Existence and blow-up of solutions for finitely degenerate semilinear parabolic equations with singular potentials

1.   Introduction

In this article, we study the initial-boundary value problem for the class of finitely degenerate semilinear parabolic equations with singular potential term as follows

where $X = \left(X_{1}, \cdots, X_{m}\right)$ is a system of real smooth vector fields defined on open set $U$ in $\mathbb{R}^n$ for $n\geq3$, $\Omega\subset\subset U$ is a bounded open domain, $X_{j} = \sum^{n}_{k = 1}a_{jk}\left(x\right)\partial_{x_{k}}$, $a_{jk}\in C^{\infty}\left(U\right)$, $j = 1, \cdots, m$, and $\left(x_1, \ldots, x_n\right)$ is the coordinate system of $U$. In general, $X_{j}$ is different from the position vector field $x = \sum^{n}_{k = 1}x_k\partial_{x_{k}}$ of $U$ in $\mathbb{R}^n$. In the whole paper, we always suppose that the system of vector fields $X$ is finitely degenerate, i.e., it satisfies the following Hörmander's condition ^[13] with $Q > 1$.

$\rm \left(H\right)$ ${X_{1}, X_{2}, \ldots, X_{m}}$ together with their commutators of length at most $Q$ can span the tangent space $T_{x}\left(U\right)$ at each point $x\in U$, where $Q$ is the Hörmander index of $U$ with respect to $X$.

The sum of square operator $\Delta_{X}: = \sum^{m}_{j = 1}X^{2}_{j}$, also called the Hörmander type operator, is finitely degenerate elliptic operator if $Q > 1$, while it is the usual elliptic operator and $m\geq n$ if $Q = 1$. Here, we further pose the following hypotheses:

$\left({\rm{H}}_{\partial \Omega}\right)$ $\partial \Omega$ is smooth and non-characteristic for the system of vector fields $X$, i.e., for any $x\in\partial \Omega$, there exists at least one vector field $X_{j}$ such that $X_{j}\left(x\right) \notin T_{x}\left(\partial \Omega\right)$.

$\left({\rm{H}}_p\right)$ $1 < p\leq\frac{\tilde{\nu}}{\tilde{\nu}-2}$, where $\tilde{\nu}\geq3$ is the generalized Métivier index (cf. Definition 2.2).

$\left({\rm{H}}_V\right)$ $\mu\in \left(0, 1/C^{2}_{*}\right)$ is constant, and the positive singular potential function $V\left(x\right)\in L^\infty\left(\Omega\right)\cap C\left(\Omega\right)$ satisfies the Hardy's inequality

for any $u$ of the Hilbert space $H^{1}_{X, 0}\left(\Omega\right)$ (cf. Section 2), where

$\left({\rm{H}}_g\right)$ $g\left(x\right)\in L^\infty\left(\Omega\right)\cap C\left(\Omega\right)$ is a non-negative weighted function.

Finitely degenerate elliptic operators originate from physical applications and mathematical problems, e.g., Lewy's example ^[19], the stochastic differential equations ^[30], $\bar{\partial}$-Neumann problem in complex geometry ^[17], Kohn Laplacian on the Heisenberg group $\mathbb{H}^{n}$ in quantum mechanics ^[4]. Hörmander ^[13] proved the hypoellipticity and the subelliptic estimates of $\Delta_{X}$, and thus $\Delta_{X}$ is still called the subelliptic operator. Bony ^[3] obtained the maximum principle and the Harnack inequality of $\Delta_{X}$, and Rothschild and Stein ^[29] gave the regularity estimates of $\Delta_{X}$. By Hörmander condition one can define a Carnot-Carathéodory metric induced by $X$, which is paid attentions by scholars in sub-Riemannian geometry ^[27]. Moreover, the Poincaré inequality ^[14], the Sobolev embedding theorem ^[6,33,39], heat kernel and Green kernel estimates ^[15] were well investigated.

Furthermore, under the Métivier's condition Métivier ^[26] studied the eigenvalues problems of $\Delta_{X}$, and defined the Métivier index $\nu$, also namely the Hausdorff dimension of $\Omega$ related to $X$. For example, let $X = \left (X_1, \ldots, X_n, Y_1, \ldots, Y_n\right)$ on the Heisenberg group $\mathbb{H}^{n}\subset \mathbb{R}^{2n+1}$, where $X_j = \partial_{x_j}+2y_j\partial_{t}$, $Y_j = \partial_{y_j}+2x_j\partial_{t}, j = 1, \ldots, n$. Then $X$ satisfies the Hörmander's condition for $Q = 2$, the Métivier's condition for $\nu = 2n+2$, and the Kohn Laplacian $\Delta_{X} = \sum^{n}_{j = 1}\left(X^{2}_{j}+Y^{2}_{j}\right)$ is a finitely degenerate elliptic operator. Unfortunately, in the finitely degenerate case, if no Métivier's condition there are no Rellich-Kondrachov compact embedding results, while such compact embedding results play an important role when one discusses the existence of solutions for the Dirichlet problem of semilinear subelliptic equations. To deal with this case, Chen and Luo ^[8] defined the generalized Métivier index $\tilde{\nu}$, also named non-isotropic dimension of $\Omega$ associated with $X$ ^[39], which is exactly the Métivier index under the Métivier's condition. Note that $X$ always has the generalized Métivier index $\tilde{\nu}$ on $\Omega$ even without the Métivier's condition. For example, the Grushin type vector fields $X = \left(\partial_{x_{1}}, \ldots, \partial_{x_{n-1}}, x_1^{i}\partial_{x_n}\right)$, $n\geq2$, $i\in \mathbb{Z}^{+}$, defined on a domain $\Omega\subset \mathbb{R}^{n}$. If $\Omega\cap\{x_1 = 0\}\neq\emptyset$, then $X$ satisfies the Hörmander's condition with $Q = i+1$, the Grushin type operator $\Delta_{X} = \sum^{m}_{j = 1}X^{2}_{j}$ is finitely degenerate, and $\tilde{\nu} = n+i$.

For the vector fields $X = \left(\partial_{x_{1}}, \partial_{x_{2}}, \cdots, \partial_{x_{n}}\right)$, $\Delta_{X}$ is exactly the usual Laplacian operator $\Delta$, and the equation in (1.1) is the heat equation with singular potentials, which has attracted attentions since the work of Baras and Goldstein ^[2] in 1984. In fact, they studied the initial-boundary value problem for the linear heat equation

with singular potential $V\left(x\right) = c/\left|x\right|^{2}$, and proved that under the initial data $u_{0} > 0$, if $c\leq\frac{\left(n-2\right)^2}{4}$ it has a global weak solution, otherwise it has no solution ^[2] and even no local solution ^[5]. Particularly, if $V\left(x\right) = 0$, the equation in (1.1) becomes

which has been popularly studied. For the initial-boundary value problem (1.5), Liu in ^[23] improved the potential wells method of Payne and Sattinger ^[28], and obtained the global existence and blow-up of solutions with subcritical initial energy in ^[24]. Then, Xu ^[34] studied this problem with critical initial energy, and Gazzola and Weth ^[12] further discussed the high initial energy. Since the family of potential wells was proposed in ^[23], it has been used to study various important and interesting nonlinear evolution equations, including hyperbolic ^{[20,22,32,38]}, the system of coupled parabolic equation ^[35], and the pseudo-parabolic equation ^[21,36,37].

On singular manifolds, Alimohammady and Kalleji ^[1] studied the initial-boundary value problem of the semilinear evolution equation as follows

with $\varrho = 1$, obtained the global existence and the finite time blow-up of weak solutions on cone type Sobolev spaces. However, for the case $k\geq 2$ the results in ^[1] are invalid, hence later Luo, Xu and Yang ^[25] considered the case $k = 2$ and $\varrho$ in some value range, proved the local existence and uniqueness of the solution by using the contraction mapping principle, and obtained the existence of global solutions and finite time blow-up of solutions on the cone-type Sobolev spaces. On the other hand, the edge-degenerate parabolic equation with singular potentials was studied by Chen and Liu ^[7].

In this article, under the assumptions ${\left(\rm H\right)}$, ${\left(\rm H_{\partial \Omega}\right)}$, ${\left(\rm H_{\it V}\right)}$, ${\left(\rm H_{\it g}\right)}$ and ${\left(\rm H_{\it p}\right)}$, by the known properties of $\Delta_{X}$ we establish the local and global existence, decay and finite time blow-up of the solutions for problem (1.1). This article is organized as follows. After introducing some notions and results on the finite degenerate vector fields in Section 2, by applying the Galerkin method and Banach fixed theorem we establish the local existence and uniqueness of the weak solution of problem (1.1) in Section 3. In Section 4, by constructing a family of potential wells, we prove some auxiliary results for it. In Section 5, by potential well method we obtain the global existence, the decay estimate and the finite time blow-up of solutions with subcritical or critical initial energy.

2.   Preliminaries

In this section, we recall some notions and properties of the finite degenerate vector fields $X$.

First, by $X$ we define the Sobolev space (cf. ^[33])

This is a Hilbert space equipped with the norm

where $\|Xu\|^2_{L^2\left(U\right)} = \sum^{m}_{i = 1}\|X_{i}u\|^2_{L^2\left(U\right)}$. We denote by $H^{1}_{X, 0}\left(\Omega\right)$ the closure of $C^\infty_{0}\left(\Omega\right)$ in $H^{1}_{X}\left(U\right)$, which is still a Hilbert space. For simplicity, from now on, we write $\|\cdot\|_{H^{1}_{X, 0}} = \|\cdot\|_{H^{1}_{X, 0}\left(\Omega\right)}$, $\|\cdot\|_{p} = \|\cdot\|_{L^p\left(\Omega\right)}$ for $1\leq p\leq\infty$, and also let $\|\cdot\| = \|\cdot\|_{L^2\left(\Omega\right)}$. Moreover, we denote by $\left(u, v\right)$ the inner product in $L^2\left(\Omega\right)$, and follow the convention that $C$ is an arbitrary positive constant, which may be different from line to line.

Next, we introduce the Métivier's condition ^[26] and the generalized Métivier index ^[8] as follows.

Definiton 2.1 (Métivier's condition). Under the Hörmander's condition $\rm{\left(H\right)}$ for the vector fields $X$, let $V_{i}\left(x\right)$ be the subspace of the tangent space at $x\in \bar{\Omega}$ spanned by all commutators of $X_{1}, \cdots, X_{m}$ with length at most $i$ for $1\leq i\leq Q$. If each $\nu_{i} = \dim V_{i}\left(x\right)$ is constant on some neighborhood of every $x\in \bar{\Omega}$, we call $X$ satisfying the Métivier's condition on $\Omega$, and define the Métivier index by

namely also the Hausdorff dimension of $\Omega$ related to the subelliptic metric induced by $X$.

Definiton 2.2 (Generalized Métivier index). Under the Hörmander's condition $\rm{\left(H\right)}$, by the notations in Definition 2.1, let $\nu_{i}\left(x\right)$ be the dimension of vector space $V_{i}\left(x\right)$ at point $x\in \bar{\Omega}$, we define the pointwise homogeneous dimension at $x$ by

Then, the generalized Métivier index of $\Omega$ is defined by

which is also named the non-isotropic dimension of $\Omega$ (cf. ^[39]).

For $Q > 1$ we see from (2.1) that $3\leq n+Q-1\leq \tilde{\nu} < nQ$, and $\tilde{\nu}$ is exactly $\nu$ under the Métivier's condition.

Now, we recall the weighted Poincaré inequality and weighted Sobolev embedding theorem related to $X$ as follows.

Proposition 2.1 (Weighted Poincaré inequality ^[16]). Under the assumptions $\rm{\left(H\right)}$ and $\rm{\left(H_{\partial \Omega}\right)}$, the first eigenvalue $\lambda_1$ of $-\Delta_{X}$ is strictly positive, and

Proposition 2.2 (Weighted Sobolev embedding theorem ^[39]). Under the assumptions $\rm{\left(H\right)}$ and $\rm{\left(H_{\partial \Omega}\right)}$, for arbitrary $u\in C^\infty (\bar{\Omega})$ we have

where $\frac{1}{p^{*}} = \frac{1}{p}-\frac{1}{\tilde{\nu}}$, $p \in[1, \tilde{\nu})$ related to the generalized Métivier index $\tilde{\nu}$, and $C = C\left(\Omega, X\right)$ is a positive constant.

Remark 2.1. As $\tilde{\nu}\geq3$, by Proposition 2.2 for $p = 2$ we see that $H^{1}_{X, 0}\left(\Omega\right)\hookrightarrow L^{q}\left(\Omega\right)$ is a bounded embedding for any $1\leq q\leq 2^*_{\tilde{\nu}}: = \frac{2\tilde{\nu}}{\tilde{\nu}-2}$.

Proposition 2.3 (compact embedding theorem, cf. ^[9]). Under the assumptions $\rm{\left(H\right)}$ and $\rm{\left(H_{\partial \Omega}\right)}$, for $1\leq q < 2^*_{\tilde{\nu}}$, the embedding

is compact.

Note from $\rm{\left(H_{\it{p}}\right)}$ and Remark 2.1 that $2 < p+1 < 2^*_{\tilde{\nu}}$. Together with the Poincaré inequality (2.3), Proposition 2.3 and $\rm{\left(H_{\it{g}}\right)}$, we can deduce the following inequality.

Lemma 2.1. Under the assumptions $\rm{\left(H\right)}$, $\rm{\left(H_{\partial \Omega}\right)}$, $\rm{\left(H_{\it{p}}\right)}$ and $\rm{\left(H_{\it{g}}\right)}$, for arbitrary $u \in H^{1}_{X, 0}\left(\Omega\right)$, we have

Thanks to Lemmas 2.1, for $1 < p\leq\frac{\tilde{\nu}}{\tilde{\nu}-2}$ we can define a positive constant

Proposition 2.4 (cf. ^[9]). Under the assumptions $\rm{\left(H\right)}$ and $\rm{\left(H_{\partial \Omega}\right)}$, the subelliptic Dirichlet problem

is well-defined, i.e., $-\Delta_{X}$ possesses a sequence of discrete eigenvalues $\{\lambda_k\}_{k\geq1}$ such that $0 < \lambda_1 < \lambda_2 \leq \lambda_3 \leq \cdots \leq \lambda_k \leq \cdots$, and $\lambda_k\rightarrow +\infty$ as $k\rightarrow +\infty$. Denote the corresponding eigenfunctions by $\{\phi_k\}_{k\geq1}$, which forms an orthonormal basis of $L^2\left(\Omega\right)$ and also an orthogonal basis of the Hilbert space $H^{1}_{X, 0}\left(\Omega\right)$.

Lemma 2.2. For $n\geq3$, $C^{\infty}_{0}\left(\Omega\backslash\{0\}\right)$ is dense in $H^{1}_{X, 0}\left(\Omega\right)$.

Proof. As $C^{\infty}_{0}\left(\Omega\right)$ is dense in $H^{1}_{X, 0}\left(\Omega\right)$, we just need to prove that

Denote by $\varphi$ a smooth function such that

Now, taking a sufficiently small $\epsilon > 0$ and defining $u_\epsilon\left(x\right) = \varphi\left(\frac{\left|x\right|}{\epsilon}\right)u\left(x\right)$ for $u\in C^{\infty}_{0}\left(\Omega\right)$, we have $u_\epsilon\left(x\right)\in C^{\infty}_{0}\left(\Omega\backslash\{0\}\right)$ and

It follows from the dominated convergence theorem that

Moreover,

Lemma 2.2 has been proved. □

By the Hardy inequality on $C^{1}_{0}\left(\Omega\backslash\{0\}\right)$ related to degenerate elliptic differential operators ^[10] and Lemma 2.2, we immediately see that there exists a positive singular potential function $V\left(x\right)\in L^\infty\left(\Omega\right)\cap C\left(\Omega\right)$ such that Hardy inequality (1.2) holds for any $u\in H^{1}_{X, 0}\left(\Omega\right)$. Therefore, the assumption $(H_V)$ is reasonable. From $(H_V)$ and the Poincaré inequality (2.3) we see that the operator $-\Delta_X-\mu V\left(x\right)$ is a positive operator on $H^{1}_{X, 0}\left(\Omega\right)$. Moreover, we have the following result.

Proposition 2.5. Under the assumptions $\rm{\left(H\right)}$, $\rm{\left(H_{\partial \Omega}\right)}$ and $\rm{\left(H_{\it{V}}\right)}$, the Dirichlet eigenvalue problem

is well-defined, i.e., $-\Delta_X-\mu V\left(x\right)$ possesses a sequence of discrete Dirichlet eigenvalues $\{\eta_k\}_{k\geq1}$ such that $0 < \eta_1 \leq \eta_2 \leq \eta_3 \leq \cdots \leq \eta_k \leq \cdots$, and $\eta_k\rightarrow +\infty$ as $k\rightarrow +\infty$. Denote the corresponding eigenfunctions by $\{\varphi_k\}_{k\geq1}$, which is an orthonormal basis of $L^2\left(\Omega\right)$ and also an orthogonal basis of the Hilbert space $H^{1}_{X, 0}\left(\Omega\right)$.

Proof. Define the bilinear form

where $L_\mu: = -\Delta_X-\mu V\left(x\right)$ is an operator defined on the Hilbert space $H^{1}_{X, 0}\left(\Omega\right)$. By combining with the Hölder inequality, (1.3) and the Poincaré inequality (2.3) we have

and

It follows from the Lax-Milgram theorem that for any $g\in H^{-1}_{X}\left(\Omega\right)$, the Dirichlet problem

has a unique solution $u\in H^{1}_{X, 0}\left(\Omega\right)$, where $H^{-1}_{X}\left(\Omega\right)$ is the dual space of $H^{1}_{X, 0}\left(\Omega\right)$ with the norm

and $L_\mu : H^{1}_{X, 0}\left(\Omega\right)\rightarrow H^{-1}_{X}\left(\Omega\right)$ is continuous. Therefore, the inverse operator $L^{-1}_\mu = (-\Delta_X-\mu V\left(x\right))^{-1}$ of $L_\mu$ is well-defined and is a continuous map from $H^{-1}_{X}\left(\Omega\right)$ into $H^{1}_{X, 0}\left(\Omega\right)$.

Since that the embedding $i: H^{1}_{X, 0}\left(\Omega\right)\rightarrow L^2\left(\Omega\right)$ is compact and the embedding $i^*:L^2\left(\Omega\right)\rightarrow H^{-1}_{X}\left(\Omega\right)$ is continuous, we deduce that

is a compact and self-adjoint operator. Therefore, $K_\mu$ possesses a sequence of discrete eigenvalues $\{\mu_k\}_{k\geq1}$ such that $\mu_k > 0$, decreasing on $k$ and $\mu_k\rightarrow 0$ as $k\rightarrow +\infty$. Denote the corresponding eigenfunctions by $\{\varphi_k\}_{k\geq1}$, then

and $\{\varphi_k\}_{k\geq1}$ form an orthonormal basis of $H^{1}_{X, 0}\left(\Omega\right)$. Proposition 2.5 has been proved. □

Finally, we give the definition of weak solutions.

Definiton 2.3 (Weak solution). A function $u = u\left(x, t\right)$ is called a weak solution of problem (1.1) on $\Omega \times [0, T)$, if $u \in L^{\infty}(0, T; H^{1}_{X, 0}\left(\Omega\right))$ with $u_t \in L^2(0, T; L^2\left(\Omega\right))$ satisfies $u(0, x) = u_{0}\left(x\right)\in H^{1}_{X, 0}\left(\Omega\right)$ and

for any $w\in H^{1}_{X, 0}\left(\Omega\right)$, $0 < t < T$, where $T$ is the maximum existence time of the solution.

3.   Local existence of the solution

In this section, we will prove the existence and uniqueness of the local solution for the problem (1.1). First, we consider the linear problem of (1.1)

For a given $T > 0$ and any $\mu \in\left (0, \frac{1}{C^{2}_{*}}\right)$, define the Banach space

equipped with the norm

By the Galerkin method we establish the local existence result of the problem (3.1) as follows.

Lemma 3.1. Under the assumptions $\rm{\left(H\right)}$, $\rm{\left(H_{\partial \Omega}\right)}$, $\rm{\left(H_{\it V}\right)}$, $\rm {\left(H_{\it g}\right)}$ and $\rm {\left(H_{\it p}\right)}$, for every $u_{0} \in H^{1}_{X, 0}\left(\Omega\right)$ and $u \in \mathcal{H}$, the problem (3.1) has a unique local solution $v \in \mathcal{H}$.

Proof. By Proposition 2.5, we see that $\{\eta_i\}_{i\geq1}$ are the eigenvalues of the positive operator $L_\mu = -\Delta_X-\mu V\left(x\right)$ of the Dirichlet eigenvalue problem

where $\|\varphi_i\| = 1$ for all $i$, and the eigenfunctions $\{\varphi_i\}_{i\geq1}$ are the orthogonal basis of both $H^{1}_{X, 0}\left(\Omega\right)$ and $L^2\left(\Omega\right)$. Let $W_m = \rm{Span}\it{\{\varphi_1, \cdots, \varphi_m\}}$, $m\in \mathbb{N^{+}}$. For each $m\in \mathbb{N^{+}}$, we can construct the approximate solutions of problem (3.1) as follows

which satisfies the following Cauchy problem in $W_m$

By taking (3.4) into (3.5), we get the Cauchy problem of the ordinary differential equation with respect to $h_{i m}\left(t\right)$ as follows

Thanks to the theory of ordinary differential equations, the problem (3.6) has a solution $h^{}_{i m}\in C^{1}\left[0, T\right]$ for each $i$. Multiplying both sides of the equation in (3.5) by $h^{'}_{i m}\left(t\right)$, summing for $i$ and integrating over $\left[0, t\right]$, one has

Next, according to the Hölder inequality, $\rm {\left(H_{\it g}\right)}$, the Sobolev embedding $H^{1}_{X, 0}\left(\Omega\right)\hookrightarrow L^{2p}\left(\Omega\right)$, the Poincaré inequality (2.3) and the Cauchy inequality with $\epsilon$, we can estimate the last term of (3.7) as follows

where the positive constant $C$ may be different from line to line. By choosing $\epsilon > 0$ such that $2C\epsilon\|g\|_{\infty} = 1$, we see from (1.3), (3.7) and (3.8) that

Let $\xrightarrow{w^{*}}$ be the weakly star convergence. By (3.9) we have a subsequence, also denoted by $\{v_m\}$, satisfying as $m\rightarrow \infty$,

These imply that

Then one has from Evans Theorem (^[11], 5.9.2. Theorem 2, p. 304) that

By Proposition 2.3 and Remark 2.1, the injection $H^{1}_{X, 0}\hookrightarrow L^{2}\left(\Omega\right)$ is continuous and compact, which together with (3.12) and Temam lemma (^[31], Section II, Lemma 3.3) shows that

It follows from (3.5) and (3.10) that

For fixed $i$, letting $m\rightarrow \infty$, taking the limit in (3.5), by (3.10)-(3.11) we get

Since $\{\varphi_i\}_{i\geq1}$ is a base of $H^{1}_{X, 0}\left(\Omega\right)$, we deduce that $v \in \mathcal{H}$ satisfies the equation in (3.1).

Finally, we prove the uniqueness of solutions. Otherwise, assume that $w_{1}$ and $w_{2}$ are two solutions of problem (3.1). Let $\tilde{w} = w_{1}-w_{2}$, there holds

Multiplying both sides of $\tilde{w}_t-\Delta_{X} \tilde{w}-\mu V\left(x\right)\tilde{w} = 0$ by $\tilde{w}_t$, and integrating it over $\Omega\times(0, t)$, we have

It follows from $\rm{\left(H_{\it V}\right)}$ that

and thus $\tilde{w} = 0$ a.e. in $\Omega$, i.e., $w_{1}\equiv w_{2}$. The conclusion follows. □

Theorem 3.1 (Local existence). Under the assumptions $\rm {\left(H\right)}$, $\rm {\left(H_{\partial \Omega}\right)}$, $\rm {\left(H_{\it V}\right)}$, $\rm {\left(H_{\it g}\right)}$ and $\rm {\left(H_{\it p}\right)}$, if $u_{0} \in H^{1}_{X, 0}\left(\Omega\right)$, there exists $T > 0$ such that the problem (1.1) has a unique weak solution

Proof. For any $T > 0$, we define the set

where

By Lemma 3.1 we can define the mapping $\Psi$ on $\mathcal{M}_{T}$, such that $\Psi\left(u\right)$ is the unique solution of the problem (3.1), i.e., $\Psi\left(u\right) = v$. We will prove that $\Psi:\mathcal{M}_{T}\rightarrow \mathcal{M}_{T}$ is a contractive mapping for small enough $T$.

First, for sufficiently small $T$ we show that $\Psi$ is a mapping from $\mathcal{M}_{T}$ to itself. For any $u\in\mathcal{M}_{T}$, similar to (3.7) and (3.8) the unique solution $v = \Psi\left(u\right)$ satisfies

It follows from (1.3) that

Then by (3.2) we obtain

Therefore, for $T$ small enough $\|u\|^{2}_{\mathcal{H}}\leq\rho^{2}$, i.e., $\Psi(\mathcal{M}_{T})\subseteq\mathcal{M}_{T}$.

Now, we will show that $\Psi$ is a contraction mapping. Let $u_{1}, u_{2}\in\mathcal{M}_{T}$ and $v_{1} = \Psi\left(u_{1}\right)$, $v_{2} = \Psi\left(u_{2}\right)$. By taking $\tilde{v}: = v_{1}-v_{2}$, we see that $\tilde{v}$ satisfies the following problem

Multiplying the equation above by $\tilde{v}_t$, and integrating it over $\Omega\times(0, t)$, we deduce

Note from Lemma 4 of ^[32] that $\left|u_{1}\right|^{p-1}u_{1}-\left|u_{2}\right|^{p-1}u_{2} \leq p \left(\left|u_{1}\right| + \left|u_{2}\right| \right)^{p-1}\left|u_{1}-u_{2}\right|$. Together with the Minkowski inequality, similar to (3.8) we have

Combining with (1.3), (3.20) and (3.21) we can deduce that

It follows from (3.2) that

By choosing $T > 0$ such that $\delta_{ T} = CT\rho^{2\left(p-1\right)} < 1$, we obtain that $\Psi$ is a contraction mapping from $\mathcal{M}_{T}$ to itself. Thanks to the Banach fixed point theorem, we get the local existence result. The proof has been completed. □

4.   Some auxiliary results of the potential wells

Under the assumptions $\rm {\left(H\right)}$, $\rm {\left(H_{\partial \Omega}\right)}$, $\rm {\left(H_{\it V}\right)}$, $\rm {\left(H_{\it g}\right)}$ and $\rm {\left(H_{\it p}\right)}$, for further discussions we construct a family of potential wells in this section, and prove some auxiliary results for it.

First, we define the potential energy functional $J$ and Nehari functional $I$ on $H^{1}_{X, 0}\left(\Omega\right)$ given by

It follows that

Define the mountain pass level

also called potential well depth. We now discuss the properties of the functionals $J$ and $I$.

Lemma 4.1. For arbitrary $u\in H^{1}_{X, 0}\left(\Omega\right)$ and $\|Xu\|\neq 0$, we have

$\rm{(1)}$ $\mathop{\lim}\limits_{\lambda \rightarrow 0}J\left(\lambda u\right) = 0$, and $\mathop{\lim}\limits_{\lambda \rightarrow +\infty}J\left(\lambda u\right) = -\infty$;

$\rm{(2)}$ $J\left(\lambda u\right)$ with respect to $\lambda$ is strictly decreasing on $[\lambda_{X}, +\infty)$, strictly increasing on $\left[0, \lambda_{X}\right]$, and thus attains the maximum at $\lambda_{X}$, where

$\rm{(3)}$

$\rm{(4)}$ $d = \frac{p-1}{2\left(p+1\right)}\left(1-\mu C^{2}_{*}\right)^{\frac{p+1}{p-1}}C^{-\frac{2\left(p+1\right)}{p-1}}_X,$ where $C_X$ is the best Sobolev constant defined in (2.4).

Proof. It follows from (4.1) that

and

Then, we have Lemma 4.1 (1) and

Hence we have a unique $\lambda_{X}: = \left(\frac{\|Xu\|^2-\int_{\Omega} \mu V\left(x\right)\left|u\right|^2dx} {\|g\left(x\right)^{\frac{1}{p+1}}~~ u\|^{p+1}_{p+1}}\right)^{\frac{1}{p-1}}$ such that $\frac{d}{d \lambda}J\left(\lambda u\right)\mid_{\lambda = \lambda_X} = 0$ and

where we used (1.3) and (2.4) in the inequality above. Together with (4.3) we immediately get remaining conclusions. □

Defining the Nehari manifold

by Lemma 4.1 we get $d > 0$, and

For any $\delta > 0$, we introduce the functionals

with the associated Nehari manifolds

and the depth of such potential wells

where $C_{*}$ is defined in (1.3). With these in mind we can prove

Lemma 4.2. Assume $u \in H^{1}_{X, 0}\left(\Omega\right)$, we obtain

$\rm{(1)}$ if $0 < \|X u\| < r\left(\delta\right)$, there holds $I_{\delta}\left(u\right) > 0$;

$\rm{(2)}$ if $I_{\delta}\left(u\right) < 0$, there holds $\|X u\| > r\left(\delta\right)$;

$\rm{(3)}$ if $I_{\delta}\left(u\right) = 0$, either $\|Xu\| = 0$ or $\|X u\|\geq r\left(\delta\right)$ holds;

$\rm{(4)}$ if $I_{\delta}\left(u\right) = 0$ and $\|Xu\|\neq0$, there hold

Proof. $\rm (1)$ As $0 < \|X u\| < r\left(\delta\right)$, by (1.3) and (2.4) there holds

By the definitions of $I_{\delta}\left(u\right)$ we have Lemma 4.2 (1).

$\rm (2)$ For $I_{\delta}\left(u\right) < 0$, we obtain that $\|Xu\|\neq0$ and

The conclusion (2) follows.

$\rm (3)$ When $I_{\delta}\left(u\right) = 0$, there holds

Thus the conclusion (3) holds.

$\rm (4)$ The last conclusion follows immediately from (3) and

□

Next, we estimate the depth $d\left(\delta\right)$ and its expression as follows.

Lemma 4.3. For the function $d\left(\delta\right)$, there hold

$\rm{(1)}$ for $\delta\in\left(0, \frac{p+1}{2}\right)$, $d\left(\delta\right)\geq b\left(\delta\right)r^2\left(\delta\right)$, where $b\left(\delta\right): = \left(1-\mu C^{2}_{*}\right)\left(\frac{1}{2}-\frac{\delta}{p+1}\right)$;

$\rm{(2)}$ for $\delta\in\left(0, \frac{p+1}{2}\right)$, $d\left(\delta\right) = \mathop{\inf}\limits_{u \in \mathcal {N}_{\delta}}J\left(u\right) = \left(\frac{1}{2}-\frac{\delta}{p+1}\right)\frac{2\left(p+1\right)}{p-1}\delta^\frac{2}{p-1}d$;

$\rm{(3)}$ $\mathop{\lim}\limits_{\delta \rightarrow 0}d\left(\delta\right) = 0$, $d\left(\frac{p+1}{2}\right) = 0$, and $d\left(\delta\right) < 0$ for $\delta\in\left(\frac{p+1}{2}, +\infty\right)$;

$\rm{(4)}$ $d\left(\delta\right)$ is strictly increasing on $0 < \delta\leq1$, decreasing on $1\leq\delta\leq\frac{p+1}{2}$ and attains the maximum $d$ at $\delta = 1$.

Proof. $\rm (1)$ For $u \in \mathcal {N}_\delta$, we have $I_{\delta}\left(u\right) = 0$ and $\|X u\|\neq0$. It follows from Lemma 4.2 (3) that

Together with (1.3) and (4.6) we see that

By combining with (4.5) we have $d\left(\delta\right)\geq b\left(\delta\right)r^2\left(\delta\right)$.

$\rm (2)$ Taking $u_{*} \in \mathcal {N}$ as the minimizer of $d = \mathop{\inf}\limits_{u \in \mathcal {N}}J\left(u\right)$, i.e., $d = J\left(u_{*}\right)$, we introduce $\lambda = \lambda\left(\delta\right)$ by

Then there holds

and thus $\lambda u_{*} \in \mathcal {N}_\delta$. Together with $I\left(u_{*}\right) = 0$, (4.1) and (4.5), we deduce

Note that

thus

for any $\delta\in\left(0, \frac{p+1}{2}\right)$.

Now, by taking $u^{*}\in \mathcal {N}_\delta$ as the minimizer of $d\left(\delta\right) = \mathop{\inf}\limits_{u \in \mathcal {N}_\delta}J\left(u\right)$, i.e., $J(u^{*}) = d\left(\delta\right)$, we determine $\lambda = \lambda\left(\delta\right)$ by

Therefore, we obtain

and thus $\lambda u^{*} \in \mathcal {N}$. Combining with (4.1), (4.4) and $I_\delta\left(u^{*}\right) = 0$, we have

Together with

we deduce

which shows

By (4.7) and (4.8) we have Lemma 4.3 (2).

The conclusions of (3) and (4) follow immediately from (2) and

□

Lemma 4.4. Assume that $u \in H^{1}_{X, 0}\left(\Omega\right)$, $J\left(u\right)\leq d\left(\delta\right)$ with $\delta\in \left(0, \frac{p+1}{2}\right)$.

$\rm{(1)}$ For $I_{\delta}\left(u\right) > 0$, there holds $\|Xu\|^2 < d\left(\delta\right)/ b\left(\delta\right)$.

$\rm{(2)}$ For $I_{\delta}\left(u\right) = 0$, there holds $\|Xu\|^2 \leq d\left(\delta\right)/ b\left(\delta\right)$.

$\rm{(3)}$ For $\|Xu\|^2 > d\left(\delta\right)/ b\left(\delta\right)$, there holds $I_{\delta}\left(u\right) < 0$.

Proof. As $\delta\in\left(0, \frac{p+1}{2}\right)$, we can see from (4.6), (1.3) and $J\left(u\right)\leq d\left(\delta\right)$ that

Then, the corresponding conclusions in Lemma 4.4 follow from the assumption of (1)-(3), respectively. □

Lemma 4.5. Suppose that $0 < J\left(u\right) < d$ for any given $u\in H^{1}_{X, 0}\left(\Omega\right)$. Denote by $\delta_1$, $\delta_2$ the two roots of $d\left(\delta\right) = J\left(u\right)$ with $\delta_1 < 1 < \delta_2$. Then the sign of $I_{\delta}\left(u\right)$ is unchangeable on $\delta \in\left(\delta_1, \delta_2\right)$.

Proof. Otherwise, we assume that $I_{\tilde{\delta}}\left(u\right) = 0$ for some $\tilde{\delta}\in \left(\delta_1, \delta_2\right)$. Note from the assumption $J\left(u\right) > 0$ that $\|X u\|\neq0$. It follows from (4.5) that $d(\tilde{\delta})\leq J\left(u\right)$, which contradicts $J\left(u\right) = d\left(\delta_1\right) = d\left(\delta_2\right) < d(\tilde{\delta})$. □

Now, we introduce the potential well

and the outer of the potential well

For each $\delta \in\left(0, \frac{p+1}{2}\right)$, by the ideas of ^[23] we can extend $W$ and $V$ respectively to the more general family of potential wells

and its outsider

From Lemma 4.3 we get the following result.

Lemma 4.6. There hold that

$\rm{(1)}$ $W_{\delta_{*}}\subset W_{\delta^{*}}$ for any $0 < \delta_{*} < \delta^{*}\leq 1$;

$\rm{(2)}$ $V_{\delta^{*}}\subset V_{\delta_{*}}$ for any $1\leq \delta_{*} < \delta^{*} < \frac{p+1}{2}$.

Moreover, by introducing

we can prove the following result.

Lemma 4.7. For $0 < \delta < \frac{p+1}{2}$, we have

where $r_1\left(\delta\right) = \min\{r\left(\delta\right), \sqrt{2d\left(\delta\right)}\}$ and $r_2\left(\delta\right) = \sqrt{d\left(\delta\right)/ b\left(\delta\right)}$.

Proof. For arbitrary $u\in B_{r_1\left(\delta\right)}$, we have $\|Xu\| < r\left(\delta\right)$. Together with 4.2 (1) we deduce that either $I_{\delta}\left(u\right) > 0$ or $\|X u\| = 0$ holds. In addition, by (4.1) there holds $J\left(u\right)\leq\frac{1}{2}\|X u\|^2$. By combining with $\|Xu\|^2 < 2d\left(\delta\right)$ we have $J\left(u\right) < d\left(\delta\right)$. Then $u\in W_{\delta}$, and thus $B_{r_1\left(\delta\right)}\subset W_{\delta}$. By Lemmas 4.2 and 4.4 the other conclusion follows. □

By Definition 2.3 we see that the weak solution $u$ satisfies the energy equality

Next, we consider the invariance of $W_\delta, V_\delta$ as follows.

Proposition 4.1. Assume that $u_{0} \in H^{1}_{X, 0}\left(\Omega\right)$, $0 < \mu < d$. Denote by $\delta_1$, $\delta_2$ the two solutions of $d\left(\delta\right) = \mu$ for $\delta_1 < 1 < \delta_2$. For any weak solution $u$ of problem (1.1) satisfying $J\left(u_{0}\right)\in(0, \mu]$, there hold that for arbitrary $t\in[0, T)$, $\delta \in\left(\delta_1, \delta_2\right)$,

$\rm{(1)}$ if $I\left(u_{0}\right) > 0$, then $u\in W_{\delta}$;

$\rm{(2)}$ if $I\left(u_{0}\right) < 0$, then $u\in V_{\delta}$.

Proof. $\rm (1)$ First, we claim $u_{0}\in W_{\delta}$ for $\delta \in \left(\delta_1, \delta_2\right)$. In fact, if $J\left(u_{0}\right)\leq\mu$ and $I\left(u_{0}\right) > 0$, we see from Lemma 4.5 that $J\left(u_{0}\right) < d\left(\delta\right)$ and $I_{\delta}\left(u_{0}\right) > 0$, and the claim follows.

Now, for arbitrary $\delta \in \left(\delta_1, \delta_2\right)$, $t\in\left(0, T\right)$ we claim $u\left(x, t\right) \in W_{\delta}$. Otherwise, there exist a first time $t_0 \in\left(0, T\right)$ and $\delta_0 \in\left(\delta_1, \delta_2\right)$ such that $u\left(x, t_0\right)\in \partial W_{\delta_0}$. This implies that either $I_{\delta_0}\left(u\left(t_0\right)\right) = 0$, $\|X u\left(t_0\right)\|\neq 0$ or $J\left(u\left(t_0\right)\right) = d\left(\delta_0\right)$ holds. By (4.10) we obtain

which implies $J\left(u\left(t_0\right)\right)\neq d\left(\delta_0\right)$. Thus $I_{\delta_0}\left(u\left(t_0\right)\right) = 0$ and $\|X u\left(t_0\right)\|\neq 0$, by (4.5) we get $J\left(u\left(t_0\right)\right)\geq d\left(\delta_0\right)$, which contradicts (4.11).

$\rm (2)$ First, we claim $u_{0}\in V_{\delta}$ for $\delta \in \left(\delta_1, \delta_2\right)$. By $J\left(u_{0}\right)\leq\mu$, $I\left(u_{0}\right) < 0$ and Lemma 4.5 we get $J\left(u_{0}\right) < d\left(\delta\right)$ and $I_{\delta}\left(u_{0}\right) < 0$, and thus the claim follows.

Next, for arbitrary $\delta \in \left(\delta_1, \delta_2\right)$ and $t\in \left(0, T\right)$ we claim $u\left(x, t\right) \in V_{\delta}$. Otherwise, there exist a first time $t_0 \in \left(0, T\right)$ and $\delta_0 \in\left(\delta_1, \delta_2\right)$ such that $I_{\delta_0}\left(u\left(t\right)\right) < 0$ for $t\in[0, t_0)$, and $u\left(x, t_0\right)\in \partial V_{\delta_0}$. This implies that

It follows from (4.11) that $J\left(u\left(t_0\right)\right)\neq d\left(\delta_0\right)$, and thus $I_{\delta_0}\left(u\left(t_0\right)\right) = 0$. Together with Lemma 4.2 there holds $\|X u\left(t\right)\|\geq r\left(\delta_0\right)$ for $0\leq t \leq t_0$. Hence, we see from (4.5) that $J\left(u\left(t_0\right)\right)\geq d\left(\delta_0\right)$, which contradicts (4.11). □

Now, by Proposition 4.1 and Lemma 4.3 we have the corollary as follows.

Corollary 4.1. Assume that $u_{0} \in H^{1}_{X, 0}\left(\Omega\right)$, $0 < J\left(u_{0}\right)\leq \mu < d$. Denote by $\delta_1$, $\delta_2$ the two solutions of $d\left(\delta\right) = \mu$ for $\delta_1 < 1 < \delta_2$. Then, both $W_{\delta}$ and $V_{\delta}$ are invariant for arbitrary $\delta \in \left(\delta_1, \delta_2\right)$, and thus

are invariant under the flow of problem (1.1).

Furthermore, we discuss the invariant manifolds of the solutions with non-positive level energy by the following results.

Proposition 4.2. For any nontrivial solutions $u$ of problem (1.1) satisfying $J\left(u_{0}\right) = 0$, we have $u\in B^{c}_{r_0}$, where

Proof. It follows from (4.10) that $J\left(u\right)\leq0$ for $0\leq t < T$. Then

which implies that either $\|X u\| = 0$ or $\|Xu\|\geq r_0$ holds. We claim $\|X u\|\equiv0$ for any $t\in[0, T)$ if $\|X u_{0}\| = 0$. If it is false, there holds $0 < \|Xu\left(t_0\right)\| < r_0$ for some $t_0\in \left(0, T\right)$, a contradiction appears. Similarly, for the case $\|Xu_{0}\|\geq r_0$ we can prove $\|Xu\|\geq r_0$ for $t\in[0, T)$. The conclusion follows. □

Proposition 4.3. Let $u_{0} \in H^{1}_{X, 0}\left(\Omega\right)$. If either $J\left(u_{0}\right) < 0$ or $J\left(u_{0}\right) = 0$, $\|Xu_{0}\|\neq0$ occurs, then $u\in V_\delta$ for any $\delta \in\left(0, \frac{p+1}{2}\right)$, where $u$ is a weak solution of problem (1.1).

Proof. It follows from (4.10) and (4.9) that

If $J\left(u_{0}\right) < 0$, there holds

This shows that

On the other hand, if $J\left(u_{0}\right) = 0$ and $\|Xu_{0}\|\neq0$ occur, by Proposition 4.2 we have $\|Xu\|\geq r_0$ for $t\in[0, T)$. By combining with (4.12), we obtain (4.13), and thus (4.14). The conclusion follows. □

Corollary 4.2. Let $u_{0} \in H^{1}_{X, 0}\left(\Omega\right)$. If either $J\left(u_{0}\right) < 0$ or $J\left(u_{0}\right) = 0$, $\|Xu_{0}\|\neq0$ occurs, then $u\in B_{r\left(\frac{p+1}{2}\right)}^{c}$, where $u$ is a weak solution of problem (1.1).

Proof. For any $\delta \in\left(0, \frac{p+1}{2}\right)$, by Proposition 4.3 and Lemma 4.2 we see that

Letting $\delta\rightarrow \frac{p+1}{2}$, we obtain $\|X u\| \geq r\left(\frac{p+1}{2}\right)$. The conclusion follows. □

Finally, for $J\left(u_{0}\right) < d$ we discuss the vacuum isolating of solutions.

Proposition 4.4. Let $u_{0} \in H^{1}_{X, 0}\left(\Omega\right)$, $\mu \in\left(0, d\right)$. Denote by $\delta_1$, $\delta_2$ the two solutions of $d\left(\delta\right) = \mu$ for $\delta_1 < 1 < \delta_2$, we have a vacuum region

for given $\mu \geq J\left(u_{0}\right)$, such that any weak solution $u$ of problem (1.1) is outside of $U_{\mu}$. Moreover, $U_{\mu}$ becomes larger and larger if $\mu$ is decreasing, and $U_{\mu}$ approximates $U_0$ as $\mu\rightarrow0$, where

Proof. For any weak solution $u$ of problem (1.1) with $J\left(u_{0}\right)\leq\mu$, it is sufficient to prove that if $\|X u\| \neq 0$, for any $\delta \in \left(\delta_1, \delta_2\right)$ there holds $u\left(t\right)\notin \mathcal {N}_{\delta}$, equivalently, $I_{\delta}\left(u\left(t\right)\right)\neq0$ for $t\in [0, T)$.

We claim $I_{\delta}\left(u_{0}\right)\neq0$. If it is false, then $I_{\delta}\left(u_{0}\right) = 0$. Together with Lemma 4.3 and (4.5) we have $d\left(\delta_1\right) = d\left(\delta_2\right) = \mu < d\left(\delta\right)\leq J\left(u_{0}\right)$, which contradicts $J\left(u_{0}\right)\leq\mu$.

Now, assume that there exists $t_1 > 0$ such that $u\left(t_1\right) \in U_{\mu}$. This shows that $u\left(t_1\right)\in \mathcal {N}_{\delta_0}$ for some $\delta_0 \in \left(\delta_1, \delta_2\right)$. Then we see from (4.11) and (4.5) that $J\left(u_{0}\right) < d\left(\delta_0\right)\leq J\left(u\left(t_1\right)\right)\leq J\left(u_{0}\right)$, which is a contradiction. Proposition 4.4 has been proved. □

5.   Global existence and blow-up in finite time of solutions

In this section, we establish the global existence, the asymptotic behavior and the finite time blow-up of solutions for problem (1.1) with subcritical or critical initial energy.

5.1.   Global existence of solutions

By the potential well method and the Galerkin method, we will show the following theorem.

Theorem 5.1 (Global existence). Under the assumptions $\rm {\left(H\right)}$, $\rm {\left(H_{\partial \Omega}\right)}$, $\rm {\left(H_{\it V}\right)}$, $\rm {\left(H_{\it g}\right)}$ and $\rm {\left(H_{\it p}\right)}$, for any $u_{0} \in H^{1}_{X, 0}\left(\Omega\right)$ satisfying $J\left(u_{0}\right)\leq d$ and $I\left(u_{0}\right)\geq 0$, there exists a global weak solution $u$ for the problem (1.1) such that $u\left(x, t\right) \in L^{\infty}\left(0, +\infty; H^{1}_{X, 0}\left(\Omega\right)\right)$ with $u_t \in L^2\left(0, +\infty; L^2\left(\Omega\right)\right)$. Moreover,

● if $J\left(u_{0}\right) < d$, there holds

where

● if $J\left(u_{0}\right) = d$ and $I\left(u_{0}\right) > 0$, for any $\varepsilon\in\left(0, d\right)$ small enough, there exists $t_\varepsilon > 0$ such that

where

For later use, we recall the following estimation.

Lemma 5.1 (cf. ^[18] Theorem 8.1). Denote by $\varphi\left(t\right): \mathbb{R}^{+}\rightarrow \mathbb{R}^{+}$ a non-increasing function. If

for some constant $C > 0$, then $\varphi\left(t\right)\leq \varphi\left(0\right)e^{1-t/C}$ for all $t$.

Proof of Theorem 5.1.. We divide our proof into the four steps as follows.

Step 1: Global existence for $J\left(u_{0}\right) < d$.

Let $\{\phi_k\left(x\right)\}_{k\geq1}$ be a base of $H^{1}_{X, 0}\left(\Omega\right)$ in Proposition 2.5. Then we can construct the approximate solutions of problem (1.1) as follows

such that

and as $m\rightarrow \infty$,

Now, multiply (5.3) by $a'_{jm}\left(t\right)$, sum for $j$, integrate with respect to $t$, we get

Together with (5.4) we obtain $J\left(u_m\left(0\right)\right)\rightarrow J\left(u_{0}\right)$ as $m\rightarrow \infty$, and thus

for $m$ large enough.

Similar to the proof of Proposition 4.1 (1), for $m$ large enough and $t\in[0, T)$, by (5.6) we have $u_m\left(x, t\right)\in W$. Together with (1.3), (4.2) and (5.6) we conclude that

which shows that $T = +\infty$,

where we used (2.4) for the penultimate inequality.

Let $\xrightarrow{w^{*}}$ be the weakly star convergence. By (5.7) and (5.8) we have a subsequence, also denoted by $\{u_m\}$, satisfying as $m\rightarrow \infty$,

Then, fix $j$ and let $m\rightarrow \infty$ in (5.3), we deduce

As $\{\phi_k\left(x\right)\}_{k\geq1}$ is a base of $H^{1}_{X, 0}\left(\Omega\right)$, and thus for any $w\in H^{1}_{X, 0}\left(\Omega\right)$ there holds

Moreover, it follows from (5.4) that $u\left(x, 0\right) = u_{0}\left(x\right)$ in $H^{1}_{X, 0}\left(\Omega\right)$. Therefore, we have a global weak solution $u\left(x, t\right) \in L^{\infty}\left(0, +\infty; H^{1}_{X, 0}\left(\Omega\right)\right)$ satisfying $u_t\left(x, t\right) \in L^2\left(0, +\infty; L^2\left(\Omega\right)\right)$.

Step 2: Asymptotic behavior for $J\left(u_{0}\right) < d$.

Now, we only need to discuss the case that $0 < J\left(u_{0}\right) < d$ and $I\left(u_{0}\right) > 0$. We see from Proposition 4.1 that $u\in W$ for $t \geq0$, which gives $I\left(u\right)\geq0$ for $t \geq0$. It follows from (1.3), (4.2) and (4.10) that

Then by (2.4) there holds

Inserting (5.10) into (4.1), by (1.3) we conclude that

where

Note from $J\left(u_{0}\right) < d$ and Lemma 4.1 (4) that $\xi > 0$.

Furthermore, by taking $w = u$ in (2.7), we deduce that

This gives that

Then, by (5.11), (5.12) and the Poincaré inequality (2.3) we get

Let $T\rightarrow +\infty$, by Lemma 5.1 we obtain (5.1).

Step 3: Global existence for $J\left(u_{0}\right) = d$.

Let $u_{0m} = \theta_mu_{0}$ for $m > 1$ and $\theta_{m} = 1-\frac{1}{m}$. We discuss the problem (1.1) with the initial condition

From Lemma 4.1 (3) and $I\left(u_{0}\right)\geq 0$ we have

The remaining proof follows from the similar proof of step 1.

Step 4: Asymptotic behavior for $J\left(u_{0}\right) = d$ and $I\left(u_{0}\right) > 0$.

It follows from the discussions above that $I\left(u\right)\geq0$ for $t\geq0$. Therefore, we only need to discuss the following two cases.

$\rm (1)$ $I\left(u\right) = -\left(u_t, u\right) > 0$ for $t\geq0$. It follows that $\|u_t\| > 0$, and thus $\int_0^t\|u_{\tau}\|^2d\tau$ is increasing for $t$ on $[0, +\infty)$. Then, for any given $\varepsilon\in\left(0, d\right)$ small enough, by (4.10) there holds

for some $t_\varepsilon > 0$. Letting the initial time $t = t_\varepsilon$, by similar proof of step 2 we obtain (5.2).

$\rm (2)$ For some $t_1 > 0$ there hold $I\left(u\left(t_1\right)\right) = 0$ and $I\left(u\right) > 0$ for $t \in[0, t_1)$. It follows that $\|u_t\| > 0$, and thus $\int_0^t\|u_{\tau}\|^2d\tau$ is strictly increasing for $0\leq t < t_1$. By (4.10) we conclude that

Together with (4.4) we deduce that $\|X u\left(t_1\right)\| = 0$. Then by $I\left(u\left(t_1\right)\right) = 0$ we get $J\left(u\left(t_1\right)\right) = 0$. By combining with

we obtain $J\left(u\left(t\right)\right) \leq 0$ for $t\geq t_1$. Together with (1.3), (2.4) and (4.1) we conclude

This shows that either $\|Xu\|\geq\left(\frac{p+1}{2C^{p+1}_{X}} \left(1-\mu C^{2}_{*}\right)\right)^{\frac{1}{p-1}}$ or $\|Xu\| = 0$ for $t\geq t_1$ holds. The former doesn't occur as $\|X u\left(t_1\right)\| = 0$, thus $\|Xu\|\equiv0$ for $t\geq t_1$. The decay estimate (5.2) follows.

Theorem 5.1 has been proved. □

Remark 5.1. If one replace the assumption " $J\left(u_{0}\right)\leq d$, $I\left(u_{0}\right)\geq0$" in Theorem 5.1 by " $0 < J\left(u_{0}\right) < d$, $I_{\delta_2}\left(u_{0}\right) > 0$" for $\delta_1$, $\delta_2$ being the two solutions of $d\left(\delta\right) = J\left(u_{0}\right)$ with $\delta_1 < 1 < \delta_2$, by Proposition 4.1 one can deduce that the problem (1.1) has a global weak solution $u \in L^{\infty}\left(0, +\infty; H^{1}_{X, 0}\left(\Omega\right)\right)$ satisfying $u_t \in L^2\left(0, +\infty; L^2\left(\Omega\right)\right)$ and $u\in W_\delta$ for $\delta\in\left(\delta_1, \delta_2\right)$, $t \in[0, +\infty)$.

Remark 5.2. If one replace the assumption " $I_{\delta_2}\left(u_{0}\right) > 0$" in Remark 5.1 by " $\|X u_{0}\| < r\left(\delta_2\right)$", by Lemmas 4.2, 4.4 and Proposition 4.1 one can deduce that the problem (1.1) has a global weak solution $u \in L^{\infty}\left(0, +\infty; H^{1}_{X, 0}\left(\Omega\right)\right)$ satisfying $u_t \in L^2\left(0, +\infty; L^2\left(\Omega\right)\right)$ and

Furthermore, there holds $\|Xu\|^2\leq \frac{d\left(\delta_1\right)}{ b\left(\delta_1\right)}, \ t \in[0, +\infty)$.

5.2.   Blow-up in finite time of solutions

In this subsection, we mainly prove the following result.

Theorem 5.2 (Blow-up). Under the assumptions $\rm {\left(H\right)}$, $\rm {\left(H_{\partial \Omega}\right)}$, $\rm {\left(H_{\it V}\right)}$, $\rm {\left(H_{\it g}\right)}$ and $\rm {\left(H_{\it p}\right)}$, for $u_{0} \in H^{1}_{X, 0}\left(\Omega\right)$ satisfying $J\left(u_{0}\right)\leq d$ and $I\left(u_{0}\right) < 0$, the weak solution $u\left(x, t\right)$ of problem (1.1) is finite time blow-up, i.e., for some $T > 0$ there holds

Proof. According to Theorem 3.1 we see that the problem (1.1) has a local weak solution $u \in C\left(\left[0, T\right]; H^{1}_{X, 0}\left(\Omega\right)\right)$. We will complete the proof of Theorem 5.2 by two steps as follows.

Step 1: Blow-up for $J\left(u_{0}\right) < d$.

By introducing

we obtain

Combining with (1.3), the Poincaré inequality (2.3), (4.2) and (4.10) we obtain

We deduce from

that

Together with (5.15), (5.16) and the Hölder inequality we see that

Next, we will prove

in the following two cases, respectively.

(1) $J\left(u_{0}\right)\leq 0$. It follows from (5.17) that

We claim $I\left(u\left(t\right)\right) < 0$ for $t > 0$. Otherwise, for some $t_0 > 0$ there hold $I\left(u\left(t_0\right)\right) = 0$ and $I\left(u\left(t\right)\right) < 0$ for $t\in[0, t_0)$. Then we see from Lemma 4.2 that $\|X u\left(t\right)\|\geq r(1)$ for $0\leq t\leq t_0$. Together with (4.4) there holds $J\left(u\left(t_0\right)\right)\geq d$, which contradicts (4.10).

Next, by (5.15) we have $\ddot{F}\left(t\right) > 0$ for $t\geq0$, which shows that

Then, for $t$ large enough we get

which together with (5.19) implies that (5.18).

(2) $0 < J\left(u_{0}\right) < d$. It follows from Proposition 4.1 that $u\left(x, t\right)\in V_{\delta}$, and thus $I_{\delta}\left(u\right) < 0$ for $t\geq 0$ and $\delta \in [1, \delta_2)$. By combining with its continuity and Lemma 4.2 we see that $\|X u\left(t\right)\|\geq r\left(\delta_2\right)$ and $I_{\delta_2}\left(u\left(t\right)\right)\leq 0$ for $t\geq 0$, where $\delta_2$ is taken to be the bigger solution of $d\left(\delta\right) = J\left(u_{0}\right)$. Then, by (5.15) we deduce that for $t\geq0$ there hold

Then for $t$ large enough we obtain

Together with (5.17) we get (5.18) again.

Finally, for any $\beta > 0$ a directly calculation shows that

Taking $\beta = \frac{p-1}{2}$, by (5.18) we obtain $\left(F^{-\frac{p-1}{2}}\left(t\right)\right)'' < 0$ for $t$ large enough, which implies that

for some $t_{1} > 0$ large enough. Therefore, for some $T \in (0, +\infty)$ there holds

This is exactly (5.14).

Step 2: Blow-up for $J\left(u_{0}\right) = d$.

By the continuities of $J\left(u\right)$ and $I\left(u\right)$ with respect to $t$, there exists a $t_0\in (0, T)$ small enough such that $J\left(u\left(t_0\right)\right) > 0$ and $I\left(u\right) < 0$ for $t\in\left[0, t_0\right]$. Then we have $\left(u_t, u\right) = -I\left(u\right) > 0$, and thus $\|u_t\left(t\right)\| > 0$, i.e., $\int_0^t\|u_{\tau}\|^2d\tau$ is strictly positive for $t\in\left[0, t_0\right]$. By (4.10) we further obtain

Let $t_0$ be the initial time. By following the similar proof of step 1, we conclude that $u$ is finite time blow-up.

Theorem 5.2 has been proved. □

Acknowledgments

This work is supported by National Nature Science Foundation of China, Grant No. 12101194.

Conflict of interest

The authors declare there is no conflict of interest.