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On decay and blow-up of solutions for a nonlinear Petrovsky system with conical degeneration

Abstract

This paper deals with a class of Petrovsky system with nonlinear damping

$$\begin{aligned} w_{tt}+\Delta _{\mathbb{B}}^{2}w-k_{2} \Delta _{\mathbb{B}}w_{t}+aw_{t} \vert w_{t} \vert ^{m-2}=bw \vert w \vert ^{p-2} \end{aligned}$$

on a manifold with conical singularity, where \(\Delta _{\mathbb{B}}\) is a Fuchsian-type Laplace operator with totally characteristic degeneracy on the boundary \(x_{1}=0\). We first prove the global existence of solutions under conditions without relation between m and p, and establish an exponential decay rate. Furthermore, we obtain a finite time blow-up result for local solutions with low initial energy \(E(0)< d\).

Introduction

Due to the frequent occurrence of high order nonlinear wave equations in many branches of engineering, physics, chemistry, material science, and other sciences, the study of wave equations plays a key role in mathematical analysis. For more details, see [1, 2]. In [3] and [4], the original Petrovsky model has the following form:

$$\begin{aligned}& w_{tt}+\Delta ^{2}w-\Delta w_{t}+w_{t} \vert w_{t} \vert ^{m-2}=w \vert w \vert ^{p-2},\quad x \in \varOmega , t>0, \end{aligned}$$
(1.1)
$$\begin{aligned}& w=0,\qquad \frac{\partial w}{\partial \nu }=0, \quad x\in \partial \varOmega , t\geq 0, \end{aligned}$$
(1.2)
$$\begin{aligned}& w(x,0)=w_{0}(x),\qquad w_{t}(x,0)=w_{1}(x), \quad x\in \bar{\varOmega }, \end{aligned}$$
(1.3)

where \(\varOmega \in \mathbb{R}^{n}\) is a bounded domain with a smooth boundary ∂Ω.

Equation (1.1) is an important physical model that appears in many applications to mathematical physics as well as in the theory of vibrating plates, geophysics, and ocean acoustics [5, 6]. Some further physical interpretations are given in [7, 8].

For Equation (1.1), many results for global existence, nonexistence, and asymptotic behavior of solutions have been obtained [311]. Li et al. [3] studied problem (1.1)–(1.3) and derived that the solution is global without the relation between m and p. Moreover, the decay estimates of the energy function and the estimates of the lifespan of solution were given. Later, under suitable conditions decay estimates of the solutions for Equation (1.1) have been established by using Nakao’s inequality in [4]. Messaoudi [9] proved the solution for problem (1.1)–(1.3) without \(\Delta w_{t}\) blows up in finite time if \(p>m\) and the energy is negative. Wu [10] proved the blow-up result for problem (1.1)–(1.3) without \(\Delta w_{t}\) if \(p>m\) and the energy is nonnegative. Recently, Chen et al. [11] proved that the solution of problem (1.1)–(1.3) without \(\Delta w_{t}\) blows up with positive initial energy and claimed that the solution blows up in finite time for even vanishing initial energy for \(m=2\). More recently, Philippin et al. [12] used a differential inequality technique to obtain a lower bound on blow-up time for Equation (1.1) without \(\Delta w_{t}\). In recent years, lower bounds for the blow-up time in a superlinear hyperbolic equation with damping term have been derived [13]. For other related works, we refer the readers to [1418] and the references therein.

In 2011 to 2012, Chen et al. established the corresponding Sobolev inequality on the cone Sobolev spaces in [19, 20]. And on this basis, they studied the initial boundary value problem of a semilinear parabolic equation on a manifold with conical singularity [21] and obtained the existence and nonexistence results by introducing a family of potential wells. Li et al. [22] proved the global existence, exponential decay, and finite time blow-up of solution for a class of semilinear pseudo-parabolic equations with conical degeneration. Recently, Alimohammady et al. [23] studied a class of semilinear degenerate hyperbolic equations on the cone Sobolev spaces

$$\begin{aligned}& w_{tt}-\Delta _{\mathbb{B}}w+V(x)w+\gamma w_{t}=g_{t}(x)w \vert w \vert ^{p-1} ,\quad x \in \operatorname{int}\mathbb{B}, t>0, \end{aligned}$$
(1.4)
$$\begin{aligned}& w(t,x)=0,\quad x\in \partial \mathbb{B}, t\geq 0, \end{aligned}$$
(1.5)
$$\begin{aligned}& w(x,0)=w_{0}(x),\qquad w_{t}(x,0)=w_{1}(x),\quad x \in \operatorname{int}\mathbb{B}, \end{aligned}$$
(1.6)

where \(\mathbb{B}=[0, 1)\times X\), X is an \((n-1)\)-dimensional closed compact manifold, which is regarded as the local model near the conical points and \(\partial \mathbb{B}=\{0\}\times X\). \(\Delta _{\mathbb{B}}=(x_{1}\partial _{x_{1}})^{2}+\partial _{x_{2}}^{2} +\cdots +\partial _{x_{n}}^{2}\).

They discussed the invariance of some sets, global existence, nonexistence, and asymptotic behavior of solutions with initial energy \(J(w_{0})< d\) by introducing a family of potential wells which was first proposed by Sattinger [24]. More works on equations with conical degeneration can be seen in the literature [2528] and the references therein.

If we consider Equation (1.1) on a manifold with conical singularity, that is, when the standard Laplace operator Δ of Equation (1.1) is replaced by Fuchsian-type Laplace operator \(\Delta _{\mathbb{B}}\), what will happened for the initial boundary value problem? For this kind of Petrovsky equation with conical degeneration, the existence and nonexistence of global solutions to both the initial boundary value problem and the initial value problem remain open.

Inspired by the ideas of [3, 4, 23] and [2931], we study the initial boundary value problem for the following Petrovsky equation:

$$\begin{aligned}& w_{tt}+\Delta _{\mathbb{B}}^{2}w-k_{2} \Delta _{\mathbb{B}}w_{t}+aw_{t} \vert w_{t} \vert ^{m-2}=bw \vert w \vert ^{p-2},\quad x\in \operatorname{int}\mathbb{B}, t>0, \end{aligned}$$
(1.7)
$$\begin{aligned}& w=0,\qquad \nabla _{\mathbb{B}}w\cdot \nu =0,\quad x\in \partial \mathbb{B}, t\geq 0, \end{aligned}$$
(1.8)
$$\begin{aligned}& w(x,0)=w_{0}(x),\qquad w_{t}(x,0)=w_{1}(x),\quad x \in \operatorname{int}\mathbb{B}, \end{aligned}$$
(1.9)

where \(w_{0}(x)\), \(w_{1}(x)\) are suitable initial data and \(k_{2}\), a, b, m, p are constants such that \(k_{2}\) and b are positive, a is nonnegative, and \(m\geq 2\), \(2< p<\frac{2n}{n-2}=p^{\ast }\), where \(p^{\ast }\) is the critical Sobolev exponents. Here \(\mathbb{B}\) is defined as above, and ν is the unit normal vector pointing toward the exterior of \(\mathbb{B}\). Moreover, the operator \(\Delta _{\mathbb{B}}\) in (1.7) is defined by \((x_{1}\partial _{x_{1}})^{2}+\partial _{x_{2}}^{2}+\cdots +\partial _{x_{n}}^{2}\), which is an elliptic operator with conical degeneration on the boundary \(x_{1}=0\) (we also called it Fuchsian-type Laplace operator), and the divergence operator \(\mathrm{div}_{\mathbb{B}}\) is defined by \(x_{1}\partial _{x_{1}}+\partial _{x_{2}}+\cdots +\partial _{x_{n}}\), the corresponding gradient operator is denoted by \(\nabla _{\mathbb{B}}= (x_{1}\partial _{x_{1}}, \partial _{x_{2}}, \ldots ,\partial _{x_{n}})\). In the neighborhood of \(\partial \mathbb{B}\) we will use coordinates \((x_{1}, x')=(x_{1}, x_{2},\ldots ,x_{n})\) for \(0\leq x_{1}<1\), \(x'\in X\).

Our main aim in this paper is to find the existence and nonexistence of solutions for problem (1.7)–(1.9) with cone degeneration by introducing a family of potential wells. Firstly, under the condition of low initial energy, we establish the existence of global solution in the cone Sobolev space by a combination of Galerkin method and potential well theory. Then, using the energy perturbation technique, we obtain the exponential decay result of the global solution. Finally, we show that the solution of the problem blows up in a finite time and give the estimates for lower and upper bounds of blow-up time. It is worth mentioning that two types of lower bounds of the blow-up time \(T_{\mathrm{max}}\) for the weak solution of (1.7)–(1.9) are given, respectively.

The rest of this article is organized as follows. In Sect. 2, we recall the cone Sobolev spaces and the corresponding properties. In Sect. 3, we establish a global existence result and show the decay rates. In Sect. 4, we prove the blow-up properties of local solution.

Preliminaries

In this section, we recall the manifold with conical singularities and the cone Sobolev spaces which were introduced in [19, 20] and introduce some lemmas and notations.

We assume that the manifold B has only one conical point on the boundary. Thus, near the conical point, we have a stretched manifold \(\mathbb{B}\) associated with B. Here \(\mathbb{B}=[0, 1)\times X\), \(\partial \mathbb{B}=\{0\}\times X\) and X is a closed compact manifold of dimension \(n-1\). Also, in the neighborhood of the conical point, we use coordinates \((x_{1}, x')=(x_{1}, x_{2},\ldots ,x_{n})\) for \(0\leq x_{1}<1\), \(x'\in X\).

Definition 2.1

Let \(\mathbb{B}=[0, 1)\times X\) be a stretched manifold of the manifold B with conical singularity. Then the cone Sobolev space \(\mathcal{H}_{p}^{m, \gamma }(\mathbb{B})\) for \(m\in \mathbb{N}\), \(\gamma \in \mathbb{R}\), and \(1< p<\infty \) is defined as

$$ \mathcal{H}_{p}^{m, \gamma }(\mathbb{B})=\bigl\{ u\in W_{\mathrm{loc}}^{m, p}(\operatorname{int} \mathbb{B})|\omega u\in \mathcal{H}_{p}^{m, \gamma }\bigl(X^{\varLambda }\bigr)\bigr\} $$

for any cut-off function ω supported by a collar neighborhood of \((0, 1)\times \partial \mathbb{B}\). Moreover, the subspace \(\mathcal{H}_{p, 0}^{m, \gamma }(\mathbb{B})\) of \(\mathcal{H}_{p}^{m, \gamma }(\mathbb{B})\) is defined by

$$ \mathcal{H}_{p, 0}^{m, \gamma }(\mathbb{B})=[\omega ] \mathcal{H}_{p, 0}^{m, \gamma }\bigl(X^{\varLambda }\bigr)+[1- \omega ]W_{0}^{m, p}(\operatorname{int}\mathbb{B}), $$

where \(X^{\varLambda }=\mathbb{R}_{+}\times X\) as the corresponding open stretched cone with the base X, \(W_{0}^{m, p}(\operatorname{int}\mathbb{B})\) denotes the closure of \(C_{0}^{\infty }(\operatorname{int}\mathbb{B})\) in Sobolev spaces \(W^{m, p}(\bar{X})\) when is a closed compact \(C^{\infty }\) manifold of dimension n that contains B as a submanifold with boundary.

Remark 2.1

([32])

We have the following properties:

  1. (1)

    \(\mathcal{H}_{p}^{m, \gamma }(\mathbb{B})\) is a Banach space for \(1\leq p<\infty \) and is a Hilbert space for \(p=2\).

  2. (2)

    \(L_{p}^{\gamma }(\mathbb{B}):=\mathcal{H}_{p}^{0, \gamma }(\mathbb{B})\).

  3. (3)

    \(L_{p}(\mathbb{B}):=\mathcal{H}_{p}^{0, 0}(\mathbb{B})\).

  4. (4)

    The embedding \(\mathcal{H}_{p}^{m, \gamma }(\mathbb{B})\hookrightarrow \mathcal{H}_{p}^{m', \gamma '}(\mathbb{B})\) is continuous if \(m\geq m'\), \(\gamma \geq \gamma '\); and is compact embedding if \(m> m'\), \(\gamma > \gamma '\).

Definition 2.2

Let \(\mathbb{B}=[0, 1)\times X\). Then \(u(x)\in L_{p}^{\gamma }(\mathbb{B})\) with \(1< p<\infty \) and \(\gamma \in \mathbb{R}\) if

$$ \bigl\Vert u(x) \bigr\Vert _{L_{p}^{\gamma }(\mathbb{B})}^{p}= \int _{\mathbb{B}}x_{1}^{n} \bigl\vert x_{1}^{-\gamma }u(x) \bigr\vert ^{p} \frac{dx_{1}}{x_{1}}\,dx'< +\infty . $$

Observe that if \(u(x)\in L_{p}^{\frac{n}{p}}(\mathbb{B})\), \(v(x)\in L_{q}^{\frac{n}{q}}( \mathbb{B})\) with \(p, q\in (1, +\infty )\) and \(\frac{1}{p}+\frac{1}{q}=1\), then we have the following Hölder inequality:

$$ \int _{\mathbb{B}} \bigl\vert u(x)v(x) \bigr\vert \frac{dx_{1}}{x_{1}}\,dx'\leq \biggl( \int _{ \mathbb{B}} \bigl\vert u(x) \bigr\vert ^{p} \frac{dx_{1}}{x_{1}}\,dx'\biggr)^{\frac{1}{p}} \biggl( \int _{ \mathbb{B}} \bigl\vert v(x) \bigr\vert ^{q} \frac{dx_{1}}{x_{1}}\,dx'\biggr)^{\frac{1}{q}}. $$
(2.1)

In the sequel, for convenience we denote

$$\begin{aligned}& (u, v)_{2}= \int _{\mathbb{B}}u(x)v(x)\frac{dx_{1}}{x_{1}}\,dx',\qquad \Vert u \Vert ^{p}_{L_{p}^{ \frac{n}{p}}(\mathbb{B})}= \int _{\mathbb{B}} \bigl\vert u(x) \bigr\vert ^{p} \frac{dx_{1}}{x_{1}}\,dx'. \\& \tilde{\mathcal{H}}_{2, 0}^{1, \frac{n}{2}}(\mathbb{B}):= \bigl\{ {u(x) \in \mathcal{H}_{2}^{1, \frac{n}{2}}(\mathbb{B})|u=0 \text{ on } \partial \mathbb{B}} \bigr\} , \\& \tilde{\mathcal{H}}_{2, 0}^{2, \frac{n}{2}}(\mathbb{B}):= \bigl\{ {u(x) \in \mathcal{H}_{2}^{2, \frac{n}{2}}(\mathbb{B})|u=\nabla _{ \mathbb{B}}u\cdot \nu =0 \text{ on } \partial \mathbb{B}} \bigr\} , \\& \Vert u \Vert _{\tilde{\mathcal{H}}_{2, 0}^{1, \frac{n}{2}}(\mathbb{B})}^{2}= \Vert u \Vert ^{2}_{L_{2}^{\frac{n}{2}}(\mathbb{B})}+ \Vert \nabla _{\mathbb{B}}u \Vert ^{2}_{L_{2}^{ \frac{n}{2}}(\mathbb{B})}, \\& \Vert u \Vert _{\tilde{\mathcal{H}}_{2, 0}^{2, \frac{n}{2}}(\mathbb{B})}^{2}= \Vert u \Vert ^{2}_{L_{2}^{\frac{n}{2}}(\mathbb{B})}+ \Vert \nabla _{\mathbb{B}}u \Vert ^{2}_{L_{2}^{ \frac{n}{2}}(\mathbb{B})}+ \Vert \Delta _{\mathbb{B}}u \Vert ^{2}_{L_{2}^{ \frac{n}{2}}(\mathbb{B})}. \end{aligned}$$

The spaces \(\tilde{\mathcal{H}}_{2, 0}^{1, \frac{n}{2}}(\mathbb{B})\), \(\tilde{\mathcal{H}}_{2, 0}^{2, \frac{n}{2}}(\mathbb{B})\) with norms \(\|u\|_{\tilde{\mathcal{H}}_{2, 0}^{1, \frac{n}{2}}(\mathbb{B})}\), \(\|u\|_{\tilde{\mathcal{H}}_{2, 0}^{2, \frac{n}{2}}(\mathbb{B})}\) are Banach spaces, where the norms \(\|u\|_{\tilde{\mathcal{H}}_{2, 0}^{1, \frac{n}{2}}(\mathbb{B})}\), \(\|u\|_{\tilde{\mathcal{H}}_{2, 0}^{2, \frac{n}{2}}(\mathbb{B})}\) are equivalent to the norms \(\|\nabla _{\mathbb{B}}u\|_{L_{2}^{\frac{n}{2}}(\mathbb{B})}\), \(\|\Delta _{\mathbb{B}}u\|_{L_{2}^{\frac{n}{2}}(\mathbb{B})}\), respectively.

Lemma 2.1

Let \(u(x), v(x)\in \tilde{\mathcal{H}}_{2, 0}^{1, \frac{n}{2}}(\mathbb{B})\). Then

$$ \int _{\mathbb{B}}v\Delta _{\mathbb{B}}u\frac{dx_{1}}{x_{1}}\,dx' =- \int _{\mathbb{B}}\nabla _{\mathbb{B}}u\cdot \nabla _{\mathbb{B}}v \frac{dx_{1}}{x_{1}}\,dx'. $$
(2.2)

Proof

Here we first suppose \(u(x), v(x)\in C_{0}^{\infty }(\mathbb{B})\). From the definition of \(\Delta _{\mathbb{B}}\), it follows that

$$ \begin{aligned}[b] &\int _{\mathbb{B}}v\Delta _{\mathbb{B}}u\frac{dx_{1}}{x_{1}}\,dx' \\ &\quad = \int _{\mathbb{B}}x_{1}\partial _{x_{1}}(x_{1} \partial _{x_{1}}u) \cdot v\frac{dx_{1}}{x_{1}}\,dx' + \int _{\mathbb{B}}\bigl(\partial ^{2}_{x_{2}}u+ \cdots +\partial ^{2}_{x_{n}}u\bigr)\cdot v \frac{dx_{1}}{x_{1}}\,dx' \\ &\quad = \int _{\mathbb{B}}\partial _{x_{1}}(x_{1} \partial _{x_{1}}u)\cdot v\,dx + \int _{\mathbb{B}}\bigl(\partial ^{2}_{x_{2}}u+ \cdots +\partial ^{2}_{x_{n}}u\bigr) \cdot v \frac{dx_{1}}{x_{1}}\,dx' \\ &\quad = \int _{\mathbb{B}}\operatorname{div} \biggl(x_{1} \partial _{x_{1}}u, \frac{\partial _{x_{2}}u}{x_{1}}, \ldots , \frac{\partial _{x_{n}}u}{x_{1}} \biggr)\cdot v\,dx \\ &\quad =- \int _{\mathbb{B}} \biggl(x_{1}\partial _{x_{1}}u, \frac{\partial _{x_{2}}u}{x_{1}}, \ldots , \frac{\partial _{x_{n}}u}{x_{1}} \biggr)\cdot \nabla v\,dx \\ &\quad =- \int _{\mathbb{B}} \bigl(x_{1}^{2}\partial _{x_{1}}u,\partial _{x_{2}}u, \ldots , \partial _{x_{n}}u \bigr)\cdot \nabla v\frac{dx_{1}}{x_{1}}\,dx' \\ &\quad =- \int _{\mathbb{B}} (x_{1}\partial _{x_{1}}u, \partial _{x_{2}}u, \ldots , \partial _{x_{n}}u )\cdot (x_{1}\partial _{x_{1}}v, \partial _{x_{2}}v, \ldots , \partial _{x_{n}}v ) \frac{dx_{1}}{x_{1}}\,dx' \\ &\quad =- \int _{\mathbb{B}}\nabla _{\mathbb{B}}u\cdot \nabla _{\mathbb{B}}v \frac{dx_{1}}{x_{1}}\,dx'. \end{aligned} $$
(2.3)

Finally, since \(C_{0}^{\infty }(\mathbb{B})\) is dense in \(\tilde{\mathcal{H}}_{2, 0}^{1, \frac{n}{2}}(\mathbb{B})\), the equation above holds in the case of \(u(x), v(x)\in \tilde{\mathcal{H}}_{2, 0}^{1, \frac{n}{2}}(\mathbb{B})\). □

Lemma 2.2

([21], Poincaré inequality)

Let \(\mathbb{B}=[0, 1)\times X\)be a bounded subspace in \(\mathbb{R}_{+}^{n}\)with \(X\subset \mathbb{R}^{n-1}\), and \(1< p<+\infty \), \(\gamma \in \mathbb{R}\). If \(u(x)\in \tilde{\mathcal{H}}_{p, 0}^{1, \gamma }(\mathbb{B})\), then

$$ \bigl\Vert u(x) \bigr\Vert _{L_{p}^{\gamma }(\mathbb{B})}\leq c_{\star } \bigl\Vert \nabla _{ \mathbb{B}}u(x) \bigr\Vert _{L_{p}^{\gamma }(\mathbb{B})}, $$
(2.4)

where \(\nabla _{\mathbb{B}}=(x_{1}\partial _{x_{1}}, \partial _{x_{2}}, \ldots ,\partial _{x_{n}})\)and the constant \(c_{\star }\)depends only on \(\mathbb{B}\).

Lemma 2.3

([21])

For \(1< p<\frac{2n}{n-2}\), the embedding \(\tilde{\mathcal{H}}_{2, 0}^{1, \frac{n}{2}}(\mathbb{B}) \hookrightarrow \tilde{\mathcal{H}}_{p, 0}^{0, \frac{n}{p}}( \mathbb{B})\)is continuous.

From Lemma 2.2 and Lemma 2.3, we obtain the following lemma.

Lemma 2.4

For \(1< p< p^{\ast }\), we have

$$ \Vert u \Vert _{L_{p}^{\frac{n}{p}}(\mathbb{B})}\leq C_{0} \Vert \Delta _{ \mathbb{B}}u \Vert _{L_{2}^{\frac{n}{2}}(\mathbb{B})} $$
(2.5)

for \(u\in \tilde{\mathcal{H}}_{2, 0}^{2, \frac{n}{2}}(\mathbb{B})\)holds, where constant \(C_{0}\)depends only on \(\mathbb{B}\)and p.

Global existence and energy decay

In this section, we discuss the global existence and decay of the solution for problem (1.7)–(1.9).

Similar to the classical case, we introduce the following functionals on cone Sobolev space \(\tilde{\mathcal{H}}_{2, 0}^{2, \frac{n}{2}}(\mathbb{B})\):

$$\begin{aligned}& J(w)=\frac{1}{2} \int _{\mathbb{B}} \vert \Delta _{\mathbb{B}}w \vert ^{2} \frac{dx_{1}}{x_{1}}\,dx'-\frac{b}{p} \int _{\mathbb{B}} \vert w \vert ^{p} \frac{dx_{1}}{x_{1}}\,dx', \end{aligned}$$
(3.1)
$$\begin{aligned}& I(w)= \int _{\mathbb{B}} \vert \Delta _{\mathbb{B}}w \vert ^{2} \frac{dx_{1}}{x_{1}}\,dx'-b \int _{\mathbb{B}} \vert w \vert ^{p} \frac{dx_{1}}{x_{1}}\,dx'. \end{aligned}$$
(3.2)

We also define the energy function as follows:

$$ E(t)=\frac{1}{2} \int _{\mathbb{B}} \vert w_{t} \vert ^{2}\frac{dx_{1}}{x_{1}}\,dx'+ \frac{1}{2} \int _{\mathbb{B}} \vert \Delta _{\mathbb{B}}w \vert ^{2} \frac{dx_{1}}{x_{1}}\,dx' -\frac{b}{p} \int _{\mathbb{B}} \vert w \vert ^{p} \frac{dx_{1}}{x_{1}}\,dx'. $$
(3.3)

Finally, we introduce the potential well

$$ W=\bigl\{ w\in \tilde{\mathcal{H}}_{2, 0}^{2, \frac{n}{2}}( \mathbb{B}))| I(w)>0 \bigr\} \cup \{0\} $$
(3.4)

and the outside sets of the corresponding potential well

$$ V=\bigl\{ w\in \tilde{\mathcal{H}}_{2, 0}^{2, \frac{n}{2}}( \mathbb{B}))| I(w)< 0 \bigr\} . $$
(3.5)

Remark 3.1

By (3.3) and Lemma 2.4, we know that

$$ E(t)\geq \frac{1}{2} \int _{\mathbb{B}} \vert \Delta _{\mathbb{B}}w \vert ^{2} \frac{dx_{1}}{x_{1}}\,dx' -\frac{b}{p} \int _{\mathbb{B}} \vert w \vert ^{p} \frac{dx_{1}}{x_{1}}\,dx'\geq g\bigl( \Vert \Delta _{\mathbb{B}}w \Vert _{L_{2}^{ \frac{n}{2}}(\mathbb{B})}\bigr), $$
(3.6)

where \(g(\lambda )=\frac{1}{2}\lambda ^{2}-\frac{bC_{0}^{p}}{p}\lambda ^{p}\) and \(C_{0}\) is given in Lemma 2.4. A direct calculation shows that \(g(\lambda )\) has the maximum value at

$$ \lambda _{1}= \biggl(\frac{1}{bC_{0}^{p}} \biggr)^{\frac{1}{p-2}} $$

and the maximum value is

$$ d=g(\lambda _{1})=\frac{p-2}{2p} \biggl( \frac{1}{bC_{0}^{p}} \biggr)^{ \frac{2}{p-2}} =\frac{p-2}{2p}\lambda _{1}^{2}>0. $$
(3.7)

By the definition of \(g(\lambda )\) and \(J(w)\), we can give another definition of d as follows:

$$ d=\inf \biggl\{ \sup_{\lambda \geq 0} J(\lambda w), w \in \tilde{\mathcal{H}}_{2, 0}^{2, \frac{n}{2}}(\mathbb{B}), \int _{ \mathbb{B}} \vert \Delta _{\mathbb{B}}w \vert ^{2}\frac{dx_{1}}{x_{1}}\,dx'\neq 0 \biggr\} >0, $$
(3.8)

and the Nehari manifold

$$ \mathcal{N}=\biggl\{ w\in \tilde{\mathcal{H}}_{2, 0}^{2, \frac{n}{2}}( \mathbb{B})\Big| I(w)=0, \int _{\mathbb{B}} \vert \Delta _{\mathbb{B}}w \vert ^{2} \frac{dx_{1}}{x_{1}}\,dx'\neq 0\biggr\} . $$
(3.9)

Similar to the results in [29], one has \(0< d=\inf_{w\in \mathcal{N}}J(w)\).

The next lemma shows that our energy functional \(E(t)\) is a nonincreasing function along the solution of (1.7)–(1.9).

Lemma 3.1

\(E(t)\) is a nonincreasing function for \(t \geq 0\) and

$$ \frac{d}{dt}E(t)=-k_{2} \int _{\mathbb{B}} \vert \nabla _{\mathbb{B}}w_{t} \vert ^{2} \frac{dx_{1}}{x_{1}}\,dx' -a \int _{\mathbb{B}} \vert w_{t} \vert ^{m} \frac{dx_{1}}{x_{1}}\,dx'\leq 0. $$
(3.10)

Proof

Multiplying (1.7) by \(w_{t}\) and integrating it over \(\mathbb{B}\times [0, t)\), we obtain

$$ E(t)-E(0)=- \int _{0}^{t} \biggl(k_{2} \int _{\mathbb{B}} \vert \nabla _{ \mathbb{B}}w_{\tau } \vert ^{2}\frac{dx_{1}}{x_{1}}\,dx' +a \int _{\mathbb{B}} \vert w_{ \tau } \vert ^{m}\frac{dx_{1}}{x_{1}}\,dx' \biggr)\,d\tau $$
(3.11)

for \(t\geq 0\). Thus, the proof is completed. □

Lemma 3.2

Assume that \(E(0)< d\). Then:

  1. (i)

    If \(\|\Delta _{\mathbb{B}}w_{0}\|_{L_{2}^{\frac{n}{2}}(\mathbb{B})}< \lambda _{1}\), then \(\|\Delta _{\mathbb{B}}w(t)\|_{L_{2}^{\frac{n}{2}}(\mathbb{B})}< \lambda _{1}\)for \(t\geq 0\).

  2. (ii)

    If \(\|\Delta _{\mathbb{B}}w_{0}\|_{L_{2}^{\frac{n}{2}}(\mathbb{B})}> \lambda _{1}\), then there exists \(\lambda _{2}>\lambda _{1}\)such that \(\|\Delta _{\mathbb{B}}w(t)\|_{L_{2}^{\frac{n}{2}}(\mathbb{B})}\geq \lambda _{2}\)for \(t\geq 0\).

Proof

From the definition of \(g(\lambda )\), we see that \(g(\lambda )\) is increasing in \((0, \lambda _{1})\), decreasing in \((\lambda _{1}, \infty )\), and \(g(\lambda )\rightarrow -\infty \) as \(\lambda \rightarrow \infty \). Since \(E(0)< d\), so there exist \(\lambda _{2}\) and \(\lambda '_{2}\) such that \(\lambda '_{2}<\lambda _{1}<\lambda _{2}\) and \(g(\lambda '_{2})=g(\lambda _{2})=E(0)\).

(i) When \(\|\Delta _{\mathbb{B}}w_{0}\|_{L_{2}^{\frac{n}{2}}(\mathbb{B})}< \lambda _{1}\), by (3.6), we have

$$ g\bigl( \Vert \Delta _{\mathbb{B}}w_{0} \Vert _{L_{2}^{\frac{n}{2}}(\mathbb{B})}\bigr) \leq E(0)=g\bigl(\lambda '_{2} \bigr). $$

It implies \(\|\Delta _{\mathbb{B}}w_{0}\|_{L_{2}^{\frac{n}{2}}(\mathbb{B})}< \lambda '_{2}\). We claim that \(\|\Delta _{\mathbb{B}}w(t)\|_{L_{2}^{\frac{n}{2}}(\mathbb{B})}< \lambda '_{2}\) for \(t>0\). If not, then there exists \(t_{0}>0\) such that \(\|\Delta _{\mathbb{B}}w(t_{0})\|_{L_{2}^{\frac{n}{2}}(\mathbb{B})}> \lambda '_{2}\). If \(\lambda '_{2}<\|\Delta _{\mathbb{B}}w(t_{0})\|_{L_{2}^{\frac{n}{2}}( \mathbb{B})}<\lambda _{2}\), then \(g(\|\Delta _{\mathbb{B}}w(t_{0})\|_{L_{2}^{\frac{n}{2}}(\mathbb{B})})> E(0)\geq E(t_{0})\). It contradicts (3.6). If \(\|\Delta _{\mathbb{B}}w(t_{0})\|_{L_{2}^{\frac{n}{2}}(\mathbb{B})} \geq \lambda _{2}\), then by the continuity of \(\|\Delta _{\mathbb{B}}w(t)\|_{L_{2}^{\frac{n}{2}}(\mathbb{B})}\), there exists \(0< t_{1}< t_{0}\) such that \(\lambda '_{2}<\|\Delta _{\mathbb{B}}w(t_{1})\|_{L_{2}^{\frac{n}{2}}( \mathbb{B})}<\lambda _{2}\), then \(g(\|\Delta _{\mathbb{B}}w(t_{1})\|_{L_{2}^{\frac{n}{2}}(\mathbb{B})})> E(0)\geq E(t_{1})\). This is a contradiction.

(ii) When \(\|\Delta _{\mathbb{B}}w_{0}\|_{L_{2}^{\frac{n}{2}}(\mathbb{B})}> \lambda _{1}\), as in case (i) we also deduce that \(\|\Delta _{\mathbb{B}}w_{0}\|_{L_{2}^{\frac{n}{2}}(\mathbb{B})}> \lambda _{1}\) implies \(\|\Delta _{\mathbb{B}}w(t)\|_{L_{2}^{\frac{n}{2}}(\mathbb{B})}\geq \lambda _{2}\) for \(t\geq 0\). □

Lemma 3.3

Suppose that \(2< p < p^{\ast }\), \(w_{1}\in L_{2}^{\frac{n}{2}}(\mathbb{B})\), and \(E(0)< d\). Let \(w_{0}\in W\)such that

$$ \beta =bC_{0}^{p} \biggl( \frac{2p}{p-2}E(0) \biggr)^{\frac{p-2}{2}}< 1. $$
(3.12)

Then \(w\in W\)for each \(t\geq 0\).

Proof

When \(w=0\), we get \(w\in W\) easily, so we just need to prove the case \(w\neq 0\). Since \(I(w_{0})>0\), it follows from the continuity of w that

$$ I(w)\geq 0 $$
(3.13)

for some interval near \(t=0\). Let \(T_{m}>0\) be a maximal time (possibly \(T_{m}=T\)) when (3.13) holds on \([0, T_{m})\).

From (3.1)–(3.2), it follows that

$$ \begin{aligned}[b] J(w)&=\frac{p-2}{2p} \int _{\mathbb{B}} \vert \Delta _{\mathbb{B}}w \vert ^{2} \frac{dx_{1}}{x_{1}}\,dx'+\frac{1}{p}I(w) \\ &\geq \frac{p-2}{2p} \int _{\mathbb{B}} \vert \Delta _{\mathbb{B}}w \vert ^{2} \frac{dx_{1}}{x_{1}}\,dx',\quad \text{on } t\in [0, T_{m}). \end{aligned} $$
(3.14)

By using (3.14), (3.3), and Lemma 3.1, we get

$$ \begin{aligned}[b] \int _{\mathbb{B}} \vert \Delta _{\mathbb{B}}w \vert ^{2}\frac{dx_{1}}{x_{1}}\,dx' & \leq \frac{2p}{p-2}J(w) \\ &\leq \frac{2p}{p-2}E(t) \\ &\leq \frac{2p}{p-2}E(0). \end{aligned} $$
(3.15)

Then, by Lemma 2.4 and (3.15), we obtain

$$ \begin{aligned}[b] b \Vert w \Vert ^{p}_{L_{p}^{\frac{n}{p}}(\mathbb{B})} &\leq bC_{0}^{p} \Vert \Delta _{\mathbb{B}}w \Vert ^{p}_{L_{2}^{\frac{n}{2}}(\mathbb{B})} \\ &\leq bC_{0}^{p} \biggl(\frac{2p}{p-2}E(0) \biggr)^{\frac{p-2}{2}} \Vert \Delta _{ \mathbb{B}}w \Vert ^{2}_{L_{2}^{\frac{n}{2}}(\mathbb{B})} \\ &=\beta \Vert \Delta _{\mathbb{B}}w \Vert ^{2}_{L_{2}^{\frac{n}{2}}(\mathbb{B})} \\ &< \Vert \Delta _{\mathbb{B}}w \Vert ^{2}_{L_{2}^{\frac{n}{2}}(\mathbb{B})} \end{aligned} $$
(3.16)

on \(t\in [0, T_{m})\). Therefore, by using (3.2), we conclude that \(I(w)>0\) for all \(t\in [0, T_{m})\). By repeating the procedure, \(T_{m}\) is extended to T. The proof is completed. □

Remark 3.2

From Lemma 3.3, we can deduce that

$$ \Vert \Delta _{\mathbb{B}}w \Vert ^{2}_{L_{2}^{\frac{n}{2}}(\mathbb{B})} \leq \frac{1}{1-\beta }I(w). $$
(3.17)

Theorem 3.1

Suppose that \(2< p< p^{\star }\), \(w_{1}\in L_{2}^{\frac{n}{2}}(\mathbb{B})\), and \(E(0)< d\), let \(w_{0}\in W\)and w satisfy the assumption of Lemma 3.3. Then problem (1.7)(1.9) admits a global weak solution \(w(x, t)\in L^{\infty }([0, T]; \tilde{\mathcal{H}}_{2, 0}^{2, \frac{n}{2}}(\mathbb{B}))\)with \(w_{t}(x, t)\in L^{2}([0, T]; \tilde{\mathcal{H}}_{2, 0}^{1, \frac{n}{2}}(\mathbb{B}))\cap L^{m}([0, T]; L_{m}^{\frac{n}{m}}( \mathbb{B})) \cap L^{\infty }([0, T]; L_{2}^{\frac{n}{2}}(\mathbb{B}))\). Moreover, \(w(t)\in W\)for \(0\leq t<\infty \).

Proof

Let \(\{\omega _{j}(x)\}\) be a system of base functions in \(\tilde{\mathcal{H}}_{2, 0}^{2, \frac{n}{2}}(\mathbb{B})\). Now we construct the following approximate solution \(w_{s}(x, t)\) of problem (1.7)–(1.9):

$$ w_{s}(x, t)=\sum_{j=1}^{s}g_{js}(t) \omega _{j}(x),\quad s=1,2,\ldots , $$

which satisfies

$$\begin{aligned}& \begin{aligned}[b] &(w_{stt}, \omega _{j})_{2}+(\Delta _{\mathbb{B}}w_{s}, \Delta _{ \mathbb{B}}\omega _{j})_{2}+k_{2}( \nabla _{\mathbb{B}}w_{st}, \nabla _{ \mathbb{B}}\omega _{j})_{2}+a\bigl(w_{st} \vert u_{st} \vert ^{m-2}, \omega _{j}\bigr)_{2} \\ &\quad =b \bigl(w_{s} \vert u_{s} \vert ^{p-2}, \omega _{j}\bigr)_{2},\quad s=1,2,\ldots , \end{aligned} \end{aligned}$$
(3.18)
$$\begin{aligned}& w_{s}(x, 0)=\sum_{j=1}^{s}g_{js}(0) \omega _{j}(x)\rightarrow w_{0}(x) \quad \text{in } \tilde{\mathcal{H}}_{2, 0}^{2, \frac{n}{2}}(\mathbb{B}), \end{aligned}$$
(3.19)
$$\begin{aligned}& w_{st}(x, 0)=\sum_{j=1}^{s}g'_{js}(0) \omega _{j}(x)\rightarrow w_{1}(x) \quad \text{in } L_{2}^{\frac{n}{2}}(\mathbb{B}). \end{aligned}$$
(3.20)

Multiplying (3.18) by \(g'_{js}(t)\), summing for j (\(j=1,2, \ldots , s\)), and integrating from 0 to t, we obtain

$$ \begin{aligned}[b] &k_{2} \int _{0}^{t} \Vert \nabla _{\mathbb{B}}w_{s\tau } \Vert ^{2}_{L_{2}^{ \frac{n}{2}}(\mathbb{B})}\,d\tau +a \int _{0}^{t} \Vert w_{s\tau } \Vert ^{m}_{L_{m}^{ \frac{n}{m}}(\mathbb{B})}\,d\tau +E\bigl(w_{s}(t) \bigr) \\ &\quad =E\bigl(w_{s}(0)\bigr), \quad 0\leq t< \infty . \end{aligned} $$
(3.21)

By (3.19) we can get \(E(w_{s}(0))\rightarrow E(w_{0})\), then for sufficiently large m, we have

$$ \begin{aligned}[b] &k_{2} \int _{0}^{t} \Vert \nabla _{\mathbb{B}}w_{s\tau } \Vert ^{2}_{L_{2}^{ \frac{n}{2}}(\mathbb{B})}\,d\tau +a \int _{0}^{t} \Vert w_{s\tau } \Vert ^{m}_{L_{m}^{ \frac{n}{m}}(\mathbb{B})}\,d\tau +E\bigl(w_{s}(t) \bigr)< d, \\ &\quad 0\leq t< \infty . \end{aligned} $$
(3.22)

From (3.22) and the proof of Lemma 3.3, we can get \(w_{s}(t)\in W\) for \(0\leq t< \infty \) and sufficiently large s. Hence, by (3.22) and

$$ E(w_{s})= \frac{1}{2} \Vert w_{st} \Vert ^{2}_{L_{2}^{\frac{n}{2}}(\mathbb{B})}+ \frac{p-2}{2p} \Vert \Delta _{\mathbb{B}}w_{s} \Vert ^{2}_{L_{2}^{\frac{n}{2}}( \mathbb{B})} +\frac{1}{p}I(w_{s}), $$
(3.23)

we obtain

$$ \begin{aligned}[b] &k_{2} \int _{0}^{t} \Vert \nabla _{\mathbb{B}}w_{s\tau } \Vert ^{2}_{L_{2}^{ \frac{n}{2}}(\mathbb{B})}\,d\tau +a \int _{0}^{t} \Vert w_{s\tau } \Vert ^{m}_{L_{m}^{ \frac{n}{m}}(\mathbb{B})}\,d\tau +\frac{1}{2} \Vert w_{st} \Vert ^{2}_{L_{2}^{ \frac{n}{2}}(\mathbb{B})} \\ &\quad {}+\frac{p-2}{2p} \Vert \Delta _{\mathbb{B}}w_{s} \Vert ^{2}_{L_{2}^{\frac{n}{2}}( \mathbb{B})}< d,\quad 0\leq t< \infty , \end{aligned} $$
(3.24)

for sufficiently large s, which yields

$$\begin{aligned}& \Vert \Delta _{\mathbb{B}}w_{s} \Vert ^{2}_{L_{2}^{\frac{n}{2}}(\mathbb{B})}< \frac{2p}{p-2}d,\quad 0\leq t< \infty , \end{aligned}$$
(3.25)
$$\begin{aligned}& \int _{0}^{t} \Vert \nabla _{\mathbb{B}}w_{s\tau } \Vert ^{2}_{L_{2}^{ \frac{n}{2}}(\mathbb{B})}\,d\tau < \frac{d}{k_{2}}, \quad 0\leq t< \infty , \end{aligned}$$
(3.26)
$$\begin{aligned}& \int _{0}^{t} \Vert w_{s\tau } \Vert ^{m}_{L_{m}^{\frac{n}{m}}(\mathbb{B})}\,d\tau < \frac{d}{a}, \quad 0\leq t< \infty , \end{aligned}$$
(3.27)
$$\begin{aligned}& \Vert w_{st} \Vert ^{2}_{L_{2}^{\frac{n}{2}}(\mathbb{B})}< 2d,\quad 0\leq t< \infty , \end{aligned}$$
(3.28)
$$\begin{aligned}& \begin{aligned}[b] \int _{\mathbb{B}} \bigl\vert \vert w_{s} \vert ^{p-2}w_{s} \bigr\vert ^{\frac{p}{p-1}} \frac{dx_{1}}{x_{1}}\,dx' &= \int _{\mathbb{B}} \vert w_{s} \vert ^{p} \frac{dx_{1}}{x_{1}}\,dx'= \Vert w_{s} \Vert ^{p}_{L_{p}^{\frac{n}{p}}(\mathbb{B})} \\ & \leq C_{0}^{p} \Vert \Delta _{\mathbb{B}}w_{s} \Vert ^{p}_{L_{2}^{\frac{n}{2}}( \mathbb{B})} \leq C_{0}^{p} \biggl(\frac{2p}{p-2}d \biggr)^{\frac{p}{2}}, \end{aligned} \end{aligned}$$
(3.29)
$$\begin{aligned}& \begin{aligned}[b] &\int _{0}^{t} \int _{\mathbb{B}} \bigl\vert \vert w_{s\tau } \vert ^{m-2}w_{s \tau } \bigr\vert ^{\frac{m}{m-1}} \frac{dx_{1}}{x_{1}}\,dx'\,d\tau \\ &\quad = \int _{0}^{t} \int _{\mathbb{B}} \vert w_{s\tau } \vert ^{m}\frac{dx_{1}}{x_{1}}\,dx'\,d\tau = \int _{0}^{t} \Vert w_{s\tau } \Vert ^{m}_{L_{m}^{\frac{n}{m}}(\mathbb{B})}\,d\tau < \frac{d}{a}. \end{aligned} \end{aligned}$$
(3.30)

Therefore, there exist w and a subsequence still denoted by \(\{w_{s}\}\) for which, as \(s\rightarrow \infty \),

$$\begin{aligned}& w_{s}\rightarrow w \quad \text{in } L^{\infty }\bigl(0, \infty ; \tilde{\mathcal{H}}_{2, 0}^{2, \frac{n}{2}}(\mathbb{B})\bigr) \text{ weakly star and a.e. in } \operatorname{int} \mathbb{B}\times [0, \infty ), \\& w_{st}\rightarrow w_{t} \quad \text{in } L^{2}\bigl(0, \infty ; \tilde{\mathcal{H}}_{2, 0}^{1, \frac{n}{2}}( \mathbb{B})\bigr) \text{ weakly}, \\& w_{st}\rightarrow w_{t}\quad \text{in } L^{m}\bigl(0, \infty ; L_{m}^{\frac{n}{m}}( \mathbb{B})\bigr) \text{ weakly}, \\& w_{st}\rightarrow w_{t} \quad \text{in } L^{\infty }\bigl(0, \infty ; L_{2}^{ \frac{n}{2}}( \mathbb{B})\bigr) \text{ weakly star}, \\& w_{s}^{p-1}\rightarrow w^{p-1} \quad \text{in } L^{\infty }\bigl(0, \infty ; L_{ \frac{p}{p-1}}^{ \frac{n(p-1)}{p}}( \mathbb{B})\bigr) \text{ weakly star}, \\& w_{st}^{m-1}\rightarrow w_{t}^{m-1} \quad \text{in } L^{\frac{m}{m-1}}\bigl(0, \infty ; L_{\frac{m}{m-1}}^{ \frac{n(m-1)}{m}}( \mathbb{B})\bigr) \text{ weakly}. \end{aligned}$$

In (3.18), we fix j, letting \(s\rightarrow \infty \) and integrating from 0 to t. Then we have

$$ \begin{aligned}[b] &(w_{t}, \omega _{j})_{2}+ \int _{0}^{t}(\Delta _{\mathbb{B}}w, \Delta _{\mathbb{B}}\omega _{j})_{2}\,d\tau +k_{2} \int _{0}^{t}(\nabla _{ \mathbb{B}}w_{\tau }, \nabla _{\mathbb{B}}\omega _{j})_{2}\,d\tau \\ &\quad {}+a \int _{0}^{t}\bigl(w_{\tau } \vert u_{\tau } \vert ^{m-2}, \omega _{j} \bigr)_{2}\,d\tau =b \int _{0}^{t}\bigl(w \vert w \vert ^{p-2}, \omega _{j}\bigr)_{2}\,d\tau +(w_{1}, \omega _{j})_{2} \end{aligned} $$
(3.31)

and

$$ \begin{aligned}[b] &(w_{t}, v)_{2}+ \int _{0}^{t}(\Delta _{\mathbb{B}}w, \Delta _{ \mathbb{B}}v)_{2}\,d\tau +k_{2} \int _{0}^{t}(\nabla _{\mathbb{B}}w_{ \tau }, \nabla _{\mathbb{B}}v)_{2}\,d\tau \\ &\quad {}+a \int _{0}^{t}\bigl(w_{\tau } \vert u_{\tau } \vert ^{m-2}, v\bigr)_{2}\,d\tau =b \int _{0}^{t}\bigl(w \vert w \vert ^{p-2}, v\bigr)_{2}\,d\tau +(w_{1}, v)_{2}, \\ &\quad \forall v \in \tilde{\mathcal{H}}_{2, 0}^{2, \frac{n}{2}}( \mathbb{B}). \end{aligned} $$
(3.32)

From (3.19) we obtain \(w(x, 0)=w_{0}(x)\) in \(\tilde{\mathcal{H}}_{2, 0}^{2, \frac{n}{2}}(\mathbb{B})\) and \(w_{t}(x, 0)=w_{1}(x)\) in \(L_{2}^{\frac{n}{2}}(\mathbb{B}), t\in (0, T)\). By density we obtain \(w\in L^{\infty }([0, T]; \tilde{\mathcal{H}}_{2, 0}^{2, \frac{n}{2}}( \mathbb{B}))\) with \(w_{t}\in L^{2}([0, T]; \tilde{\mathcal{H}}_{2, 0}^{1, \frac{n}{2}}( \mathbb{B}))\cap L^{m}([0, T]; L_{m}^{\frac{n}{m}}(\mathbb{B})) \cap L^{\infty }([0, T]; L_{2}^{\frac{n}{2}}(\mathbb{B}))\) is a global weak solution of problem (1.7)–(1.9). It is obvious that \(w(t)\in W\) for \(0\leq t< \infty \). □

Now, we use the following “modified” functional:

$$ G(t)=E(t)+\varepsilon \biggl( \int _{\mathbb{B}}ww_{t} \frac{dx_{1}}{x_{1}}\,dx'+ \frac{k_{2}}{2} \Vert \nabla _{\mathbb{B}}w \Vert ^{2}_{L_{2}^{ \frac{n}{2}}(\mathbb{B})} \biggr). $$

Lemma 3.4

Let w satisfy the assumption of Theorem 3.1. For ε small enough, we have

$$ \alpha _{1}G(t)\leq E(t)\leq \alpha _{2}G(t) $$
(3.33)

holds for two positive constants \(\alpha _{1}\)and \(\alpha _{2}\).

Proof

Making use of (3.23), straightforward computations lead to

$$ \begin{aligned}[b] G(t)&=E(t)+\varepsilon \biggl( \int _{\mathbb{B}}ww_{t} \frac{dx_{1}}{x_{1}}\,dx'+ \frac{k_{2}}{2} \int _{\mathbb{B}} \vert \nabla _{ \mathbb{B}}w \vert ^{2}\frac{dx_{1}}{x_{1}}\,dx' \biggr) \\ &\leq E(t)+\frac{\varepsilon }{2} \int _{\mathbb{B}} \vert w \vert ^{2} \frac{dx_{1}}{x_{1}}\,dx' +\frac{\varepsilon }{2} \int _{\mathbb{B}} \vert w_{t} \vert ^{2} \frac{dx_{1}}{x_{1}}\,dx' \\ &\quad {} +\frac{\varepsilon k_{2}}{2} \int _{\mathbb{B}} \vert \nabla _{ \mathbb{B}}w \vert ^{2}\frac{dx_{1}}{x_{1}}\,dx' \\ &\leq E(t)+\frac{\varepsilon }{2}\bigl(k_{2}+c_{\star }^{2} \bigr) \int _{ \mathbb{B}} \vert \nabla _{\mathbb{B}}w \vert ^{2}\frac{dx_{1}}{x_{1}}\,dx' + \frac{\varepsilon }{2} \int _{\mathbb{B}} \vert w_{t} \vert ^{2} \frac{dx_{1}}{x_{1}}\,dx' \\ &\leq E(t)+\frac{\varepsilon }{2}C_{1} \int _{\mathbb{B}} \vert \Delta _{ \mathbb{B}}w \vert ^{2}\frac{dx_{1}}{x_{1}}\,dx' +\frac{\varepsilon }{2} \int _{ \mathbb{B}} \vert w_{t} \vert ^{2}\frac{dx_{1}}{x_{1}}\,dx' \\ &\leq E(t)+\frac{\varepsilon }{2}\frac{2p C_{1}}{p-2}E(t)+\varepsilon E(t) \\ &\leq \frac{1}{\alpha _{1}}E(t), \end{aligned} $$
(3.34)

and in the same way, we get

$$ \begin{aligned}[b] G(t)&\geq E(t)-\frac{\varepsilon }{2}C_{1} \int _{\mathbb{B}} \vert \Delta _{ \mathbb{B}}w \vert ^{2}\frac{dx_{1}}{x_{1}}\,dx' -\frac{\varepsilon }{2} \int _{ \mathbb{B}} \vert w_{t} \vert ^{2}\frac{dx_{1}}{x_{1}}\,dx' \\ &\geq E(t)-\frac{\varepsilon }{2}\frac{2p C_{1}}{p-2}E(t)-\varepsilon E(t) \\ &\geq \frac{1}{\alpha _{2}}E(t) \end{aligned} $$
(3.35)

for ε small enough. □

Theorem 3.2

Suppose that \(2\leq m< m^{\ast }=\frac{n-2}{2n}\). Let \(w(x, t)\)satisfy the assumption of Theorem 3.1. Then we have the following decay estimates:

$$ E(t)\leq Ke^{-kt}, \quad t \geq 0, $$
(3.36)

where K and k are positive constants which will be defined later.

Proof

From the definition of \(G(t)\), we get

$$\begin{aligned} G'(t)&=-k_{2} \int _{\mathbb{B}} \vert \nabla _{\mathbb{B}}w_{t} \vert ^{2} \frac{dx_{1}}{x_{1}}\,dx' -a \int _{\mathbb{B}} \vert w_{t} \vert ^{m} \frac{dx_{1}}{x_{1}}\,dx'+\varepsilon \int _{\mathbb{B}} \vert w_{t} \vert ^{2} \frac{dx_{1}}{x_{1}}\,dx' \\ &\quad {} +\varepsilon \int _{\mathbb{B}}w(w_{tt}-k_{2}\Delta _{ \mathbb{B}}w_{t})\frac{dx_{1}}{x_{1}}\,dx' \\ &=-k_{2} \int _{\mathbb{B}} \vert \nabla _{\mathbb{B}}w_{t} \vert ^{2} \frac{dx_{1}}{x_{1}}\,dx' -a \int _{\mathbb{B}} \vert w_{t} \vert ^{m} \frac{dx_{1}}{x_{1}}\,dx'+\varepsilon \int _{\mathbb{B}} \vert w_{t} \vert ^{2} \frac{dx_{1}}{x_{1}}\,dx' \\ &\quad {} +\varepsilon \int _{\mathbb{B}}b \vert w \vert ^{p} \frac{dx_{1}}{x_{1}}\,dx' -\varepsilon \int _{\mathbb{B}}aww_{t} \vert u_{t} \vert ^{m-2} \frac{dx_{1}}{x_{1}}\,dx' \\ &\quad {} -\varepsilon \int _{\mathbb{B}}w\Delta ^{2}_{\mathbb{B}}w \frac{dx_{1}}{x_{1}}\,dx' \\ &=-k_{2} \Vert \nabla _{\mathbb{B}}w_{t} \Vert ^{2}_{L_{2}^{\frac{n}{2}}( \mathbb{B})} -a \Vert w_{t} \Vert ^{m}_{L_{m}^{\frac{n}{m}}(\mathbb{B})}+ \varepsilon \Vert w_{t} \Vert ^{2}_{L_{2}^{\frac{n}{2}}(\mathbb{B})} + \varepsilon \biggl( \frac{p}{2} \Vert w_{t} \Vert ^{2}_{L_{2}^{\frac{n}{2}}( \mathbb{B})} \\ &\quad {} +\frac{p}{2} \Vert \Delta _{\mathbb{B}}w \Vert ^{2}_{L_{2}^{ \frac{n}{2}}(\mathbb{B})} -pE(t) \biggr) -\varepsilon \int _{\mathbb{B}}aww_{t} \vert w_{t} \vert ^{m-2} \frac{dx_{1}}{x_{1}}\,dx' \\ &\quad {} -\varepsilon \Vert \Delta _{\mathbb{B}}w \Vert ^{2}_{L_{2}^{ \frac{n}{2}}(\mathbb{B})} \\ &=-k_{2} \Vert \nabla _{\mathbb{B}}w_{t} \Vert ^{2}_{L_{2}^{\frac{n}{2}}( \mathbb{B})} -a \Vert w_{t} \Vert ^{m}_{L_{m}^{\frac{n}{m}}(\mathbb{B})} + \varepsilon \biggl(\frac{p}{2}+1 \biggr) \Vert w_{t} \Vert ^{2}_{L_{2}^{\frac{n}{2}}( \mathbb{B})} \\ &\quad {} +\varepsilon \biggl(\frac{p}{2}-1 \biggr) \Vert \Delta _{ \mathbb{B}}w \Vert ^{2}_{L_{2}^{\frac{n}{2}}(\mathbb{B})} -\varepsilon pE(t) \\ &\quad {} -\varepsilon \int _{\mathbb{B}}aww_{t} \vert w_{t} \vert ^{m-2} \frac{dx_{1}}{x_{1}}\,dx'. \end{aligned}$$
(3.37)

Using Lemma 2.2, we obtain

$$ \begin{aligned}[b] G'(t)&\leq \biggl[ \varepsilon \biggl(\frac{p}{2}+1 \biggr)c_{\star }^{2}-k_{2} \biggr] \Vert \nabla _{\mathbb{B}}w_{t} \Vert ^{2}_{L_{2}^{\frac{n}{2}}( \mathbb{B})} -a \Vert w_{t} \Vert ^{m}_{L_{m}^{\frac{n}{m}}(\mathbb{B})} \\ &\quad {} + \biggl(\frac{p}{2}-1 \biggr)\varepsilon \Vert \Delta _{\mathbb{B}}w \Vert ^{2}_{L_{2}^{\frac{n}{2}}(\mathbb{B})} - \varepsilon pE(t) \\ &\quad {} -\varepsilon a \int _{\mathbb{B}}ww_{t} \vert u_{t} \vert ^{m-2} \frac{dx_{1}}{x_{1}}\,dx'. \end{aligned} $$
(3.38)

Then, we will show that from the estimate of the last term in (3.38), by Young’s inequality and the proof of Lemma 3.3, we obtain

$$\begin{aligned}& \biggl\vert \int _{\mathbb{B}}ww_{t} \vert u_{t} \vert ^{m-2}\frac{dx_{1}}{x_{1}}\,dx' \biggr\vert \leq \theta \Vert w_{t} \Vert ^{m}_{L_{m}^{\frac{n}{m}}(\mathbb{B})}+c( \theta ) \Vert w \Vert ^{m}_{L_{m}^{\frac{n}{m}}(\mathbb{B})}, \end{aligned}$$
(3.39)
$$\begin{aligned}& \Vert w \Vert ^{m}_{L_{m}^{\frac{n}{m}}(\mathbb{B})}\leq C^{m}_{0} \biggl( \frac{2p}{p-2}E(0) \biggr)^{\frac{m-2}{2}} \Vert \Delta _{\mathbb{B}}w \Vert ^{2}_{L_{2}^{ \frac{n}{2}}(\mathbb{B})}. \end{aligned}$$
(3.40)

Then by exploiting (3.38)–(3.40), we arrive at

$$ \begin{aligned}[b] G'(t)&\leq \biggl[ \varepsilon \biggl(\frac{p}{2}+1\biggr)c_{\star }^{2}-k_{2} \biggr] \Vert \nabla _{\mathbb{B}}w_{t} \Vert ^{2}_{L_{2}^{\frac{n}{2}}(\mathbb{B})} +( \varepsilon \theta -1)a \Vert w_{t} \Vert ^{m}_{L_{m}^{\frac{n}{m}}(\mathbb{B})} \\ &\quad {} +\varepsilon \biggl[a c(\theta )C^{m}_{0}\biggl( \frac{2p}{p-2}E(0)\biggr)^{\frac{m-2}{2}}+\biggl(\frac{p}{2}-1 \biggr) \biggr] \Vert \Delta _{ \mathbb{B}}w \Vert ^{2}_{L_{2}^{\frac{n}{2}}(\mathbb{B})} \\ &\quad {} -\varepsilon pE(t) \\ & \leq \biggl[\varepsilon \biggl(\frac{p}{2}+1\biggr)c_{\star }^{2}-k_{2} \biggr] \Vert \nabla _{\mathbb{B}}w_{t} \Vert ^{2}_{L_{2}^{\frac{n}{2}}( \mathbb{B})} +(\varepsilon \theta -1)a \Vert w_{t} \Vert ^{m}_{L_{m}^{ \frac{n}{m}}(\mathbb{B})} \\ &\quad {} +\varepsilon \biggl[a c(\theta )C^{m}_{0}\biggl( \frac{2p}{p-2}E(0)\biggr)^{\frac{m-2}{2}}+\biggl(\frac{p}{2}-1 \biggr) \biggr] E(t)- \varepsilon pE(t) \\ & = \biggl[\varepsilon \biggl(\frac{p}{2}+1\biggr)c_{\star }^{2}-k_{2} \biggr] \Vert \nabla _{\mathbb{B}}w_{t} \Vert ^{2}_{L_{2}^{\frac{n}{2}}( \mathbb{B})} +(\varepsilon \theta -1)a \Vert w_{t} \Vert ^{m}_{L_{m}^{ \frac{n}{m}}(\mathbb{B})} \\ &\quad {} -\varepsilon \biggl\{ p- \biggl[ac(\theta )C^{m}_{0} \biggl( \frac{2p}{p-2}E(0)\biggr)^{\frac{m-2}{2}}+\biggl( \frac{p}{2}-1\biggr) \biggr] \biggr\} E(t). \end{aligned} $$
(3.41)

Choose ε so small that \(\varepsilon (\frac{p}{2}+1)c_{\star }^{2}-k_{2} \leq 0\), \(\varepsilon \theta -1 \leq 0\). And choose suitable θ such that \(p-[ac(\theta )C^{m}_{0}(\frac{2p}{p-2}E(0))^{\frac{m-2}{2}}+( \frac{p}{2}-1)] >0\). Then from the above inequality, we obtain

$$ G'(t)\leq - \varepsilon \biggl\{ p- \biggl[ac(\theta )C^{m}_{0} \biggl( \frac{2p}{p-2}E(0) \biggr)^{\frac{m-2}{2}}+\biggl( \frac{p}{2}-1\biggr) \biggr] \biggr\} E(t). $$
(3.42)

Then, by the relation between \(E(t)\) and \(G(t)\), we get

$$ G'(t)\leq - \varepsilon \alpha _{1} \biggl\{ p- \biggl[ac(\theta )C^{m}_{0}\biggl( \frac{2p}{p-2}E(0) \biggr)^{\frac{m-2}{2}}+\biggl(\frac{p}{2}-1\biggr) \biggr] \biggr\} G(t). $$
(3.43)

We take ε small enough such that

$$ G(0)=E(0)+\varepsilon \biggl( \int _{\mathbb{B}}w_{0}w_{1} \frac{dx_{1}}{x_{1}}\,dx'+ \frac{k_{2}}{2} \Vert \nabla _{\mathbb{B}}w_{0} \Vert ^{2}_{L_{2}^{ \frac{n}{2}}(\mathbb{B})} \biggr)>0. $$

Integrating (3.43), we obtain

$$ G(t)\leq G(0)e^{-kt},\quad t\geq 0, $$

where

$$ k=\varepsilon \alpha _{1} \biggl\{ p- \biggl[ac(\theta )C^{m}_{0} \biggl( \frac{2p}{p-2}E(0) \biggr)^{\frac{m-2}{2}}+\biggl(\frac{p}{2}-1\biggr) \biggr] \biggr\} >0. $$

By using (3.33) again, we get

$$ E(t)\leq Ke^{-kt},\quad t\geq 0, $$

where \(K=\alpha _{2}G(0)\). This completes the proof. □

Finite time blow-up of solution

In this section, we show that the solution of problem (1.7)–(1.9) blows up in finite time if \(p>m\) and \(E(0)< d\). For this purpose, we first give the following lemma which will be used later.

Lemma 4.1

Suppose that \(2< p< p^{\star }\), \(E(0)< d\), \(w_{1}\in L_{2}^{\frac{n}{2}}(\mathbb{B})\). Let \(w_{0}\in V\), then we have

$$\begin{aligned}& w(t)\in V,\quad \forall t\in [0, T), \end{aligned}$$
(4.1)
$$\begin{aligned}& d< \frac{p-2}{2p} \Vert \Delta _{\mathbb{B}}w \Vert ^{2}_{L_{2}^{\frac{n}{2}}( \mathbb{B})},\quad \forall t\in [0, T). \end{aligned}$$
(4.2)

Proof

Let \(w_{0}\in V\), we have to prove that \(w(t)\in V\) for all \(t\in [0, T)\). We argue by contradiction. Assume that there exists \(t_{0}\in [0, T)\) such that \(w(t_{0})\notin V\). This implies that

$$ \bigl\Vert \Delta _{\mathbb{B}}w(t_{0}) \bigr\Vert ^{2}_{L_{2}^{\frac{n}{2}}(\mathbb{B})} \geq b \bigl\Vert w(t_{0}) \bigr\Vert ^{p}_{L_{p}^{\frac{n}{p}}(\mathbb{B})}. $$

By the continuity of \(w(t)\), there exists at least one \(\bar{t}\in (0, t_{0}]\) such that

$$ \bigl\Vert \Delta _{\mathbb{B}}w(\bar{t}) \bigr\Vert ^{2}_{L_{2}^{\frac{n}{2}}( \mathbb{B})}= b \bigl\Vert w(\bar{t}) \bigr\Vert ^{p}_{L_{p}^{\frac{n}{p}}(\mathbb{B})}. $$

Let

$$ \tilde{t}=\inf \bigl\{ \bar{t}\in (0, t_{0}]: \bigl\Vert \Delta _{\mathbb{B}}w( \bar{t}) \bigr\Vert ^{2}_{L_{2}^{\frac{n}{2}}(\mathbb{B})}= b \bigl\Vert w(\bar{t}) \bigr\Vert ^{p}_{L_{p}^{ \frac{n}{p}}(\mathbb{B})} \bigr\} . $$

In particular, the regularity of \(w(t)\) implies that \(\tilde{t}\in (0, t_{0}]\). Thus, we know

$$ \bigl\Vert \Delta _{\mathbb{B}}w(\tilde{t}) \bigr\Vert ^{2}_{L_{2}^{\frac{n}{2}}( \mathbb{B})}= b \bigl\Vert w(\tilde{t}) \bigr\Vert ^{p}_{L_{p}^{\frac{n}{p}}(\mathbb{B})} $$

and \(w(t)\in V\) for all \(t\in [0, \tilde{t})\). We have two cases to consider.

First case: \(\|\Delta _{\mathbb{B}}w(\tilde{t})\|^{2}_{L_{2}^{\frac{n}{2}}( \mathbb{B})}= 0\).

In this case, by the continuity of \(w(t)\), we have

$$ \underset{t\rightarrow \tilde{t}^{-}}{\lim } \bigl\Vert \Delta _{\mathbb{B}}w(t) \bigr\Vert ^{2}_{L_{2}^{\frac{n}{2}}(\mathbb{B})}=0. $$
(4.3)

On the other hand, the fact that \(w(t)\in V\) for all \(t\in [0, \tilde{t})\) implies that \(\|\Delta _{\mathbb{B}}w(t)\|^{2}_{L_{2}^{\frac{n}{2}}(\mathbb{B})} \neq 0\) and

$$ \bigl\Vert \Delta _{\mathbb{B}}w(t) \bigr\Vert ^{2}_{L_{2}^{\frac{n}{2}}(\mathbb{B})}< b \bigl\Vert w(t) \bigr\Vert ^{p}_{L_{p}^{\frac{n}{p}}(\mathbb{B})},\quad t\in [0, \tilde{t}). $$
(4.4)

By Lemma 2.4, we get

$$ \Vert w \Vert ^{p}_{L_{p}^{\frac{n}{p}}(\mathbb{B})}\leq C_{0}^{p} \Vert \Delta _{ \mathbb{B}}w \Vert ^{p}_{L_{2}^{\frac{n}{2}}(\mathbb{B})},\quad t\in [0, \tilde{t}). $$
(4.5)

Then, by (4.4), (4.5), we have

$$ \underset{t\rightarrow \tilde{t}^{-}}{\lim } \bigl\Vert \Delta _{\mathbb{B}}w(t) \bigr\Vert _{L_{2}^{\frac{n}{2}}(\mathbb{B})}> \biggl( \frac{1}{bC_{0}^{p}} \biggr)^{ \frac{1}{p-2}}. $$

This contradicts (4.3).

Second case: \(\|\Delta _{\mathbb{B}}w(\tilde{t})\|^{2}_{L_{2}^{\frac{n}{2}}( \mathbb{B})}\neq 0\).

In this case, by recalling (3.8), we know that \(J(w(\tilde{t}))\geq d\). Thus, \(E(\tilde{t})\geq d\), which contradicts the fact that \(E(t)\leq E(0)< d\). Hence, in either case we conclude that \(w(t)\in V\) for all \(t\in [0, T)\). Since

$$ J(\lambda w)=\frac{1}{2}\lambda ^{2} \Vert \Delta _{\mathbb{B}}w \Vert ^{2}_{L_{2}^{ \frac{n}{2}}(\mathbb{B})} - \frac{b}{p}\lambda ^{p} \Vert w \Vert ^{p}_{L_{p}^{ \frac{n}{p}}(\mathbb{B})}, $$

we obtain

$$ \frac{d}{d\lambda } J(\lambda w)=\lambda \Vert \Delta _{\mathbb{B}}w \Vert ^{2}_{L_{2}^{ \frac{n}{2}}(\mathbb{B})} -b\lambda ^{p-1} \Vert w \Vert ^{p}_{L_{p}^{ \frac{n}{p}}(\mathbb{B})} $$

and

$$ \frac{d^{2}}{d\lambda ^{2}} J(\lambda w)= \Vert \Delta _{\mathbb{B}}w \Vert ^{2}_{L_{2}^{ \frac{n}{2}}(\mathbb{B})} -b(p-1)\lambda ^{p-2} \Vert w \Vert ^{p}_{L_{p}^{ \frac{n}{p}}(\mathbb{B})}. $$

Let \(\frac{d}{d\lambda } J(\lambda w)=0\), which implies

$$ \bar{\lambda }_{1}=0, \bar{\lambda }_{2}= \biggl( \frac{ \Vert \Delta _{\mathbb{B}}w \Vert ^{2}_{L_{2}^{\frac{n}{2}}(\mathbb{B})}}{b \Vert w \Vert ^{p}_{L_{p}^{\frac{n}{p}}(\mathbb{B})}} \biggr)^{\frac{1}{p-2}}. $$

An elementary calculation shows

$$ \frac{d^{2}}{d\lambda ^{2}} J(\bar{\lambda }_{1} w)>0,\qquad \frac{d^{2}}{d\lambda ^{2}} J(\bar{\lambda }_{2} w)< 0. $$

So we have

$$ \sup_{\lambda \geq 0} J(\lambda w)=J(\bar{\lambda }_{2} w)= \frac{p-2}{2p} \frac{( \Vert \Delta _{\mathbb{B}}w \Vert ^{2}_{L_{2}^{\frac{n}{2}}(\mathbb{B})})^{\frac{p}{p-2}}}{(b \Vert w \Vert ^{p}_{L_{p}^{\frac{n}{p}}(\mathbb{B})})^{\frac{2}{p-2}}}. $$

By \(I(u)<0\), we have

$$ \begin{aligned}[b] d&\leq \sup_{\lambda \geq 0} J(\lambda w)=J(\bar{\lambda }_{2} w)= \frac{p-2}{2p} \frac{( \Vert \Delta _{\mathbb{B}}w \Vert ^{2}_{L_{2}^{\frac{n}{2}}(\mathbb{B})})^{\frac{p}{p-2}}}{(b \Vert w \Vert ^{p}_{L_{p}^{\frac{n}{p}}(\mathbb{B})})^{\frac{2}{p-2}}} \\ &< \frac{p-2}{2p} \Vert \Delta _{\mathbb{B}}w \Vert ^{2}_{L_{2}^{\frac{n}{2}}( \mathbb{B})}. \end{aligned} $$
(4.6)

 □

Lemma 4.2

Let \(2< p< p^{\star }\). Then there exists a positive constant C depending only on \(\mathbb{B}\)such that

$$ \Vert w \Vert ^{s}_{L_{p}^{\frac{n}{p}}(\mathbb{B})}\leq C \bigl( \bigl\Vert \Delta _{ \mathbb{B}}w(t) \bigr\Vert ^{2}_{L_{2}^{\frac{n}{2}}(\mathbb{B})}+ \Vert w \Vert ^{p}_{L_{p}^{ \frac{n}{p}}(\mathbb{B})} \bigr), \quad \textit{with } 2\leq s\leq p, $$
(4.7)

for any \(w\in \tilde{\mathcal{H}}_{2, 0}^{2, \frac{n}{2}}(\mathbb{B})\).

Proof

If \(\|w\|_{L_{p}^{\frac{n}{p}}(\mathbb{B})}\leq 1\), then \(\|w\|^{s}_{L_{p}^{\frac{n}{p}}(\mathbb{B})}\leq \|w\|^{2}_{L_{p}^{ \frac{n}{p}}(\mathbb{B})}\leq C_{0}^{2}\|\Delta _{\mathbb{B}}w(t)\|^{2}_{L_{2}^{ \frac{n}{2}}(\mathbb{B})}\) by Lemma 2.4. If \(\|w\|_{L_{p}^{\frac{n}{p}}(\mathbb{B})}> 1\), then \(\|w\|^{s}_{L_{p}^{\frac{n}{p}}(\mathbb{B})}\leq \|w\|^{p}_{L_{p}^{ \frac{n}{p}}(\mathbb{B})}\). Therefore (4.7) follows. □

Now we introduce the following auxiliary function:

$$ H(t)=d_{1}-E(t),\quad t\geq 0, $$
(4.8)

where \(d_{1}=\frac{E(0)+d}{2}>0\).

From Lemma 4.1 and Lemma 4.2, we obtain the following corollary.

Corollary 4.1

Let the assumption of Lemma 4.2hold. Then we have

$$ \Vert w \Vert ^{s}_{L_{p}^{\frac{n}{p}}(\mathbb{B})}\leq C \bigl( \bigl\vert H(t) \bigr\vert + \Vert w_{t} \Vert ^{2}_{L_{2}^{ \frac{n}{2}}(\mathbb{B})}+ \Vert w \Vert ^{p}_{L_{p}^{\frac{n}{p}}(\mathbb{B})} \bigr),\quad \textit{with } 2\leq s\leq p, $$
(4.9)

for any \(w\in \tilde{\mathcal{H}}_{2, 0}^{2, \frac{n}{2}}(\mathbb{B})\).

Theorem 4.1

Suppose that \(2< p< p^{\star }\)and \(p>m\geq 2\), \(w_{1}\in L_{2}^{\frac{n}{2}}(\mathbb{B})\), \(w_{0}\in V\). If one of the following is satisfied:

  1. (1)

    \(0\leq E(0)< d\)and \(\|\Delta _{\mathbb{B}}w_{0}\|_{L_{2}^{\frac{n}{2}}(\mathbb{B})}> \lambda _{1}\);

  2. (2)

    \(E(0)<0\),

then the local solution w of problem (1.7)(1.9) blows up in finite time; that is, the maximum existence time \(T_{\mathrm{max}}\)of w is finite and

$$ \underset{T\rightarrow T^{-}_{\mathrm{max}}}{\lim } \bigl[ \Vert \Delta _{\mathbb{B}}w \Vert ^{2}_{L_{2}^{\frac{n}{2}}(\mathbb{B})} + \Vert w \Vert ^{p}_{L_{p}^{ \frac{n}{p}}(\mathbb{B})}+ \Vert w_{t} \Vert ^{2}_{L_{2}^{\frac{n}{2}}( \mathbb{B})} \bigr]=+\infty . $$

Moreover, the lifespan \(T_{\mathrm{max}}\)is estimated by \(0< T_{\mathrm{max}}\leq \frac{1-\alpha }{\varGamma \alpha [L(0)]^{\alpha /(1-\alpha )}}\), here \(L(0)\)and Γ are given in (4.30) and (4.36) respectively. α is a constant given in (4.26).

Proof

(1) For \(0\leq E(0)< d\), from (4.8), it follows that

$$ H'(t)=-E'(t)=k_{2} \int _{\mathbb{B}} \vert \nabla _{\mathbb{B}}w_{t} \vert ^{2} \frac{dx_{1}}{x_{1}}\,dx' +a \int _{\mathbb{B}} \vert w_{t} \vert ^{m} \frac{dx_{1}}{x_{1}}\,dx'\geq 0. $$
(4.10)

Thus, we have

$$ H(t)\geq H(0)=d_{1}-E(0)> 0,\quad t\geq 0. $$
(4.11)

Let

$$ A(t)= \int _{\mathbb{B}}w(t)w_{t}(t)\frac{dx_{1}}{x_{1}}\,dx'. $$
(4.12)

By differentiating (4.12) and using (1.7), (4.8), we obtain

$$\begin{aligned} & A'(t) \\ &\quad = \int _{\mathbb{B}} \vert w_{t} \vert ^{2}\frac{dx_{1}}{x_{1}}\,dx'+ \int _{ \mathbb{B}}ww_{tt}\frac{dx_{1}}{x_{1}}\,dx' \\ &\quad = \int _{\mathbb{B}} \vert w_{t} \vert ^{2}\frac{dx_{1}}{x_{1}}\,dx'+ \int _{ \mathbb{B}}w \bigl[-\Delta _{\mathbb{B}}^{2}w +k_{2}\Delta _{\mathbb{B}}w_{t}-aw_{t} \vert w_{t} \vert ^{m-2} \\ &\qquad {} +bw \vert w \vert ^{p-2}\bigr]\frac{dx_{1}}{x_{1}}\,dx' \\ &\quad = \Vert w_{t} \Vert ^{2}_{L_{2}^{\frac{n}{2}}(\mathbb{B})}- \Vert \Delta _{ \mathbb{B}}w \Vert ^{2}_{L_{2}^{\frac{n}{2}}(\mathbb{B})} -k_{2} \int _{ \mathbb{B}}\nabla _{\mathbb{B}}w\cdot \nabla _{\mathbb{B}}w_{t} \frac{dx_{1}}{x_{1}}\,dx' \\ &\qquad {} +b \Vert w \Vert ^{p}_{L_{p}^{\frac{n}{p}}(\mathbb{B})} -a \int _{ \mathbb{B}}ww_{t} \vert u_{t} \vert ^{m-2}\frac{dx_{1}}{x_{1}}\,dx' \\ &\quad =\biggl(1+\frac{p}{2}\biggr) \Vert w_{t} \Vert ^{2}_{L_{2}^{\frac{n}{2}}(\mathbb{B})} +\biggl( \frac{p}{2}-1\biggr) \Vert \Delta _{\mathbb{B}}w \Vert ^{2}_{L_{2}^{\frac{n}{2}}( \mathbb{B})} \\ &\qquad {} -a \int _{\mathbb{B}}ww_{t} \vert w_{t} \vert ^{m-2} \frac{dx_{1}}{x_{1}}\,dx' -k_{2} \int _{\mathbb{B}}\nabla _{\mathbb{B}}w \cdot \nabla _{\mathbb{B}}w_{t}\frac{dx_{1}}{x_{1}}\,dx' \\ &\qquad {} +pH(t)-pd_{1}. \end{aligned}$$
(4.13)

Moreover,

$$ \begin{aligned}[b] & \biggl(\frac{p}{2}-1 \biggr) \Vert \Delta _{\mathbb{B}}w \Vert ^{2}_{L_{2}^{ \frac{n}{2}}(\mathbb{B})}-pd_{1} \\ &\quad =\biggl(\frac{p}{2}-1\biggr) \frac{\lambda _{2}^{2}-\lambda _{1}^{2}}{\lambda _{2}^{2}} \Vert \Delta _{ \mathbb{B}}w \Vert ^{2}_{L_{2}^{\frac{n}{2}}(\mathbb{B})} +\biggl( \frac{p}{2}-1\biggr) \lambda _{1}^{2} \frac{ \Vert \Delta _{\mathbb{B}}w \Vert ^{2}_{L_{2}^{\frac{n}{2}}(\mathbb{B})}}{\lambda _{2}^{2}}-pd_{1} \\ &\quad \geq c_{1} \Vert \Delta _{\mathbb{B}}w \Vert ^{2}_{L_{2}^{\frac{n}{2}}( \mathbb{B})}+c_{2}, \end{aligned} $$
(4.14)

where \(\lambda _{2}\) is given in Lemma 3.2, \(c_{1}=(\frac{p}{2}-1) \frac{\lambda _{2}^{2}-\lambda _{1}^{2}}{\lambda _{2}^{2}}\) and \(c_{2}=(\frac{p}{2}-1)\lambda _{1}^{2}-pd_{1}\). By Lemma 3.2(ii), we have \(c_{1}>0\), and by (3.7), we see that

$$ \begin{aligned}[b] c_{2}&= \biggl( \frac{p}{2}-1 \biggr)\lambda _{1}^{2}-pd_{1} \\ &= \biggl(\frac{p}{2}-1 \biggr)\lambda _{1}^{2}- \frac{p(d+E(0))}{2} \\ &=pd-\frac{p(d+E(0))}{2} \\ &=\frac{p(d-E(0))}{2}>0. \end{aligned} $$
(4.15)

Thus, by (4.13)–(4.15), we arrive at

$$ \begin{aligned}[b] A'(t)&> \biggl(1+ \frac{p}{2} \biggr) \Vert w_{t} \Vert ^{2}_{L_{2}^{\frac{n}{2}}( \mathbb{B})} +c_{1} \Vert \Delta _{\mathbb{B}}w \Vert ^{2}_{L_{2}^{\frac{n}{2}}( \mathbb{B})} -a \int _{\mathbb{B}}ww_{t} \vert w_{t} \vert ^{m-2} \frac{dx_{1}}{x_{1}}\,dx' \\ &\quad {} -k_{2} \int _{\mathbb{B}}\nabla _{\mathbb{B}}w\cdot \nabla _{ \mathbb{B}}w_{t}\frac{dx_{1}}{x_{1}}\,dx' +pH(t). \end{aligned} $$
(4.16)

We estimate the right-hand side of the above equation. By Hölder’s inequality and the inequality \(\|w\|_{L_{m}^{\frac{n}{m}}(\mathbb{B})}\leq C \|w\|_{L_{p}^{ \frac{n}{p}}(\mathbb{B})}\), we obtain

$$ \begin{aligned}[b] \biggl\vert \int _{\mathbb{B}}ww_{t} \vert w_{t} \vert ^{m-2}\frac{dx_{1}}{x_{1}}\,dx' \biggr\vert & \leq \Vert w \Vert _{L_{m}^{\frac{n}{m}}(\mathbb{B})} \Vert w_{t} \Vert ^{m-1}_{L_{m}^{ \frac{n}{m}}(\mathbb{B})} \\ &\leq C \Vert w \Vert _{L_{p}^{\frac{n}{p}}(\mathbb{B})} \Vert w_{t} \Vert ^{m-1}_{L_{m}^{ \frac{n}{m}}(\mathbb{B})} \\ &=C \Vert w \Vert ^{1-\frac{p}{m}}_{L_{p}^{\frac{n}{p}}(\mathbb{B})} \Vert w \Vert ^{ \frac{p}{m}}_{L_{p}^{\frac{n}{p}}(\mathbb{B})} \Vert w_{t} \Vert ^{m-1}_{L_{m}^{ \frac{n}{m}}(\mathbb{B})}. \end{aligned} $$
(4.17)

Note that from (4.8) and (4.2) we get

$$ \begin{aligned}[b] H(t)&=d_{1}-E(t) \\ &< d-\frac{1}{2} \Vert w_{t} \Vert ^{2}_{L_{2}^{\frac{n}{2}}(\mathbb{B})} - \frac{1}{2} \Vert \Delta _{\mathbb{B}}w \Vert ^{2}_{L_{2}^{\frac{n}{2}}( \mathbb{B})} + \frac{b}{p} \Vert w \Vert ^{p}_{L_{p}^{\frac{n}{p}}(\mathbb{B})} \\ &< \frac{p-2}{2p} \Vert \Delta _{\mathbb{B}}w \Vert ^{2}_{L_{2}^{\frac{n}{2}}( \mathbb{B})} -\frac{1}{2} \Vert w_{t} \Vert ^{2}_{L_{2}^{\frac{n}{2}}( \mathbb{B})} - \frac{1}{2} \Vert \Delta _{\mathbb{B}}w \Vert ^{2}_{L_{2}^{ \frac{n}{2}}(\mathbb{B})} \\ &\quad {} +\frac{b}{p} \Vert w \Vert ^{p}_{L_{p}^{\frac{n}{p}}(\mathbb{B})} \\ &\leq \frac{b}{p} \Vert w \Vert ^{p}_{L_{p}^{\frac{n}{p}}(\mathbb{B})}. \end{aligned} $$
(4.18)

Thus, by (4.11) and (4.18), we see that

$$ 0< H(0)\leq H(t)< \frac{b}{p} \Vert w \Vert ^{p}_{L_{p}^{\frac{n}{p}}(\mathbb{B})}, \quad t\geq 0. $$
(4.19)

Then, using (4.19), we have from (4.17) that

$$ \biggl\vert \int _{\mathbb{B}}ww_{t} \vert u_{t} \vert ^{m-2}\frac{dx_{1}}{x_{1}}\,dx' \biggr\vert \leq C \biggl(\frac{p}{b}H(t) \biggr)^{\frac{1}{p}(1-\frac{p}{m})} \Vert w \Vert ^{\frac{p}{m}}_{L_{p}^{\frac{n}{p}}(\mathbb{B})} \Vert w_{t} \Vert ^{m-1}_{L_{m}^{ \frac{n}{m}}(\mathbb{B})}. $$
(4.20)

Hence, by Young’s inequality and (4.10), we obtain

$$ \begin{aligned}[b] & a \biggl\vert \int _{\mathbb{B}}ww_{t} \vert w_{t} \vert ^{m-2} \frac{dx_{1}}{x_{1}}\,dx' \biggr\vert \\ &\quad \leq c_{3}H(t)^{-\alpha ^{\ast }} \biggl(\frac{a\theta ^{m}}{m} \Vert w \Vert ^{p}_{L_{p}^{ \frac{n}{p}}(\mathbb{B})}+ \frac{a(m-1)}{m}\theta ^{-m/(m-1)} \Vert w_{t} \Vert ^{m}_{L_{m}^{\frac{n}{m}}(\mathbb{B})} \biggr) \\ &\quad \leq c_{4}H(t)^{-\alpha ^{\ast }} \bigl(\theta ^{m} \Vert w \Vert ^{p}_{L_{p}^{ \frac{n}{p}}(\mathbb{B})}+ a\theta ^{-m/(m-1)} \Vert w_{t} \Vert ^{m}_{L_{m}^{ \frac{n}{m}}(\mathbb{B})} \bigr), \end{aligned} $$
(4.21)

where \(c_{3}=C (\frac{p}{b} )^{\frac{1}{p}-\frac{1}{m}}\), \(\alpha ^{\ast }=\frac{1}{m} -\frac{1}{p}>0\), \(\theta >0\), and \(c_{4}=c_{3}\max \{\frac{a}{m}, \frac{m-1}{m} \}\).

Letting \(0<\alpha <\alpha ^{\ast }\) and by (4.19), we see that

$$ \begin{aligned}[b] & a \biggl\vert \int _{\mathbb{B}}ww_{t} \vert w_{t} \vert ^{m-2} \frac{dx_{1}}{x_{1}}\,dx' \biggr\vert \\ &\quad \leq c_{4} \bigl[\theta ^{m}H(0)^{-\alpha ^{\ast }} \Vert w \Vert ^{p}_{L_{p}^{ \frac{n}{p}}(\mathbb{B})} +\theta ^{-m/(m-1)}H(t)^{-\alpha ^{\ast }}H'(t) \bigr] \\ &\quad \leq c_{4} \bigl[\theta ^{m}H(0)^{-\alpha ^{\ast }} \Vert w \Vert ^{p}_{L_{p}^{ \frac{n}{p}}(\mathbb{B})} +\theta ^{-m/(m-1)}H(0)^{\alpha -\alpha ^{ \ast }}H(t)^{-\alpha }H'(t) \bigr]. \end{aligned} $$
(4.22)

Using Young’s inequality again, we obtain

$$\begin{aligned} & \int _{\mathbb{B}}\nabla _{\mathbb{B}}w\cdot \nabla _{ \mathbb{B}}w_{t}\frac{dx_{1}}{x_{1}}\,dx' \\ &\quad =- \int _{\mathbb{B}}\Delta _{\mathbb{B}}w\cdot w_{t} \frac{dx_{1}}{x_{1}}\,dx' \\ &\quad \leq \biggl\vert \int _{\mathbb{B}}\Delta _{\mathbb{B}}w\cdot w_{t} \frac{dx_{1}}{x_{1}}\,dx' \biggr\vert \\ &\quad \leq \frac{1}{2} \bigl( \Vert \Delta _{\mathbb{B}}w \Vert ^{2}_{L_{2}^{ \frac{n}{2}}(\mathbb{B})} + \Vert w_{t} \Vert ^{2}_{L_{2}^{\frac{n}{2}}( \mathbb{B})} \bigr). \end{aligned}$$
(4.23)

Then (4.16) becomes

$$ \begin{aligned}[b] & A'(t) \\ &\quad > \biggl(1+\frac{p}{2} \biggr) \Vert w_{t} \Vert ^{2}_{L_{2}^{\frac{n}{2}}( \mathbb{B})} +c_{1} \Vert \Delta _{\mathbb{B}}w \Vert ^{2}_{L_{2}^{\frac{n}{2}}( \mathbb{B})} -c_{4}\theta ^{m}H(0)^{-\alpha ^{\ast }} \Vert w \Vert ^{p}_{L_{p}^{ \frac{n}{p}}(\mathbb{B})} \\ &\qquad {} -c_{4}\theta ^{-m/(m-1)}H(0)^{\alpha -\alpha ^{\ast }}H(t)^{- \alpha }H'(t) -\frac{k_{2}}{2} \Vert \Delta _{\mathbb{B}}w \Vert ^{2}_{L_{2}^{ \frac{n}{2}}(\mathbb{B})} \\ &\qquad {} -\frac{k_{2}}{2} \Vert w_{t} \Vert ^{2}_{L_{2}^{\frac{n}{2}}(\mathbb{B})} +pH(t). \end{aligned} $$
(4.24)

Now, we define

$$ L(t)=H^{1-\alpha }(t)+\varepsilon A(t),\quad t\geq 0, $$
(4.25)

where ε is small to be specified later and

$$ 0< \alpha \leq \frac{p-2}{2p}. $$
(4.26)

By differentiating (4.25), by Lemma 2.2 and (4.24), we see that

$$ \begin{aligned}[b] L'(t)&=(1-\alpha )H^{-\alpha }(t)H'(t)+\varepsilon A'(t) \\ &>(1-\alpha )H^{-\alpha }(t)H'(t)+\varepsilon \biggl[ \biggl(1+\frac{p}{2}- \frac{k_{2}}{2} \biggr) \Vert w_{t} \Vert ^{2}_{L_{2}^{\frac{n}{2}}(\mathbb{B})} \\ &\quad {} + \biggl(c_{1}-\frac{k_{2}}{2} \biggr) \Vert \Delta _{\mathbb{B}}w \Vert ^{2}_{L_{2}^{ \frac{n}{2}}(\mathbb{B})} -c_{4}\theta ^{m}H(0)^{-\alpha ^{\ast }} \Vert w \Vert ^{p}_{L_{p}^{\frac{n}{p}}(\mathbb{B})} \\ &\quad {} -c_{4}\theta ^{-m/(m-1)}H(0)^{\alpha -\alpha ^{\ast }}H(t)^{- \alpha }H'(t) +pH(t) \biggr]. \end{aligned} $$
(4.27)

Letting

$$ a_{1}=\min \biggl\{ \frac{p}{2}, c_{1}- \frac{k_{2}}{2}, 1+\frac{p}{2}- \frac{k_{2}}{2} \biggr\} >0 $$

and decomposing \(\varepsilon pH(t)\) in (4.27) by

$$ \varepsilon pH(t)=2a_{1}\varepsilon H(t)+(p-2a_{1}) \varepsilon H(t). $$

Thus, by (4.8) and (3.3), we obtain

$$ \begin{aligned}[b] L'(t) &> \bigl(1- \alpha -c_{4}\varepsilon \theta ^{-m/(m-1)}H(0)^{ \alpha -\alpha ^{\ast }} \bigr) H^{-\alpha }(t)H'(t) \\ &\quad {} +\varepsilon \biggl(1+\frac{p}{2}-\frac{k_{2}}{2} \biggr) \Vert w_{t} \Vert ^{2}_{L_{2}^{ \frac{n}{2}}(\mathbb{B})} +\varepsilon \biggl(c_{1}-\frac{k_{2}}{2} \biggr) \Vert \Delta _{\mathbb{B}}w \Vert ^{2}_{L_{2}^{\frac{n}{2}}(\mathbb{B})} \\ &\quad {} -c_{4}\varepsilon \theta ^{m}H(0)^{-\alpha ^{\ast }} \Vert w \Vert ^{p}_{L_{p}^{ \frac{n}{p}}(\mathbb{B})} +2a_{1} \varepsilon \biggl(-\frac{1}{2} \Vert w_{t} \Vert ^{2}_{L_{2}^{\frac{n}{2}}(\mathbb{B})} \\ &\quad {} -\frac{1}{2} \Vert \Delta _{\mathbb{B}}w \Vert ^{2}_{L_{2}^{\frac{n}{2}}( \mathbb{B})} +\frac{b}{p} \Vert w \Vert ^{p}_{L_{p}^{\frac{n}{p}}(\mathbb{B})} \biggr) +(p-2a_{1})\varepsilon H(t) \\ &= \bigl(1-\alpha -c_{4}\varepsilon \theta ^{-m/(m-1)}H(0)^{\alpha - \alpha ^{\ast }} \bigr) H^{-\alpha }(t)H'(t) \\ &\quad {} +\varepsilon \biggl(1+\frac{p}{2}-\frac{k_{2}}{2}-a_{1} \biggr) \Vert w_{t} \Vert ^{2}_{L_{2}^{\frac{n}{2}}(\mathbb{B})} \\ &\quad {} +\varepsilon \biggl(c_{1}-\frac{k_{2}}{2}-a_{1} \biggr) \Vert \Delta _{ \mathbb{B}}w \Vert ^{2}_{L_{2}^{\frac{n}{2}}(\mathbb{B})} \\ &\quad {} +\varepsilon \biggl[\frac{2a_{1}b}{p}-c_{4}\theta ^{m}H(0)^{- \alpha ^{\ast }} \biggr] \Vert w \Vert ^{p}_{L_{p}^{\frac{n}{p}}(\mathbb{B})} \\ &\quad {} +(p-2a_{1})\varepsilon H(t). \end{aligned} $$
(4.28)

Now, we choose \(\theta >0\) small such that

$$ \frac{2a_{1}b}{p}-c_{4}\theta ^{m}H(0)^{-\alpha ^{\ast }} \geq \frac{a_{1}b}{2p}, $$

and we pick ε small enough so that

$$ 1-\alpha -c_{4}\varepsilon \theta ^{-m/(m-1)}H(0)^{\alpha -\alpha ^{ \ast }} \geq 0. $$

Then (4.28) becomes

$$ L'(t)>c_{5}\varepsilon \bigl( \Vert w_{t} \Vert ^{2}_{L_{2}^{\frac{n}{2}}( \mathbb{B})}+ \Vert \Delta _{\mathbb{B}}w \Vert ^{2}_{L_{2}^{\frac{n}{2}}( \mathbb{B})} + \Vert w \Vert ^{p}_{L_{p}^{\frac{n}{p}}(\mathbb{B})}+H(t) \bigr), $$
(4.29)

here \(c_{5}=\min \{\frac{a_{1}b}{2p}, c_{1}-\frac{k_{2}}{2}-a_{1}, 1+ \frac{p}{2}-\frac{k_{2}}{2}-a_{1}, p-2a_{1} \}\). Thus \(L(t)\) is a nondecreasing function on \(t\geq 0\), and we take ε small enough such that

$$ L(0)=H^{1-\alpha }(0)+\varepsilon \int _{\mathbb{B}}w_{0}w_{1} \frac{dx_{1}}{x_{1}}\,dx'>0. $$
(4.30)

Hence, we have

$$ L(t)>0, \quad \forall t\geq 0. $$
(4.31)

Next we estimate the second term in (4.25) as follows:

$$ \biggl\vert \int _{\mathbb{B}}ww_{t}\frac{dx_{1}}{x_{1}}\,dx' \biggr\vert \leq \Vert w \Vert _{L_{2}^{ \frac{n}{2}}(\mathbb{B})} \Vert w_{t} \Vert _{L_{2}^{\frac{n}{2}}(\mathbb{B})} \leq C \Vert w \Vert _{L_{p}^{\frac{n}{p}}(\mathbb{B})} \Vert w_{t} \Vert _{L_{2}^{ \frac{n}{2}}(\mathbb{B})}. $$

So we have

$$ \biggl\vert \int _{\mathbb{B}}ww_{t}\frac{dx_{1}}{x_{1}}\,dx' \biggr\vert ^{1/(1- \alpha )}\leq C \Vert w \Vert ^{1/(1-\alpha )}_{L_{p}^{\frac{n}{p}}(\mathbb{B})} \Vert w_{t} \Vert ^{1/(1-\alpha )}_{L_{2}^{\frac{n}{2}}(\mathbb{B})}. $$

Again Young’s inequality gives

$$ \biggl\vert \int _{\mathbb{B}}ww_{t}\frac{dx_{1}}{x_{1}}\,dx' \biggr\vert ^{1/(1- \alpha )}\leq C \bigl[ \Vert w \Vert ^{\mu _{1}/(1-\alpha )}_{L_{p}^{\frac{n}{p}}( \mathbb{B})}+ \Vert w_{t} \Vert ^{\mu _{2}/(1-\alpha )}_{L_{2}^{\frac{n}{2}}( \mathbb{B})} \bigr]. $$
(4.32)

We take \(\mu _{1}=\frac{2(1-\alpha )}{1-2\alpha }\), \(\mu _{2}=2(1-\alpha )\) to get \(\mu _{1}/(1-\alpha )=2/(1-2\alpha )\leq p\) by condition (4.26). Therefore (4.32) becomes

$$ \biggl\vert \int _{\mathbb{B}}ww_{t}\frac{dx_{1}}{x_{1}}\,dx' \biggr\vert ^{1/(1- \alpha )}\leq C \bigl[ \Vert w \Vert ^{s}_{L_{p}^{\frac{n}{p}}(\mathbb{B})}+ \Vert w_{t} \Vert ^{2}_{L_{2}^{\frac{n}{2}}(\mathbb{B})} \bigr], $$
(4.33)

where \(s=2/(1-2\alpha )\leq p\). By using Corollary 4.1, we obtain

$$ \biggl\vert \int _{\mathbb{B}}ww_{t}\frac{dx_{1}}{x_{1}}\,dx' \biggr\vert ^{1/(1- \alpha )}\leq C \bigl[H(t)+ \Vert w \Vert ^{p}_{L_{p}^{\frac{n}{p}}(\mathbb{B})}+ \Vert w_{t} \Vert ^{2}_{L_{2}^{\frac{n}{2}}(\mathbb{B})} \bigr],\quad \forall t\geq 0. $$
(4.34)

Consequently, we have

$$ \begin{aligned}[b] L^{1/(1-\alpha )}(t) &= \biggl(H^{1-\alpha }(t)+\varepsilon \int _{ \mathbb{B}}ww_{t}\frac{dx_{1}}{x_{1}}\,dx' \biggr)^{1/(1-\alpha )} \\ &\leq 2^{\alpha /(1-\alpha )} \biggl(H(t)+ \biggl\vert \int _{\mathbb{B}}ww_{t} \frac{dx_{1}}{x_{1}}\,dx' \biggr\vert ^{1/(1-\alpha )} \biggr) \\ &\leq C \bigl[H(t)+ \Vert \Delta _{\mathbb{B}}w \Vert ^{2}_{L_{2}^{\frac{n}{2}}( \mathbb{B})}+ \Vert w \Vert ^{p}_{L_{p}^{\frac{n}{p}}(\mathbb{B})}+ \Vert w_{t} \Vert ^{2}_{L_{2}^{ \frac{n}{2}}(\mathbb{B})} \bigr]. \end{aligned} $$
(4.35)

We then combine (4.29) and (4.35) to arrive at

$$ L'(t)\geq \varGamma L^{1/(1-\alpha )}(t), $$
(4.36)

where Γ is a constant dependent on C, \(c_{3}\) and ε only (and hence is independent of the solution w). A simple integration of (4.36) over \((0, t)\) then yields

$$ L^{\alpha /(1-\alpha )}(t)\geq \frac{1}{L^{-\alpha /(1-\alpha )}(0)-\varGamma t\alpha /(1-\alpha )}. $$
(4.37)

Since \(L(0)>0\), (4.37) shows that \(L(t)\) becomes infinite in a finite time \(T_{\mathrm{max}}\leq T^{\ast }= \frac{1-\alpha }{\varGamma \alpha [L(0)]^{\alpha /(1-\alpha )}}\).

(2) For \(E(0)<0\), we set

$$ H(t)=-E(t), $$

instead of (4.8). Then, applying the same arguments as in part (1), we have the result. □

Theorem 4.2

Under the assumption of Theorem 4.1, let \(w(x, t)\)be a blow-up solution of problem (1.7)(1.9). Then a lower bound T for the lifespan \(t^{\star }\)of w is given by

$$ T:= \int _{\phi (0)}^{+\infty } \frac{ds}{\bar{c}_{2}s^{\frac{\alpha (p-1)}{p(\alpha -1)}}+\bar{c}_{3}} \leq t^{\star }, $$
(4.38)

with

$$ \phi (0)=\frac{1}{2} \int _{\mathbb{B}} \vert w_{1} \vert ^{2}\frac{dx_{1}}{x_{1}}\,dx'+ \frac{1}{2} \int _{\mathbb{B}} \vert \Delta _{\mathbb{B}}w_{0} \vert ^{2} \frac{dx_{1}}{x_{1}}\,dx' + \frac{b}{p} \int _{\mathbb{B}} \vert w_{0} \vert ^{p} \frac{dx_{1}}{x_{1}}\,dx', $$

where \(1<\alpha <2\)and \(\bar{c}_{2}\), \(\bar{c}_{3}\)are positive constants to be determined later.

Proof

Now we want to derive a lower bound for the lifespan \(t^{\star }\) of the blow-up solution. To this end, we introduce the auxiliary function

$$ \phi (t)=\frac{1}{2} \int _{\mathbb{B}} \vert w_{t} \vert ^{2}\frac{dx_{1}}{x_{1}}\,dx'+ \frac{1}{2} \int _{\mathbb{B}} \vert \Delta _{\mathbb{B}}w \vert ^{2} \frac{dx_{1}}{x_{1}}\,dx' +\frac{b}{p} \int _{\mathbb{B}} \vert w \vert ^{p} \frac{dx_{1}}{x_{1}}\,dx' $$
(4.39)

and compute a value \(T>0\) such that \(\phi (t)\) remains bounded for \(t\in [0, T]\). Clearly, T is a lower bound for \(t^{\star }\). Differentiating (4.39) and making use of the second Green’s formula, we obtain in view of (1.7)

$$ \begin{aligned}[b] \phi '(t)&= \int _{\mathbb{B}}w_{t}w_{tt} \frac{dx_{1}}{x_{1}}\,dx'+ \int _{ \mathbb{B}} \Delta _{\mathbb{B}}w\cdot \Delta _{\mathbb{B}}w_{t} \frac{dx_{1}}{x_{1}}\,dx' \\ &\quad {} +b \int _{\mathbb{B}} \vert w \vert ^{p-2}ww_{t} \frac{dx_{1}}{x_{1}}\,dx' \\ &= \int _{\mathbb{B}}w_{t}\bigl[2b \vert w \vert ^{p-2}w-aw_{t} \vert u_{t} \vert ^{m-2}+k_{2} \Delta _{\mathbb{B}}w_{t} \bigr]\frac{dx_{1}}{x_{1}}\,dx' \\ &=2b \int _{\mathbb{B}} \vert w \vert ^{p-2}ww_{t} \frac{dx_{1}}{x_{1}}\,dx'-a \int _{ \mathbb{B}} \vert w_{t} \vert ^{m}\frac{dx_{1}}{x_{1}}\,dx' \\ &\quad {} -k_{2} \int _{\mathbb{B}} \vert \nabla _{\mathbb{B}}w_{t} \vert ^{2} \frac{dx_{1}}{x_{1}}\,dx' \\ &\leq 2b \int _{\mathbb{B}} \vert w_{t} \vert \vert w \vert ^{p-1}\frac{dx_{1}}{x_{1}}\,dx'-k_{2} \int _{\mathbb{B}} \vert \nabla _{\mathbb{B}}w_{t} \vert ^{2}\frac{dx_{1}}{x_{1}}\,dx'. \end{aligned} $$
(4.40)

Now we make use of Hölder’s inequality to the first term on the right-hand side of (4.40) to obtain

$$ \begin{aligned}[b] \int _{\mathbb{B}} \vert w_{t} \vert \vert w \vert ^{p-1}\frac{dx_{1}}{x_{1}}\,dx'&\leq \biggl( \int _{\mathbb{B}} \vert w \vert ^{p} \frac{dx_{1}}{x_{1}}\,dx' \biggr)^{\frac{p-1}{p}} \biggl( \int _{\mathbb{B}} \vert w_{t} \vert ^{p}\frac{dx_{1}}{x_{1}}\,dx' \biggr)^{ \frac{1}{p}} \\ &= \Vert w \Vert _{L_{p}^{\frac{n}{p}}}^{p-1} \Vert w_{t} \Vert _{L_{p}^{\frac{n}{p}}}. \end{aligned} $$
(4.41)

Then

$$ \phi '(t)\leq 2b \Vert w \Vert _{L_{p}^{\frac{n}{p}}}^{p-1} \Vert w_{t} \Vert _{L_{p}^{ \frac{n}{p}}}-k_{2} \int _{\mathbb{B}} \vert \nabla _{\mathbb{B}}w_{t} \vert ^{2} \frac{dx_{1}}{x_{1}}\,dx'. $$
(4.42)

In the rest of the proof, we apply Young’s inequality to the first term on the right-hand side of (4.42) with exponents α and \(\frac{\alpha }{\alpha -1}\), where \(1<\alpha <2\) is a constant. Thus we obtain

$$ \begin{aligned}[b] 2b \Vert w \Vert _{L_{p}^{\frac{n}{p}}}^{p-1} \Vert w_{t} \Vert _{L_{p}^{\frac{n}{p}}}& \leq \bar{c}_{1} \Vert w \Vert _{L_{p}^{\frac{n}{p}}}^{ \frac{\alpha (p-1)}{\alpha -1}}+ \Vert w_{t} \Vert _{L_{p}^{\frac{n}{p}}}^{ \alpha } \\ &\leq \bar{c}_{2}\phi (t)^{\frac{\alpha (p-1)}{p(\alpha -1)}}+c_{ \star }^{\alpha } \Vert \nabla _{\mathbb{B}}w_{t} \Vert _{L_{2}^{\frac{n}{2}}}^{ \alpha } \end{aligned} $$
(4.43)

with \(\bar{c}_{1}=(2b)^{\frac{\alpha }{\alpha -1}}\alpha ^{- \frac{1}{\alpha -1}}\frac{\alpha -1}{\alpha }\) and \(\bar{c}_{2}=\bar{c}_{1}(\frac{p}{b})^{ \frac{\alpha (p-1)}{p(\alpha -1)}}\). Because of \(\alpha <2\), we can use Young’s inequality with exponents \(\frac{2}{\alpha }\) and \(\frac{2}{2-\alpha }\) to have

$$ c_{\star }^{\alpha } \Vert \nabla _{\mathbb{B}}w_{t} \Vert _{L_{2}^{\frac{n}{2}}}^{ \alpha }\leq k_{2} \Vert \nabla _{\mathbb{B}}w_{t} \Vert _{L_{2}^{\frac{n}{2}}}^{2}+ \bar{c}_{3} $$

with \(\bar{c}_{3}=c_{\star }^{\frac{2\alpha }{2-\alpha }}( \frac{\alpha }{2k_{2}})^{\frac{\alpha }{2-\alpha }}\frac{2-\alpha }{2}\). Inserting this in (4.43) yields

$$ 2b \Vert w \Vert _{L_{p}^{\frac{n}{p}}}^{p-1} \Vert w_{t} \Vert _{L_{p}^{\frac{n}{p}}} \leq \bar{c}_{2}\phi (t)^{\frac{\alpha (p-1)}{p(\alpha -1)}}+k_{2} \Vert \nabla _{\mathbb{B}}w_{t} \Vert _{L_{2}^{\frac{n}{2}}}^{2}+ \bar{c}_{3}. $$
(4.44)

Inequality (4.44) along with (4.42) implies that

$$ \phi '(t)\leq \bar{c}_{2}\phi (t)^{\frac{\alpha (p-1)}{p(\alpha -1)}}+ \bar{c}_{3}. $$
(4.45)

Then

$$ \frac{d\phi }{\bar{c}_{2}\phi (t)^{\frac{\alpha (p-1)}{p(\alpha -1)}}+\bar{c}_{3}} \leq dt. $$
(4.46)

Integrating (4.46) from 0 to \(t^{\star }\), we obtain

$$ \int _{\phi (0)}^{\phi (t)} \frac{ds}{\bar{c}_{2}s^{\frac{\alpha (p-1)}{p(\alpha -1)}}+\bar{c}_{3}} \leq t^{\star }. $$
(4.47)

Thus, we obtain the desired result. □

In the following theorem, by means of a first order differential inequality technique, we obtain a lower bound for the blow-up time which is different from (4.38).

Theorem 4.3

Suppose that the conditions of Theorem 4.1hold. Let \(w(x, t)\)be a blow-up solution of problem (1.7)(1.9). Then a lower bound for the lifespan \(t^{\star }\)of w is given by

$$ \tilde{T}:=\bigl\{ (p-2)b\kappa ^{\frac{1}{2}}\bigl(\psi (0) \bigr)^{\frac{p-2}{2}}\bigr\} ^{-1}< t^{\star }, $$
(4.48)

with

$$ \psi (0)= \int _{\mathbb{B}} \vert w_{1} \vert ^{2}\frac{dx_{1}}{x_{1}}\,dx'+ \int _{ \mathbb{B}} \vert \Delta _{\mathbb{B}}w_{0} \vert ^{2}\frac{dx_{1}}{x_{1}}\,dx', $$

where \(\kappa =C_{0}^{2(p-1)}\).

Proof

We introduce the auxiliary function

$$ \psi (t)= \int _{\mathbb{B}} \vert w_{t} \vert ^{2}\frac{dx_{1}}{x_{1}}\,dx'+ \int _{ \mathbb{B}} \vert \Delta _{\mathbb{B}}w \vert ^{2}\frac{dx_{1}}{x_{1}}\,dx' $$
(4.49)

and compute a value \(\tilde{T}>0\) such that \(\psi (t)\) remains bounded for \(t\in [0, \tilde{T}]\). Clearly is a lower bound for \(t^{\star }\). Differentiating (4.49) and making use of the second Green’s formula, we obtain in view of (1.7)

$$ \begin{aligned}[b] \psi '(t)&=2 \int _{\mathbb{B}}w_{t}w_{tt} \frac{dx_{1}}{x_{1}}\,dx'+2 \int _{\mathbb{B}} \Delta _{\mathbb{B}}w\cdot \Delta _{\mathbb{B}}w_{t} \frac{dx_{1}}{x_{1}}\,dx' \\ &=2 \int _{\mathbb{B}}w_{t}\bigl[w_{tt}+\Delta _{\mathbb{B}}^{2}w\bigr] \frac{dx_{1}}{x_{1}}\,dx' \\ &=2b \int _{\mathbb{B}} \vert w \vert ^{p-2}ww_{t} \frac{dx_{1}}{x_{1}}\,dx'-2a \int _{ \mathbb{B}} \vert w_{t} \vert ^{m}\frac{dx_{1}}{x_{1}}\,dx' \\ &\quad {} -2k_{2} \int _{\mathbb{B}} \vert \nabla _{\mathbb{B}}w_{t} \vert ^{2} \frac{dx_{1}}{x_{1}}\,dx' \\ &\leq 2b \int _{\mathbb{B}} \vert w_{t} \vert \vert w \vert ^{p-1}\frac{dx_{1}}{x_{1}}\,dx'. \end{aligned} $$
(4.50)

Making use of the Schwarz inequality leads to

$$ \psi '(t)\leq 2b \biggl( \int _{\mathbb{B}} \vert w_{t} \vert ^{2} \frac{dx_{1}}{x_{1}}\,dx' \int _{\mathbb{B}} \vert w \vert ^{2(p-1)} \frac{dx_{1}}{x_{1}}\,dx' \biggr)^{\frac{1}{2}}. $$
(4.51)

Applying the Poincaré inequality, we obtain

$$ \begin{aligned}[b] \int _{\mathbb{B}} \vert w \vert ^{2(p-1)} \frac{dx_{1}}{x_{1}}\,dx'&= \Vert w \Vert _{L_{2(p-1)}^{ \frac{n}{2(p-1)}}}^{2(p-1)} \\ &\leq C_{0}^{2(p-1)} \Vert \Delta _{\mathbb{B}}w \Vert _{L_{2}^{\frac{n}{2}}}^{2(p-1)} \\ &\leq C_{0}^{2(p-1)}\bigl(\psi (t)\bigr)^{p-1}. \end{aligned} $$
(4.52)

Moreover, we have

$$ \int _{\mathbb{B}} \vert w_{t} \vert ^{2}\frac{dx_{1}}{x_{1}}\,dx'< \psi (t). $$
(4.53)

From (4.51)–(4.53), we obtain the differential inequality

$$ \psi '(t)< 2b\kappa ^{\frac{1}{2}}\psi (t)^{\frac{p}{2}}, $$
(4.54)

where \(\kappa =C_{0}^{2(p-1)}\), then (4.54) can be rewritten as

$$ \bigl(\psi ^{\frac{2-p}{2}}(t) \bigr)'>-(p-2)b\kappa ^{\frac{1}{2}}. $$
(4.55)

Integrating (4.55) from 0 to t, we obtain

$$ \bigl(\psi ^{\frac{2-p}{2}}(t) \bigr)> \psi ^{\frac{2-p}{2}}(0)-(p-2)b\kappa ^{ \frac{1}{2}}t. $$
(4.56)

Inequality (4.56) shows that \(\psi (t)\) remains bounded for

$$ t< \tilde{T}:= \frac{(\psi (0))^{\frac{2-p}{2}}}{(p-2)b\kappa ^{\frac{1}{2}}}. $$
(4.57)

From the discussion above in Theorem 3.1 and Theorem 4.1, we immediately obtain a specifying result of the global existence and nonexistence of solutions for problem (1.7)–(1.9) as follows. □

Remark 4.1

Suppose that \(2< p< p^{\star }\), \(w_{1}\in L_{2}^{\frac{n}{2}}(\mathbb{B})\), and \(0< E(0)< d\), then problem (1.7)–(1.9) admits a global weak solution without relation between m and p provided \(I(w_{0})>0\) and w satisfies the assumption of Lemma 3.3; problem (1.7)–(1.9) does not admit any global solution provided \(p>m\geq 2\), \(I(w_{0})<0\), and \(\|\Delta _{\mathbb{B}}w_{0}\|_{L_{2}^{\frac{n}{2}}(\mathbb{B})}> \lambda _{1}\).

From the discussion above in Theorem 4.1 and Theorem 4.2, we give the bounds for blow-up time for problem (1.7)–(1.9) under the initial condition \(I(w_{0})<0\).

Remark 4.2

Suppose that \(2< p< p^{\star }\), \(p>m\geq 2\) and \(E(0)< d\), \(w_{1}\in L_{2}^{\frac{n}{2}}(\mathbb{B})\), then problem (1.7)–(1.9) does not admit any global solution provided \(I(w_{0})<0\). Furthermore, the corresponding upper and lower bounds of blow-up time \(T_{\mathrm{max}}\) are given by the following form:

$$ \int _{\phi (0)}^{+\infty } \frac{ds}{\bar{c}_{2}s^{\frac{\alpha (p-1)}{p(\alpha -1)}}+\bar{c}_{3}} \leq T_{\mathrm{max}} \leq \frac{1-\alpha }{\varGamma \alpha [L(0)]^{\alpha /(1-\alpha )}}. $$
(4.58)

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Acknowledgements

The authors thank the referees for valuable comments and suggestions which improved the presentation of this manuscript.

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This work was supported by the NSF of China (Grant Nos. 11701116, 11801108, and 11871172).

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Yu, J., Shang, Y. & Di, H. On decay and blow-up of solutions for a nonlinear Petrovsky system with conical degeneration. Bound Value Probl 2020, 141 (2020). https://doi.org/10.1186/s13661-020-01438-w

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MSC

  • 35B44
  • 35K30
  • 35K59

Keywords

  • Petrovsky system
  • Cone Sobolev spaces
  • Global existence
  • Decay rate
  • Blow-up