A paradigmatic approach to investigate restricted hyper totient graphs

1.   Introduction and main results

In this paper, we consider the following plate equation with Hardy-Hénon potential and polynomial nonlinearity:

where $\Omega\subset \mathbb{R}^n$ is a bounded domain with smooth boundary $\partial \Omega$, $\nu$ is the unit outer normal to the boundary $\partial \Omega$ at $x$, $\alpha\in(-\infty, n)$ and $\sigma\in \mathbb{R}$ are constants, and

The initial value $(u_0, u_1)\in H_0^2( \Omega)\times L^2( \Omega)$. Here,

Plate equations have been investigated for many years due to their importance in some physical areas such as vibration and elasticity theories of solid mechanics. For instance, in the case when $\sigma$ is identically zero, equation (1.1) becomes an equation with polynomial nonlinearity which arises in aeroelasticity modeling (see, for example, [14,15]), and the problem with (or without) damping, memory, time-delay etc. were studied extensively (see [17,21,25,26,27,33,34,37,40,55,44] and references therein for the topics on well-posedness, global existence, finite time blow-up, global attractor etc.).

The potential term $|x|^{-\alpha}* u$ is known in the literature as Hardy potential if $\alpha>0$, while if $\alpha<0$ it is known as Hénon potential. This type of potential is important in analyzing many aspects of physical phenomena with singular poles (at origin). For example, the Efimov states (the circumstances that the two-particle attraction is so weak that any two bosons can not form a pair, but the three bosons can be stable bound states): see e.g., [16]; effects on dipole-bound anions in polar molecules: see e.g., [4,7,28]; capture of matter by black holes (via near-horizon limits): see e.g., [9,19]; the motions of cold neutral atoms interacting with thin charged wires (falling in the singularity or scattering): see e.g., [5,12]; the renormalization group of limit cycle in nonrelativistic quantum mechanics: see e.g., [6,8]; and so on. The are a lot of studies of evolution equations with this type of potentials, see, for example, [1,2,3,45,32,50] for parabolic equation, [11,49,52,39] for wave equations, and [24,22,23,42,47,48,53] for Schrödinger equation. However, as far as we know, there seems little studies of fourth-order plate equation with Hardy-Hénon potentials.

Motivated by the previous studies, in this paper, we will consider a fourth-order plate equation with Hardy-Hénon potential and polynomial nonlinearity, i.e, problem (1.1). We mainly concern with the well-posedness, and the conditions on global existence and finite time blow-up. To state the main results of this paper, we first introduce some notations used in this paper:

● Let $(X, \|\cdot\|_X)$ and $(Y, \|\cdot\|_Y)$ be two Banach space such that $X\hookrightarrow Y$ continuously. Then we denotes by $C_{X \rightarrow Y}$ the optimal constant of the embedding, i.e.,

● The norm of the Lebesgue space $L^p( \Omega)$, $1\leq p\leq \infty$ is denoted by $\|\cdot\|_p$. Especially, we denote $\|\cdot\|_2$ by $\|\cdot\|$ for simplicity.

● The inner product of the Hilbert space $L^2( \Omega)$ is denoted by $(\cdot, \cdot)$.

● The norm of the Sobolev space $H_0^2( \Omega)$ is denoted by $\|\cdot\|_{H^2}$ and

Definition 1.1. Assume $\alpha\in(-\infty, n)$ and $\sigma\in \mathbb{R}$. Let $T>0$, $u_0\in H_0^2( \Omega)$, and $u_1\in L^2( \Omega)$. By a weak solution to problem (1.1), we mean a function

such that $u(0) = u_0$, $u_t(0) = u_1$, and

holds for any $\phi\in H_0^2( \Omega)$ and $0<t\leq T$.

The local-well posedness of solutions to problem (1.1) is the following theorem:

Theorem 1.2. Assume $p$ satisfies (1.2), $\alpha\in(-\infty, n)$ and $\sigma\in \mathbb{R}$. Let $(u_0, u_1)\in H_0^2( \Omega)\times L^2( \Omega)$. Then there exists a positive constant $T$ depending only on $\|u_0\|_{H^2}+\|u_1\|$ such that problem (1.1) admits a weak solution

The solution $u$ can be extended to a maximal weak solution in $[0, {T_{\max}})$ such that either

1. ${T_{\max}} = \infty$, i.e., the problem admits a global weak solution; or

2. ${T_{\max}}<\infty$, and

i.e., the solution blows up at a finite time ${T_{\max}}$.

Furthermore, then energy $E(t)$ is conservative, i.e.,

where

and

Moreover, we have $(u_t, u)\in C^1[0, {T_{\max}})$, and

Here $J: H_0^2( \Omega) \rightarrow \mathbb{R}$ is a functional defined by

and $I: H_0^2( \Omega) \rightarrow \mathbb{R}$ is a functional defined by

Based on Theorem 1.2, we study the conditions on global existence and finite time blow-up. The first result is about the case that the initial energy is non-positive.

Theorem 1.3. Assume $p$ satisfies (1.2), $\alpha\in(-\infty, n)$ and $\sigma\in \mathbb{R}$. Let $(u_0, u_1)\in H_0^2( \Omega)\times L^2( \Omega)$ satisfy

1. $E(0)<0$; or

2. $E(0) = 0$ and $(u_0, u_1)>0$.

Then the weak solution got in Theorem 1.2 blows up in finite time, i.e., ${T_{\max}}<\infty$. Moreover, ${T_{\max}}$ satisfies the following estimates:

1. If $|\sigma|\leq\sigma^*$, then

where

2. If $|\sigma|>\sigma^*$ and $(u_0, u_1)\geq0$, then

Here,

and

where

Remark 1. There are two remarks on the above theorem.

1. Firstly, by Lemma 2.2, if $\alpha\in(-\infty, n)$, $\sigma^*$ is well-defined and

where $C_{H_0^2 \rightarrow L^1}$ is the optimal constant of the embedding $H_0^2( \Omega)\hookrightarrow L^1( \Omega)$, $C_{H_0^2 \rightarrow L^{\frac{2n}{2n- \alpha}}}$ is the optimal constant of the embedding $H_0^2( \Omega)\hookrightarrow L^{\frac{2n}{2n- \alpha}}( \Omega)$. So Theorem 1.3 makes sense.

2. Secondly, for $|\sigma|>\sigma^*$ and $(u_0, u_1)<0$. Due to technique reasons, we only show the solution will blow up in finite time, but the upper bound of the blow-up time ${T_{\max}}$ is not given. We left the study of this problem as an open question.

Theorem 1.3 is above the case $E(0)\leq0$. In order to derive some results for the case $E(0)>0$, we use the potential well method (see, for example, [10,20,35,36,41,43,51]). Let

where $J$ is the functional defined by (1.9), and $\mathcal{N}$ is the Nehari manifold defined as

Here $I$ is the functional defined by (1.10).

Remark 2. If we assume $p$ satisfies (1.2), $\alpha\in(-\infty, n)$, and $\sigma\in(-\sigma^*, \sigma^*)$, where $\sigma^*$ is the positive constant defined by (1.11), by Lemma 2.4, $d$ is a positive constant and

where $C_{H_0^2 \rightarrow L^p}$ is the optimal constant of the embedding $H_0^2( \Omega)\hookrightarrow L^p( \Omega)$.

Theorem 1.4. (Global existence for $0<E(0)\leq d$). Assume $p$ satisfies (1.2), $\alpha\in(-\infty, n)$, and $\sigma\in(-\sigma^*, \sigma^*)$, where $\sigma^*$ is the positive constant defined by (1.11). Let $u$ be the weak solution got in Theorem 1.2 with $(u_0, u_1)\in H_0^2( \Omega)\times L^2( \Omega)$ satisfying $I(u_0)>0$ and $0<E(0)\leq d$, where $E(0), I, d$ are defined in (1.7), (1.10), (1.13) respectively. Then $u$ exists globally, i.e., ${T_{\max}} = \infty$.

Theorem 1.5. (Blow-up for $0<E(0)\leq d$) Assume $p$ satisfies (1.2), $\alpha\in(-\infty, n)$, and $\sigma\in(-\sigma^*, \sigma^*)$, where $\sigma^*$ is the positive constant defined by (1.11). Let $u$ be the weak solution got in Theorem 1.2 with $(u_0, u_1)\in H_0^2( \Omega)\times L^2( \Omega)$ satisfying

1. $E(0)< d, I(u_0)<0$; or

2. $E(0) = d, I(u_0)<0, (u_0, u_1)>0$,

where $E(0), I, d$ are defined in (1.7), (1.10), (1.13) respectively. Then $u$ blows up in finite time, i.e., ${T_{\max}}<\infty$. Moreover, ${T_{\max}}$ satisfies the following estimates:

where

The organization of the rest of this paper is as follows: In Section 2, we give some preliminaries, which will be used in this paper; In Section 3, we study the well-posed of solutions by semigroup theory and prove Theorem 1.2; In Section 4, we study the conditions on global existence and finite time blow-up and prove Theorems 1.3, 1.4 and 1.5.

2.   Preliminaries

The following well-known Hardy-Littlewood-Sobolev inequality can be found in [31]:

Lemma 2.1. Let $q, r>1$ and $0< \theta<n$ with

Let $u\in L^q \left( \mathbb{R}^n \right)$ and $v\in L^r \left( \mathbb{R}^n \right)$. Then there exists a sharp positive constant $C$ depending only on $n, \alpha$ and $q$ such that

If

then

If $u\in L^{q}( \Omega)$ and $v\in L^{r}( \Omega)$ with $q$ and $r$ satisfying the assumptions, by valuing them $0$ in $\mathbb{R}^n\setminus \Omega$, it also holds

Lemma 2.2. Let $\alpha\in(-\infty, n)$ and $\sigma^*$ be the constant defined in (1.11). Then $\sigma^*$ is well-defined and

where $C_{H_0^2 \rightarrow L^1}$ is the optimal constant of the embedding $H_0^2( \Omega)\hookrightarrow L^1( \Omega)$, $C_{H_0^2 \rightarrow L^{\frac{2n}{2n- \alpha}}}$ is the optimal constant of the embedding $H_0^2( \Omega)\hookrightarrow L^{\frac{2n}{2n- \alpha}}( \Omega)$, $\kappa$ is the constant given in (2.15) with $\theta = n- \alpha$, and

Proof. $\underline{\text{ If }\alpha \in (-\infty ,0]}$, we have,

$\underline{\text{ If }\alpha \in \left( 0,n \right)}$, by Hardy-Littlewood-Sobolev inequality (see (2.16) of Lemma 2.1) with $q = r = \frac{2n}{2n- \alpha}$ and $H_0^2( \Omega)\hookrightarrow L^{\frac{2n}{2n- \alpha}}( \Omega)$ with constant $C_{H_0^2 \rightarrow L^{\frac{2n}{2n- \alpha}}}$,

Lemma 2.3. [29,30] Suppose $F(t) \in C^{2}[0, T)$ is a nonnegative function satisfying

where $0<T \leq+\infty$ and $r$ is a positive constant. If $F(0)>0$ and $F^{\prime}(0)>0,$ then

and $F(t) \rightarrow+\infty$ as $t \rightarrow T$.

Lemma 2.4. Assume $p$ satisfies (1.2), $\alpha\in(-\infty, n)$, and $\sigma\in(-\sigma^*, \sigma^*)$, where $\sigma^*$ is the positive constant defined by (1.11). Let $d$ be the constant defined in (1.13). Then we have

and

where $C_{H_0^2 \rightarrow L^p}$ is the optimal constant of the embedding $H_0^2( \Omega)\hookrightarrow L^p( \Omega)$.

Proof. Firstly, we prove (2.19). For any $\phi\in H_0 ^2 (\Omega) \setminus\{0\}$, since $\sigma\in[0, \sigma^*)$, by means of a simple calculation, we find there exists a unique positive constant $\hat \lambda$ defined by

such that

Then,

On the other hand, for any $\phi\in \mathcal{N}$, we have $\hat \lambda = 1$. Then

Then (2.19) follows from the above two inequalities.

Secondly, we prove (2.20), by (1.11), we have

Lemma 2.5. Assume $p$ satisfies (1.2), $\alpha\in(-\infty, n)$, and $\sigma\in(-\sigma^*, \sigma^*)$, where $\sigma^*$ is the positive constant defined by (1.11). Let

where $d$ is the positive constant defined in (1.13). Then

and

where $C_{H_0^2 \rightarrow L^p}$ is the optimal constant of the embedding $H_0^2( \Omega)\hookrightarrow L^p( \Omega)$. Moreover, $\mathcal{W} = \mathcal{W}_1$, and $\mathcal{V}_1 = \mathcal{V}$, where

and

Here,

Proof. Step 1. We prove (2.26). By the definition of $J$ and $I$ (see (1.9) and (1.10)), we have

For any $\phi\in \mathcal{W}$, since $I(\phi)>0$ and $J(\phi)<d$, (2.26) follows from the above inequality.

Step 2. We prove (2.27). For any $\phi\in V$, by the definition of $I$ in (1.10), $|\sigma|<\sigma^*$, it follows from (1.11) and $I(\phi)<0$ that

which implies (2.27).

Step 3. We show that $\mathcal{V}_1 = \mathcal{V}$.

Firstly, for any $\phi\in \mathcal{V}_1$, by (1.9) and the definition of $\mathcal{V}_1$, we get

which implies

Thus, by the definition of $I$ in (1.10) and the above two inequalities,

which, together with $J(\phi)<d$, implies $\phi\subset \mathcal{V}$, then $\mathcal{V}_1\subset \mathcal{V}$.

Secondly, we prove $\mathcal{V}\subset \mathcal{V}_1$. For any $\phi\in \mathcal{V}$, since $|\sigma|<\sigma^*$, it follow from (2.27) and (2.31) that

Then, in view of $I(\phi)<0$, i.e.,

we get

which, together with the second line of (2.23), implies

Since $J(\phi)<d$, the above inequality infers that $\phi\in \mathcal{V}_1$, and then $\mathcal{V}\subset \mathcal{V}_1$.

Step 4. We show that $\mathcal{W} = \mathcal{W}_1$.

Firstly we prove $\mathcal{W}\subset\mathcal{W}_1$. In fact, for any $\phi\in \mathcal{W}$, if $\phi = 0$, it is obvious that $\phi\in \mathcal{W}_1$; if $\phi \neq0$, in view of the definition of $\mathcal{W}$, we get

The above two inequalities imply

which, together with $J(u)<d$, implies $\phi\in \mathcal{W}_1$, then $\mathcal{W}\subset \mathcal{W}_1$.

Secondly we prove $\mathcal{W}_1\subset\mathcal{W}$ by contradiction argument. If there exists $\phi\in \mathcal{W}_1\setminus \mathcal{W}$. Then we have

If $I(\phi)<0$, the by (2.32), we get $\phi\in \mathcal{V}$ (see (2.25)), and then $\phi\in \mathcal{V}_1$ (since $\mathcal{V} = \mathcal{V}_1$ has been proved in Step 3), which, together with (2.29), contradicts (2.33); If $I(\phi) = 0$, then by $\phi\neq0$, we get $\phi\in \mathcal{N}$ (see (1.14)), and then $J(\phi)\geq d$ (see (1.13)), which contradicts (2.32). So, $\phi\in \mathcal{W}$, and then $\mathcal{W}_1\subset \mathcal{W}$.

Lemma 2.6. Assume $p$ satisfies (1.2), $\alpha\in(-\infty, n)$, and $\sigma\in(-\sigma^*, \sigma^*)$, where $\sigma^*$ is the positive constant defined by (1.11). Let

be the maximal weak solution to (1.1) with initial value $(u_0, u_1)\in H_0^2( \Omega)\times L^2( \Omega)$ got in Theorem 1.2.

1. If there exists a $t_0\in [0, T_{\max})$ such that $E(t_0)<d$, then $u(t)\in \mathcal{W}$ for $t\in[t_0, {T_{\max}})$ provided that $u(t_0) \in \mathcal{W}$;

2. If there exists a $t_0\in [0, T_{\max})$ such that either $E(t_0)<d$ or $E(t_0) = d$ and $(u_t, u)_{t = t_0}\geq 0$, then $u(t)\in \mathcal{V}$ for $t\in[t_0, {T_{\max}})$ provided that $u(t_0) \in \mathcal{V}$,

where $E(t)$ is the energy functional defined in (1.6), $\mathcal{W}$ and $\mathcal{V}$ is is the sets defined in (2.24) and (2.25) respectively, $d$ constant defined in (1.13).

Proof. Firstly, we proof the first part by contradiction argument. Actually, if the conclusion is incorrect, by using $u\in C \left([0, {T_{\max}});H_0^2( \Omega) \right)\cap C^1 \left([0, {T_{\max}});L^2( \Omega) \right)$ and $u(t_0)\in \mathcal{W}$, there must exist a $t_1\in(t_0, {T_{\max}})$ such that

Since the energy is conservative (see (1.5)), $E(t) = \frac{1}{2}\left\| u_t\right\|^{2}+J(u)(t)$ (see (1.6)), and $E(0)<d$, we get

Then by $u(t_1)\in \partial \mathcal{W}$ and the definition of $\mathcal{W}$ (see (2.24)), it follows $I(u)(t_1) = 0$ and $u(t_1)\neq0$. So $u(t_1)\in \mathcal{N}$ (see the definition of $\mathcal{N}$ in (1.14)), then by the definition of $d$ (see (1.13)), it follows $J(u)(t_1)\geq d$, which contradicts (2.35).

Secondly, we proof the second part. In the case $E\left(t_{0}\right)<d$, in view of (2.27), the proof is similar to the first part. We only prove the case

in detail. Arguing by contradiction, if the conclusion is incorrect, by $u\left(t_{0}\right) \in \mathcal{V}$ and $u \in$ $C\left(\left[0, T_{\max }\right) ; H_{0}^{2}\right),$ we obtain that there must exist a $t_{1} \in\left(t_{0}, T_{\max }\right)$ such that $u(\cdot, t) \in \mathcal{V}$, $t \in\left[t_{0}, t_{1}\right)$ and $u(t_1)\in \partial \mathcal{V}$, i.e. (see (2.25)),

$(i)$: $J\left(u\right)\left(t_{1}\right)<d, I\left(u\right)\left(t_{1}\right) = 0$; or

$(ii)$: $J\left(u\right)\left(t_{1}\right) = d, I\left(u\right)\left(t_{1}\right) \leq 0$.

Due to $u(t) \in \mathcal{V}$ for any $t \in\left[t_{0}, t_{1}\right)$ and $u \in C\left(\left[0, T_{\max }\right) ; H_0 ^{2}\right)$, by (2.27), we get

If $(i)$ is true, by using $I\left(u\left(t_{1}\right)\right) = 0$ and (2.36), we have $u\left(t_{1}\right) \in \mathcal{N}$ (see (1.14)), which implies $J\left(u\right)\left(t_{1}\right) \geq d$ (see (1.13)), a contradiction.

If $(ii)$ is true, by (1.5) and $E(t) = \frac{1}{2}\left\|u_t\right\|^{2}+J(u)(t)$ (see (1.6)), we get

Combining (2.37) and $J\left(u\right)\left(t_{1}\right) = d,$ we have

Utilizing Cauchy-Schwartz's inequality, we obtain that

Integrating (1.8) over $[0, t]$, we obtain

By (2.40), we get

Then,

Since $I(u)(t)<0$ (by using $u(t)\in \mathcal{V}$ for $t\in [t_0, t_1)$) and $(u_t, u)|_{t = t_0}\geq0$, we get from the above equality that

which contradicts (2.39).

3.   Local-well posedness

In this section, we study local well-posedness of solutions to (1.1) by semigroup theory. To this end, first, we introduce some fundamental theory on semigroup theory.

Suppose that $H$ is a Hilbert space with inner product $(\cdot, \cdot)_H$ and norm

Suppose $F$ is a nonlinear operator from $H$ into $H$. $F$ is said to satisfy the local Lipschitz condition if for any positive constant $M>0$, there is a positive constant $L_M$ depending only on $M$ such that when $U, V\in H$, $\|U\|_H\leq M$ and $\|V\|_H\leq M$,

Consider the following abstract semilinear evolution equation

where $A:D(A)\rightarrow H$ is a densely defined linear operator on $H$, i.e., $A$ is linear and $D(A)$ is dense in $H$, where $D(A) = \{\Phi\in H: A\Phi\in H\}$.

First, we introduce the Lumer-Phillips theorem (see, for example, [38,Theorem 1.2.3] and [54,Lemma 2.2.3]):

Lemma 3.1. The necessary and sufficient conditions for $A$ generating a contraction $C_0$-semigroup $\{e^{tA}\}_{t\geq0}$ on $H$ are

1. $(A\Phi, \Phi)_H\leq0$ for all $\Phi\in D(A)$, and

2. $R(I-A) = H$.

Here $R(I+A) = \{\Phi+A\Phi:\Phi\in D(A)\}$ is the range of the operator.

Next, we introduce the local well-posedness results for (3.42), which can be found in [54,Theorems 2.5.4 and 2.5.5]:

Lemma 3.2. Suppose that $A$ generates a contraction $C_0$-semigroup $\{e^{tA}\}_{t\geq0}$ on $H$, and $F$ is a nonlinear operator from $H$ into $H$ satisfying the local Lipschitz condition. Then for any $U_0\in H$, there is a positive constant $T$ depending only on $\|U_0\|_H$ such that problem (3.42) admits a unique local mild solution $U(t)$, i.e., $U\in C([0, T], H)$ and satisfies

The solution $U$ can be extended to a maximal mild solution in $[0, {T_{\max}})$ such that either

1. ${T_{\max}} = \infty$, i.e., the problem admits a global mild solution; or

2. ${T_{\max}}<\infty$, and

i.e., the solution blows up at a finite time ${T_{\max}}$.

Furthermore, if $u_0\in D(A)$, then $u\in C \left([0, {T_{\max}});D(A) \right)\cap C^1 \left([0, {T_{\max}});H \right)$ is classical solution.

By introduction $U = (u, v): = (u, u_t)$, $U_0 = (u_0, u_1)$, and

where $I$ is the identity operator, (1.1) can be equivalently written as the following system

In the next lemma, we show $A$ generates a contraction semigroup $\{e^{tA}\}_{t\geq0}$ on $H_0^2( \Omega)\times L^2( \Omega)$.

Lemma 3.3. Let $A$ be the operator defined in (3.44). Then $A$ generates a contraction semigroup $\{e^{tA}\}_{t\geq0}$ on $H_0^2( \Omega)\times L^2( \Omega)$.

Proof. Let $H: = H_0 ^2( \Omega)\times L^2( \Omega)$, then $H$ is a Hilbert space with inner produce $(\cdot, \cdot)_H$ defined as

where $\Psi = (\varphi_1, \varphi_2), \; \Psi = (\psi_1, \psi_2)\in H$. Then

Let $A$ be the linear operator defined in (3.44), then

Next we show $A$ generates a $C_0$-semigroup on $H$ by using Lemma 3.1. It is obvious $D(A)$ is dense in $H$, and for any $\Phi = (\varphi_1, \varphi_2)\in D(A)$, we have

Next we show $R(I-A) = H$. Fixed any $f = (f_1, f_2)\in H$, since $f_1+f_2\in L^2( \Omega)$, by standard theory of elliptic equation, the following problem

admits a unique solution $u\in H^4( \Omega)\cap H_0^2( \Omega)$. Let $v = u-f_1\in H_0^2( \Omega)$. Then $U = (u, v)$ satisfies

which implies $R(I-A) = H$. Then by Lemma 3.1, $A$ generates a contraction $C_0$-semigroup on $H$.

Next, we show (3.45) admits a mild solution.

Lemma 3.4. Assume $\alpha\in(-\infty, n)$ and $\sigma\geq0$. Let $H$ be the Hilbert space defined in Lemma 3.3, and $U_0 = (u_0, u_1)\in H = H_0^2( \Omega)\times L^2( \Omega)$. Then there exists a positive constant $T$ depending only on $\|U_0\|_H = \|u_0\|_{H^2}+\|u_1\|$ such that problem (3.45) admits a unique mild solution $U(t)$, i.e., $U = (u, u_t)\in C \left([0, T]; H \right)$ and satisfies

The solution $U$ can be extended to a maximal weak solution in $[0, {T_{\max}})$ such that either

1. ${T_{\max}} = \infty$, i.e., the problem admits a global mild solution; or

2. ${T_{\max}}<\infty$, and

i.e., the solution blows up at a finite time ${T_{\max}}$.

Furthermore, it holds

Proof. Let $F$ be the nonlinear function defined in (3.44). In view of Lemmas 3.2 and 3.3, to prove local existence, uniqueness, and extension of mild solutions, we only need to show $F:H\rightarrow H$ satisfying the local Lipschitz condition.

$\underline{\text{First,we}\ \text{show}\ F(H)\subset H}$. For any $U = (u, v)\in H$, by (3.44), to prove $F(U)\in H$, we only need to prove,

Since $H_0^2( \Omega)\hookrightarrow L^{2(p-1)}( \Omega)$ (see (1.2)), it is obvious $|u|^{p-2}u\in L^2( \Omega)$.

Next we show $\left|x\right|^{-\alpha}* u\in L^2( \Omega)$. According the range of $\alpha$, we divide the proof into two cases: $\alpha\leq 0$ and $\alpha\in(0, n)$.

Case 1. $\alpha\leq 0$. Since $\Omega$ is bound, we have

Then, by Hölder's inequality,

Case 2. $0< \alpha<n$. Since $\alpha<n$, it follows $\frac{2n}{2n- \alpha}<2$. For any $\phi\in L^2( \Omega)$, there exists a positive $C_ \Omega$ depending only on $\Omega$ such that $\|\phi\|_{\frac{2n}{2n- \alpha}}\leq C_ \Omega\|\phi\|$. Since

by using (2.16) with $q = r = \frac{2n}{2n- \alpha}$, $\theta = n- \alpha$, we get

Then we get

So (3.51) is true.

$\underline{\text{Next}\ \text{we}\ \text{show}\ F\ \text{is}\ \text{locally}\ \text{Lipschitz}\ \text{continuous}}$. Let $U_1 = (u_1, v_1)\in H$ and $U_2 = (u_2, v_2)\in H$ be such that

where $M$ is a positive constant. Let

Then, by (3.52) and (3.53),

Case 1. $n = 1, 2, 3, 4$. Since $H_0^2( \Omega)\hookrightarrow L^\infty( \Omega)$ with optimal constant $C_{H_0^2 \rightarrow L^\infty}$, in view of (3.54), we have

Case 2. $n\geq 5$. Since $H_0^2( \Omega)\hookrightarrow L^\frac{2n}{n-4} ( \Omega)$ with optimal constant $C_{H_0^2 \rightarrow L^\frac{2n}{n-4}}$ and $H_0^2( \Omega)\hookrightarrow L^\frac{n(p-2)}{2}( \Omega)$ with optimal constant $C_{H_0^2 \rightarrow L^\frac{n(p-2)}{2}}$, in view of (3.54), we have

In view of the above three inequalities, we get $F$ is locally Lipschitz continuous. Then the local existence and extension of mild solutions follows.

Next we prove (3.50). Suppose firstly $U_0\in D(A) = (H^4( \Omega) \cap H_0 ^2( \Omega))\times H_0 ^2( \Omega)$, then by Lemma 3.2, $U\in C \left([0, {T_{\max}});D(A) \right)\cap C^1 \left([0, {T_{\max}});H \right)$ is a classical solution. Then it follows from (3.45) and (3.47) that

For fixed $t\in[0, {T_{\max}})$, integrating the above equality over $[0, t]$, we get (3.50).

In general case $U_0\in H$, since $D(A)$ is dense in $H$, we approximate $U_0$ by a sequence $\{U_{n0}\}_{n = 1}^\infty$, and then we pass to the limit to obtain (3.50).

Proof of Theorem 1.2. Step 1. Existence of maximal weak solution. By Lemma 3.4 and Definition 1.1, to show the existence of maximal weak solution, we only need to prove the mild solution $U = (u, u_t)$ got in Lemma 3.4 satisfies (1.4).

We denote the inner produce of the Hilbert space $L^2( \Omega)\times L^2( \Omega)$ by $\left(\left(\cdot, \cdot \right)\right)$, i.e.,

Since $C_0^\infty( \Omega)$ is dense in $H_0^2( \Omega)$, by density arguments, we only need to prove (1.4) with $\phi\in C_0^\infty( \Omega)$. Let $U(t) = (u, u_t)\in C \left([0, {T_{\max}}); H_0^2( \Omega)\times L^2( \Omega) \right)$ be the mild solution of (3.45) got in Lemma 3.4 and $\Phi = (0, \phi)$. For fixed $t\in[0, {T_{\max}})$, by using (3.49), we get

We differentiate to obtain

Now, using the standard properties of the semigroup (see for example, [54,Chapter 2]), we obtain

where

is the adjoint operator of $A$; and

Then it follows from (3.55)-(3.57) and (3.49) that

Since $U = (u, u_t)$ and $\Phi = (0, \phi)$, we have

Then it follows from (3.58) that

Since $u\in C \left([0, {T_{\max}}); H_0^2( \Omega) \right)$, integrating by parts, we get

Note $\phi_t = 0$, integrating in time over $[0, t]$ for any $t\in(0, {T_{\max}})$, we obtain (1.4).

Step 2. Proof of $(u_t, u)$ in $C^1[0, {T_{\max}})$ and the equality (1.8). Since

by taking $\phi = u(t)$ in (3.59), we get

i.e, $(u_t, u)$ in $C^1[0, {T_{\max}})$ and (1.8) holds.

Step 3. Proof of the equality (1.5). The energy identity (1.5) follows from (3.50) directly. In fact by using $U = (u, u_t)$, (3.44), and (3.46), we have

Then by (3.50), we get (1.5).

4.   Global existence and finite time blow-up

Proof of Theorem 1.3. Let $u\in C \left([0, {T_{\max}});H_0^2( \Omega) \right)\cap C^1 \left([0, {T_{\max}});L^2( \Omega) \right)$ be the maximal weak solution got in Theorem 1.2. By $E(0)\leq0$ and (1.5), it holds,

By the definitions of $J$ and $I$ (see (1.9) and (1.10)), we get

By (1.5) and (1.6), it follows $J(u) = E(0)-\frac{1}{2}\|u_t\|^2$. Then, by the above two inequalities, we get

and

By (1.11), we get

In the following we divide the proof into two cases: $|\sigma|\leq\sigma^*$ and $|\sigma|>\sigma^*$.

Case 1. $|\sigma|\leq\sigma^*$. It follows from (4.61) and (4.62) that

Let

where $\beta\geq0$ and $\gamma\geq0$ are two constants to be determined later. Then by using (1.8) and (4.63), we have

By Cauchy-Schwartz's inequality,

Then by (4.66) and (4.67), it follows

Subcase 1. $E(0)\leq 0$ and $(u_0, u_1)>0$. We take $\beta = \gamma = 0$, then $h(0) = \|u_0\|^2>0$ and $h'(0) = 2(u_0, u_1)>0$. Then, it follows from Lemma 2.3 that $h$ blows up at a finite time $\hat T$, $\hat T\geq {T_{\max}}$ (by Theorem 1.2), and

Subcase 2. $E(0)<0$ and $(u_0, u_1) = 0$. We take

then $h(0) = \|u_0\|^2+ \beta\gamma^2 = 2\|u_0\|^2>0$, $h'(0) = 2 \beta\gamma = 2\sqrt{-2E(0)}\|u_0\|>0$. Then, it follows from Lemma 2.3 that $h$ blows up at a finite time $\hat T$, $\hat T\geq {T_{\max}}$ (by Theorem 1.2), and

Subcase 3. $E(0)<0$ and $(u_0, u_1)<0$. We take

then,

Then, it follows from Lemma 2.3 that $h$ blows up at a finite time $\hat T$, $\hat T\geq {T_{\max}}$ (by Theorem 1.2), and

Case 2. $|\sigma|>\sigma^*$. Firstly we estimate $\int_ \Omega\int_ \Omega|x-y|^{- \alpha}u(x, t)u(y, t)dxdy$.

$\underline{\text{If}\ \alpha \le 0}$, let $R = \sup_{x, y\in \Omega}|x-y|$, then by Hölder's inequality, we get

$\underline{\text{If }0<\alpha <n}$, since

and $\frac{2n}{2n- \alpha}<p$, by using (2.16) with $q = r = \frac{2n}{2n- \alpha}$, $\theta = n- \alpha$ and Hölder's inequality, we get

where $\kappa$ is the positive constant defined in (2.15).

Let

By (4.69) and (4.70), we get

Subcase 1. $E(0)<0$ and $(u_0, u_1)\geq0$; or $E(0) = 0$ and $(u_0, u_1)>0$. By (1.8), (4.60), and $E(0)\leq 0$, we get

Then

and then $\|u\|\geq\|u_0\|$. Then it follows from the Hölder's inequality that

which, together with (4.71) implies

In view of (1.11), (4.61) and (4.72), we obtain

where

It follows $(\frac{\Lambda}{2}\times(4.60)+\frac{1}{p}\times(4.73))\times\frac{2p}{p\Lambda+2}$ that

Let $h$ be the function defined in (4.64). Then by using (1.8) and (4.75), we have

and

Then it follows from (4.67) that

$\underline{\text{If}\ E(0)\le 0\ \text{and}\ \left( {{u}_{0}},{{u}_{1}} \right)>0}$, we take $\beta = \gamma = 0$, then $h(0) = \|u_0\|^2>0$ and $h'(0) = 2(u_0, u_1)>0$. Then, it follows from Lemma 2.3 that $h$ blows up at a finite time $\hat T$, $\hat T\geq {T_{\max}}$ (by Theorem 1.2), and

$\underline{\text{If }E(0)<0\ \text{and}\ \left( {{u}_{0}},{{u}_{1}} \right)=0}$, we take

then $h(0) = \|u_0\|^2+ \beta\gamma^2 = 2\|u_0\|^2>0$, $h'(0) = 2 \beta\gamma = 2\sqrt{-2E(0)}\|u_0\|>0$. Then, it follows from Lemma 2.3 that $h$ blows up at a finite time $\hat T$, $\hat T\geq {T_{\max}}$ (by Theorem 1.2), and

Subcase 2. $E(0)<0$ and $(u_0, u_1)<0$. We prove $u$ blows up by contradiction argument. Assume $u$ exists globally, i.e., ${T_{\max}} = \infty$.

By (4.60), $-I(u)(t)\geq-2E(0)>0$, $t\in[0, \infty)$. Then it follows (1.8),

Integrating this inequality from $0$ to $t$ we get

Let $t_0 = \frac{(u_0, u_1)}{2E(0)}$, then $(u(t_0), u_t(t_0))\geq0$. Moreover, $E(t_0) = E(0)<0$. We come back to subcase 1. Then $\|u\|$ will become infinite in finite time, which contradicts ${T_{\max}} = \infty$.

Proof of Theorem 1.4. Let $u\in C \left([0, {T_{\max}});H_0^2( \Omega) \right)\cap C^1 \left([0, {T_{\max}});L^2( \Omega) \right)$ be the weak solution got in Theorem 1.2 with $(u_0, u_1)\in H_0^2( \Omega)\times L^2( \Omega)$ satisfying $I(u_0)>0$ and $0<E(0)\leq d$.

Step 1. $E(0)<d$; or $E(0) = d$ and $\|u_1\|>0$. By (1.7), $J(u_0)<d$. Since, we also have $I(u_0)>0$, we get $u_0\in \mathcal{W}$ (see (2.24) for the definition of $\mathcal{W}$). Then, by Lemma 2.6, $u(t)\in \mathcal{W}$ for all $t\in[0, {T_{\max}})$. Since $\mathcal{W} = \mathcal{W}_1$ (see Lemma 2.5), it follows from the definition of $\mathcal{W}_1$ (see (2.28)) that

By (1.11), it follows

which, together with (4.79) and $|\sigma|<\sigma^*$, implies

Since $E(t) = E(0)$ (see (1.5)), $E(t) = \frac12\|u_t\|^2+J(u)(t)$ (see (1.6)), $2J(u)-I(u) = \frac{p-2}{p}\|u\|_p^p$ (see (2.30)), $I(u)(t)>0$ (since $u(t)\in \mathcal{W}$) for all $t\in[0, {T_{\max}})$, and $E(0)\leq d$, it holds

By (4.80), (4.81), and Theorem 1.2, we get ${T_{\max}} = \infty$.

Step 2. $E(0) = d$ and $\|u_1\| = 0$. Since $E(t) = E(0)$ (see (1.5)), $E(t) = \frac12\|u_t\|^2+J(u)(t)$ (see (1.6)) and $E(0) = \frac{1}{2}\|u_1\|^2+J(u_0)$ (see (1.7)), we get

If $J(u(t))\equiv d$ for all $t\in[0, {T_{\max}})$, then by (1.6), $\|u_t\|\equiv0$ for $t\in[0, {T_{\max}})$, then $u(t)\equiv u_0$, so by Theorem 1.2, ${T_{\max}} = \infty$; If there exists $t_1\in[0, {T_{\max}})$ such that $J(u)(t_1)<d$, we claim

● there exists a constant $\sigma>0$ sufficient small, and a sequence $\{t_n\}_{n = 1}^\infty$ such that $\sigma\geq t_n\downarrow0$ as $n\uparrow\infty$ and $J(u)(t_n)<d$ for $n = 1, 2, \cdots$.

In fact, if the claim is not true, there must exists a constant $\delta>0$ such that $J(u)(t) = d$ for $t\in[0, \delta]$. Then by analysis as the first case, $u(t)\equiv u_0$ for $t\in[0, \delta]$; and then $u(t)\equiv u_0$ for $t\in[\delta, 2\delta]$, $[2\delta, 3\delta]$, etc. So $u(t)\equiv u_0$ for all $t\in[0, {T_{\max}})$, then $J(u)(t_1) = J(u_0) = d$, a contradiction. So the claim is true, since $I(u_0)>0$ and $I(u)(t)\in C[0, {T_{\max}})$, $\lim_{n\uparrow\infty}I(u)(t_n) = I(u_0)>0$. So for $n$ large enough, we have $J(u)(t_n)<d$ and $I(u)(t_n)>0$, we come back to step 1.

Proof of Theorem 1.5. Let $u\in C \left([0, {T_{\max}});H_0^2( \Omega) \right)\cap C^1 \left([0, {T_{\max}});L^2( \Omega) \right)$ be the weak solution got in Theorem 1.2 with $(u_0, u_1)\in H_0^2( \Omega)\times L^2( \Omega)$.

If $E(0)<d$, by (1.7), we get $J(u_0)<d$; If $E(0) = d$ and $(u_0, u_1)>0$, we must have $\|u_1\|>0$, and by (1.7) again, we get $J(u_0)<d$. Now we prove $\|u_1\|>0$ by contradiction argument. In fact if $\|u_1\| = 0$, by Cauchy-Schwartz's inequality, we have $0<(u_0, u_1)\leq\|u_0\|\|u_1\| = 0$, a contradiction.

Since $J(u_0)<d$ and $I(u_0)<0$, we get $u_0\in \mathcal{V}$ (see (2.25)). Then by Lemma 2.6, $u(t)\in \mathcal{V}$ for $t\in[0, {T_{\max}})$, where $\mathcal{V}$ is the set defined in (2.25). Since $\mathcal{V} = \mathcal{V}_1$ (see Lemma 2.5), by (2.29) and (4.61),

Let $h(t)$ be the function defined in (4.64), with $\beta\geq0$ and $\gamma\geq0$ to be determined later. Then by using (1.8) and (4.82), we have

Then it follows from (4.67) that

Case 1. $E(0)\leq d$ and $(u_0, u_1)>0$. We take $\beta = \gamma = 0$, then $h(0) = \|u_0\|^2>0$ and $h'(0) = 2(u_0, u_1)>0$. Then, it follows from Lemma 2.3 that $h$ blows up at a finite time $\hat T$, $\hat T\geq {T_{\max}}$ (by Theorem 1.2), and

Case 2. $E(0)<d$ and $(u_0, u_1) = 0$. We take

then $h(0) = \|u_0\|^2+ \beta\gamma^2 = 2\|u_0\|^2>0$, $h'(0) = 2 \beta\gamma = 2\sqrt{2(d-E(0))}\|u_0\|>0$. Then, it follows from Lemma 2.3 that $h$ blows up at a finite time $\hat T$, $\hat T\geq {T_{\max}}$ (by Theorem 1.2), and

Case 3. $E(0)<d$ and $(u_0, u_1)<0$. We take

then,

Then, it follows from Lemma 2.3 that $h$ blows up at a finite time $\hat T$, $\hat T\geq {T_{\max}}$ (by Theorem 1.2), and

Acknowledgments

The authors would like to thank the referees for the comments and valuable suggestions.