\frac{2T}{(2-q)C(\Omega)}}$.

From (2.1), (3.12), (3.13), and (3.15), we have

for all $k, n \in N^+$.

Therefore, we can derive

Combining (3.9) and (3.16), we have

Step 3: Pass to the limit

By (3.17), (3.19), and the Aubin-Lions-Simon Lemma (see ^[25], Corollary 4), we get

as $k, n\rightarrow +\infty$. Thus, $u^k_n(x, 0) \rightarrow u(x, 0)$ in $L^2(\Omega)$. Combining (3.3) with $u_{n0} \to {u_0}\left(x \right)\; in\; W_0^{1, p}(\Omega)$, we observe that $u(x, 0) = u_0$ in $W_0^{1, p}(\Omega)$.

From (3.21), we have $u^k_n \rightarrow u$ a.e.$(x, t) \in \Omega \times (0, T)$. This implies

It follows from (3.14) and the Hölder inequality that

That means

On the other hand, there is

From Lemma 2.1 and Lemma 2.2, we have

where $B_1$ is the best constant of the Sobolev embedding $W_{0}^{1, p}(\Omega)\hookrightarrow L ^{{\frac{p}{p-1}}\left({q - 1 + \mu } \right)}(\Omega)$. Here we choose $0 < \mu\leq p\left(1+\frac{p-1}{N-p}\right)-q$, $q < p\left(1+\frac{p-1}{N-p}\right)$, and we know that

By (3.17)–(3.19), (3.22), (3.23), there exist functions $u$, $\chi$ and a subsequence of $\left\{u^k_n\right\}^{\infty}_{n, k = 1}$ which we still denote by $\left\{u^k_n\right\}^{\infty}_{n, k = 1}$ such that

Next, by the method of Browder and Minty in the theory of monotone operators, we obtain $\chi = {{{\left| {\nabla u} \right|}^{p - 2}}\nabla u}$.

By (3.24)–(3.27), passing to the limit in (3.2) as $n, k \rightarrow +\infty$, it follows that $u$ satisfies the initial condition $u(0) = u_0$,

for all $\omega \in W_0^{1, p}(\Omega)$, and for a.e. $t \in [0, T ]$.

Step 4. Uniqueness

Suppose there are two solutions $u_1$ and $u_2$ to Problem (1.1), and we have

and

Let $w = u_1 - u_2$ and $w(0) = 0$, then by subtracting (3.29) and (3.30), we can derive

Let $v = w$, and we recall the following elementary vector inequalities that are used frequently: for all $a, b \in \mathbb{R}^N$, we have $0 \le \left({p - 1} \right)\frac{{{{\left| {a - b} \right|}^2}}}{{{{\left({\left| a \right| + \left| b \right|} \right)}^{2 - p}}}} \le \left({{{\left| a \right|}^{p - 2}}a - {{\left| b \right|}^{p - 2}}b} \right) \cdot \left({a - b} \right)$, if $1 < p < 2$. So, we obtain

Integrating it on $[0, t]$, we obtain

where $F(s) = \left|s\right|^{q-2}s \ln\left|s\right|$. Combining with (3.9), we get

By the locally Lipschitz continuity of $F : R^N \rightarrow R$, the uniqueness follows from Gronwall's inequality.

Step 5: Energy equality

We multiply (1.1) with $u_t$ and integrate over $\Omega\times(0, t)$ to obtain the equality

The proof of Theorem 3.1 is complete. □

4.   Global existence and decay rate

4.1.   Global existence

In this section, we are concerned with the existence of a global weak solution to Problem (1.1).

Theorem 4.1. Assume that $u_0 \in W^+$, $\max\left\{\frac{2N}{N+2}, 1\right\} < p \leq q < p(1 +\frac{2}{N})$, and then Problem (1.1) admits a global solution $u \in L^\infty(0, \infty; X_0)$ with $\left|x\right|^{-\frac{ s}{2}}u_t \in L^2(0, \infty; L^2(\Omega))$ and $u(t) \in W^+$ for $0 \leq t \leq \infty$.

Proof. Now, we prove Theorem 4.1. In order to prove the existence of global weak solutions, we consider the following two steps:

Step 1. The initial data $u_{0}\in W_{1}^{+}$

From (3.32), we know that

where $T_{max}$ is the maximal existence time of solution $u(t)$. We shall prove that $T_{max} = +\infty$. Next, we will show that

Indeed, assume that (4.2) does not hold and let $t_*$ be the smallest time for which $u(t_*) \notin W_1^+$. Then, by the continuity of $u(t)$, one has $u(t_*)\in \partial W_1^+$. Hence, it follows that

or

Nevertheless, it is clear that (4.3) could not occur by (4.1) while if (4.4) holds then, by the definition of $d$, we have

which also contradicts with (4.1). Hence, (4.2) is valid.

Next, it is discussed in two cases.

Case 1: $p < q$

As a consequence, it follows from this fact and the definition of functional $J(u(t))$ that

and

This estimate allows us to take $T_{max} = +\infty$. So, we can conclude that there is a unique global weak solution $u(t)\in W_{1}^{+}$ of Problem (1.1) which satisfies that

Case 2: $p = q$

Similar to Case 1, we can derive

By Lemma 2.5, we have

From (2.2), (4.2), and the above inequality, we know that

Combining the two cases above, we know that the estimate allows us to take $T_{max} = +\infty$. It means that there is a unique global weak solution $u(t)\in W_{1}^{+}$ of Problem (1.1).

Step 2. The initial data $u_{0}\in W_{2}^{+}$

First, we choose a sequence $\left\{ {{\theta _m}} \right\}_{m = 1}^\infty \subset (0, 1)$ such that $\mathop {\lim }\limits_{m \to \infty } {\theta _m} = 1$. Then we consider the following problem

where $u_{0m} = \theta_{m}u_{0}$. First of all, we claim that $u_{0m}\in W_1^+$, and then $J(u_{0m}) < d$ and $I(u_{0m}) > 0$. In fact, from $u_0 \in W_1^+$, $\mathop {\lim }\limits_{m \to \infty } {\theta _m} = 1$ and $I(u_0) > 0$, and we can see

On the other hand, by direct calculations, we obtain

which implies that $J({\theta _m}u_0)$ is strictly increasing with respect to ${\theta _m}$ and

Since $u_{0m} \rightarrow u_0$ as $m \rightarrow +\infty$, our result can be derived by the same processes as the proof of Step 1.

Theorem 4.1 is complete. □

4.2.   Decay estimates

Theorem 4.2. Let $u(t)$ be the solution of Problem (1.1) and $p, q$ satisfy

If $u_0 \in W_1^+$, then there exist positive constants $c_2$ such that

Especially, if $p = 2$, then there exist positive constants $c_4$ such that

Proof. We are now in a position to prove the algebraic decay results. Thanks to $u_{0}\in W_{1}^{+}$, we get $u(t)\in W_{1}^{+}$. From (2.3), we have

By (4.9), through a direct calculation, we arrive at

where $\lambda_0 = \max\{(\lambda^*)^p, (\lambda^*)^q\}$. Combining with (4.9), we get

which means that $\lambda^* > 1$, $\lambda^*\geq\left(\frac{d}{J(u_0)}\right)^{\frac{1}{p}}$.

From (2.2), we have

Namely,

where ${c_1} = {1 - {{\left({\frac{d}{{J\left({{u_0}} \right)}}} \right)}^{1 - \frac{q}{p}}}}$, $p < q$.

According to Lemma 2.6, and (2.2), we obtain

By (4.12) and (4.13), we get

Let $T\rightarrow +\infty$ in (4.14), and by the virtue of Lemma 2.7, it follows that

Theorem 4.2 is complete. □

5.   Blow-up phenomena of weak solutions

In this section, we present the blow-up phenomena of the solutions to (1.1) including infinite and finite time blow-up, and give some bounders of the blow-up. For simplicity, we shall write $L(t) = \frac{1}{2}{\left\| {{\left| x \right|^{-\frac{s}{2}}}}{u(t)} \right\|_2^2 }$ in the sequel.

5.1.   Infinite blow-up

This subsection is devoted to infinite blow-up for Problem (1.1).

Theorem 5.1. (Infinite Blow-Up). Let $u_0 \in W^-$ and $p, q$ satisfy $1 < p \leq q < 2$. Then $u(t)$ blows up in infinite time.

Proof. We divide the proof into 2 steps.

Step 1: $u_0\in W_1^-$

We claim that $u(t) \in W_1^-$ for all $t \in [0, T_{max})$ provided that $u_0\in W_1^-$. Let $u(t)$ be the weak solution of Problem (1.1) with $u_{0}\in W_{1}^{-}$, which means that $u_0\neq0$ and $J({u_{0}}) < d, \; I(u_0) < 0$.

If $J(u_0) < 0$. By the energy equality, we arrive at

It means that $u(t)\neq0$, $J(u(t)) < d$, and $I(u(t)) < 0$, which implies that $u(t)\in W_1^-$.

If $0 < J(u_0) < d$. From the energy equality, we obtain

which means that $u(x, t)\neq 0$. Next, we will show that $I(u(t)) < 0$ for all $t\in[0, T_{\rm max})$. Otherwise, by the continuity of $I(u)$, there would exist a $t_{*}\in(0, T_{\rm max})$ such that $I(u(t)) < 0$, $t\in[0, t_{*})$ and $I(u(t_{*})) = 0.$ It means that $u(t_*)\in N$. Then, from the definition of $d$, it holds that $J(u(t_{*}))\geq d$ which contradicts (5.2). Then $u(t)\in W_{1}^{-}$ for all $t\in[0, T_{\rm max})$.

From Lemma 2.4(ⅳ), as $I(u(t)) < 0$, there is a $\lambda^* < 1$ such that $I(\lambda^*u) = 0$. Then

Then, by taking the derivative of $L(t)$, we obtain

Combining (5.4) and $L(t)-L(0) = \int_0^t{L^\prime(t)}dt$, we can derive

Now, we prove that $u(t)$ cannot blow up in finite time. Arguing by contradiction, we assume that $u(t)$ blows up in finite time, which implies that

Meantime, by (5.4), we have

Next, combining $\ln \left| {u\left(x \right)} \right| < \frac{{{{\left| {u\left(x \right)} \right|}^\delta }}}{\delta }$ a.e. $x\in\Omega, \forall\; \delta > 0$ and (2.2), and taking $\delta = 2 - q$, we obtain

On the other hand, from (5.4) and (5.6), we can see that there exists a $t_1 \in (0, T_{max})$ such that

where $\left|x\right| < L$. Then by combining (5.7), (5.8), and (5.9) we can derive

which means that

Through a direct calculation, we have

Then by virtue of Gronwall's inequality, we get

which implies that

That contradicts with (5.6). Therefore, $T_{max} = +\infty$ and $u(t)$ blows up in infinite time.

Step 2: $u_0\in W_2^-$

First of all, $u_0\in W_2^-$ means that $I(u(t)) < 0, J(u(t)) = d$, $\forall t \in [0, T_{max}]$. We claim that $u(t) \in W_2^-$ for all $t \in [0, T_{max})$ provided that $u_0\in W_2^-$. Otherwise, by continuity, there would exist a $t_1 \in [0, T_{max})$ such that $I(u(t)) < 0$ for $t \in [0, t_1)$ and $I(u(t_1)) = 0$. Recalling the definition of $d$, it is clear that $J(u(t_1))\geq d$. On the other hand, from $\int_\Omega {\left| {u \cdot \frac{{{u_t}}}{{{{\left| x \right|}^s}}}} \right|} dx = - I\left({u\left(t \right)} \right) > 0, \; t \in \left[ {t_1, {T_{\max }}} \right],$ we know that $u_t \neq 0$ and $\int_0^{{t_1}} {\left\| {{{\left| x \right|}^{ - \frac{s}{2}}}{u_\tau }\left(\tau \right)} \right\|_2^2d\tau > 0}$, $t_1\in [0, T_{max}]$. Meanwhile, it follows from the energy equality that

which contradicts with $J(u(t_1))\geq d$. Therefore, there exists a $t_2\in[0, T_{max}]$ such that $I(u(t_2)) < 0$ and $J(u(t_2)) < d$. If we take $t_2$ as the initial time, then similar to Step 1, we can obtain that the weak solution $u(t)$ of Problem (1.1) blows up in infinite time. □

5.2.   Finite time blow-up

Theorem 5.2. (Finite Blow-Up). Let $u_0 \in W^-$ and $p, q$ satisfy $2 < p \leq q < p\left(\frac{2}{N}+1\right)$. Then $u(t)$ blows up in finite time. Moreover,

Proof. We shall apply the first-order differential inequality technique to show the finite time blow-up result for Problem (1.1) with negative initial energy. For this, set $K(t) = -J(u(t))$. Then $L(0) > 0, K(0) > 0$. From Problem (1.1), it follows that

which means that $K\left(t \right) \ge K\left(0 \right) = - J\left({{u_0}} \right) > 0$ for all $t \in [0, T^*)$. Recalling (2.2) and (2.3), we obtain, for any $t\in[0, T^*)$, that

Making use of the Holder inequality and Cauchy-Schwarz inequality, we arrive at

which then implies

Therefore,

Integrating (5.15) over $[0, t]$ for any $t \in (0, T^*)$ and noticing that $q > 2$, one has

or equivalently

It is obvious that (5.16) cannot hold for all $t > 0$. Therefore, $T^* < +\infty$. Moreover, it can be inferred from (5.16) that

The proof is complete. □

For the case of $J(u_0) \geq 0$, we obtain blow-up results when the initial energy is 'subcritical' and when the initial Nehari functional is negative which means that $u_0\in W^-$. More precisely, we have the following theorem.

Theorem 5.3. Assume that $2 < p \leq q < p\left(\frac{2}{N}+1\right)$, $u_0 \in W^-$. Then the weak solution $u(t)$ to Problem (1.1) blows up in finite time. Furthermore, if $u_0\in W_1^-$, then $T_{max}$ can be estimated from above as follows:

where $\beta, b$ are constants that will be determined in the proof.

Proof. We will divide the proof into two cases.

Case 1: $u_0\in W_1^-$

We claim that $u(t) \in W_1^-$ for all $t \in [0, T_{max})$ provided that $u_0\in W_1^-$. Otherwise, by continuity, there would exist a $t_0 \in [0, T_{max})$ such that $I(u(t)) > 0$ for $t \in [0, t_0)$ and $I(u(t_0)) = 0$. Recalling the definition of $d$, it is clear that $J(u(t_0))\geq d$, which contradicts with $J(u(t))\leq J(u_0) < d$.

From Lemma 2.4(ⅳ), as $I(u(t)) < 0$, there is a $\lambda^* < 1$ such that $I(\lambda^*u) = 0$. Then

We show that $T_{max} < +\infty$. For any $T \in [0, T_{max})$, define the positive function

where $\beta > 0, b > 0$. By direct computations,

Applying the Cauchy-Schwarz inequality, Young inequality, and Holder's inequality to yields

Therefore, by recalling (5.19) and (5.20), and noticing the nonnegativity of $f(t)$, we arrive at

Choosing $\theta = \frac{{q - 2}}{2}$ and recalling (5.17) lets us obtain

In view of (5.18) and (5.23), we get, for any $t\in(0, T_{max})$ and $\beta \in \left({0, \frac{{q\left({d - J\left({{u_0}} \right)} \right)}}{{q - 1}}} \right]$, that

Therefore, Lemma 2.8 guarantees that $F(0) > 0$ and $F'(0) = \beta b > 0$, and then $\exists\; {T_1}\; :\; 0 < {T_1} < \frac{{2F(0)}}{{\left({q - 2} \right){F^\prime }(0)}}$, such that $F(t)\rightarrow \infty, \; t\rightarrow T_{1}$

where $b > \max \left\{ {0, \frac{{\left\| {{\left| x \right|^{-\frac{s}{2}}}}{u(0)} \right\|_2^2 }}{{\left({q - 2} \right)\beta }}} \right\}$.

Case 2: $u_0\in W_2^-$

By similar arguments as those in the proof of case 1, when $u_0\in W_2^-$, by continuity, we see that there exists a $t_2 > 0$ such that $I(u(t_2)) < 0$ and $\left\|\left|x\right|^{-\frac{s}{2}}u_t\right\|^2_2 > 0$ for all $t\in[0, t_2)$. From the energy equality, we get

The remainder of the proof is the same as in Case 1. □

In the following, we shall derive a lower bound for the blow-up time $T_*$.

Theorem 5.4. Assume $2 < p \leq q < p\left(\frac{2}{N}+1\right)$. Let $u(t)$ be a weak solution to Problem (1.1) that blows up at $T_*$. Then

where $C_L > 0, \alpha > 1$ are two constants that will be determined in the proof.

Proof. Combining (2.2) and (3.8), we have

where $\alpha > 1$. As ${{{\left({e\mu } \right)}^{ - 1}}{C_G}\varepsilon - 1} < 0$, recalling the definition of $L(t)$, we get

where $C_L = {\left({e\mu } \right)^{ - 1}}{C_G}C(\varepsilon){C(\Omega)}$. Integrating (5.25) over $[0, t)$, we get

Since $\alpha > 1$, letting $t \rightarrow T_*$ in the above inequality and recalling that $\lim _{t\rightarrow T_*} L(t) = +\infty$, we obtain

The proof is complete. □

6.   Extinction phenomenon

In this section, we present the result of extinction for Problem (1.1).

Theorem 6.1. (Extinction). Assuming $\frac{2N}{N+2} < p < q < 2$ and

then the weak solution of Problem (1.1) becomes extinct in finite time. Furthermore, we have the following estimates

and

The extinction time is

where $\epsilon_0, \; \delta, \; C_p$ are given in the following.

Proof. Multiplying (1.1) by $u(t)$ and integrating over $\Omega$, we have

Meanwhile, by $\frac{2N}{N+2} < p < 2$, we have $0 \leq s \leq 2 < (N + 2)-\frac{2N}{p}$, which implies that $1 < \frac{2N}{N-s+2} < p$ in Lemma 2.6, and we can see that there exists a constant $C_p > 0$ such that

Combining (2.4), (3.9), (6.1), (6.2), and H$\rm\ddot{o}$lder's inequality, we deduce that there exists a $0 < \delta \leq 2 - q$ such that

Then by Lemma 2.9, we know that

We assume that

and then we can see that

and

where

and

The proof is complete. □

Author contributions

Xiulan Wu: Methodology, Writing-original draft, Writing-review & editing; Yanxin Zhao and Xiaoxin Yang: Methodology, Writing-original draft.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The authors express their heartfelt thanks to the editors and referees who have provided some important suggestions. This work is supported by the Natural Science Foundation of Jilin Province, Free Exploration Basic Research (No. YDZJ202201ZYTS584).

Conflict of interest

The authors declare there is no conflict of interest.