In this section we shall prove the global existence and finite time blow up of solutions and give some sharp conditions for global existence and finite time blow up of solutions for problem
(1.1) which are completely different from those given in [2, 3, 4, 5], [17, 18, 23].

#### Theorem 4.1

*Let p satisfy* (*A*), *φ*_{0} ∈ *H*. *Assume that* ∥*φ*_{0}∥ = 0 *or E*(*φ*_{0}) ≤ *d*, *I*(*φ*_{0}) > 0. *Then problem (1.1) admits a unique global solution φ*(*t*) ∈ *C*([0, ∞); *H͠*) *such that*

∥*φ*(*t*)∥ = ∥*φ*_{0}∥ = 0 *for* 0 ≤ *t* < ∞ *if* ∥*φ*_{0}∥ = 0.

*or*

*φ*(*t*) ∈ *W for* 0 ≤ *t* < ∞ *if E*(*φ*_{0}) ≤ *d*, *I*(*φ*_{0}) > 0.

#### Proof

Firstly from Proposition 1.1 it follows that problem (1.1) admits a unique local solution *φ*(*t*) ∈ *C*([0, *T*); *H*) satisfying (1.5), (1.6). Next we prove *T* = +∞.

If ∥*φ*_{0}∥ = 0, then by (1.5) we have ∥*φ*(*t*)∥ = ∥*φ*_{0}∥ = 0, 0 ≤ *t* < *T*, which gives ∥|*x*|*φ*(*t*)∥ = 0 and ∥∇*φ*(*t*)∥ = 0, i.e. ∥*φ*(*t*)∥_{H} = 0 for 0 ≤ *t* < *T*. Hence by Proposition 1.1 we get *T* = +∞.

If *E*(*φ*_{0}) ≤ *d*, *I*(*φ*_{0}) > 0, then

$$\begin{array}{}{\displaystyle E({\phi}_{0})=\frac{1}{2}\parallel |x|{\phi}_{0}{\parallel}^{2}+\frac{n(p-1)-4}{2n(p-1)}\parallel \mathrm{\nabla}{\phi}_{0}{\parallel}^{2}+\frac{1}{p+1}I({\phi}_{0})>0.}\end{array}$$

Consequently, from Theorem 3.1 and Theorem 3.2 we have *φ*(*t*) ∈ *W* for 0 ≤ *t* < *T*. Thus, from

$$\begin{array}{}\begin{array}{rl}& \frac{1}{2}\parallel |x|\phi {\parallel}^{2}+\frac{n(p-1)-4}{2n(p-1)}\parallel \mathrm{\nabla}\phi {\parallel}^{2}+\frac{1}{p+1}I(\phi )\\ =& \frac{1}{2}\parallel |x|\phi {\parallel}^{2}+J(\phi )=E({\phi}_{0}),\phantom{\rule{thinmathspace}{0ex}}0\le t<T,\end{array}\end{array}$$

we get

$$\begin{array}{}{\displaystyle \parallel |x|\phi {\parallel}^{2}+\parallel \mathrm{\nabla}\phi {\parallel}^{2}\le \frac{2n(p-1)}{n(p-1)-4}E({\phi}_{0}),\phantom{\rule{thinmathspace}{0ex}}0\le t<T}\end{array}$$

and

$$\begin{array}{}{\displaystyle \parallel |x|\phi {\parallel}^{2}+\parallel \mathrm{\nabla}\phi {\parallel}^{2}+\parallel \phi {\parallel}^{2}\le \frac{2n(p-1)}{n(p-1)-4}E({\phi}_{0})+\parallel {\phi}_{0}{\parallel}^{2},\phantom{\rule{thinmathspace}{0ex}}0\le t<T,}\end{array}$$

which gives *T* = +∞.

□

Since *I*(*φ*_{0}) > 0 gives *E*(*φ*_{0}) > 0, the following corollary is the improvement of Theorem 4.1.

#### Corollary 4.2

If in Theorem 4.1 the assumption “*E*(*φ*_{0}) ≤ *d*, *I*(*φ*_{0}) > 0” is replaced by “0 < *E*(*φ*_{0}) < *d*, *I*_{δ2}(*φ*_{0}) > 0”, where *δ*_{1} < *δ*_{2} are two roots of equation *d*(*δ*) = *E*(*φ*_{0}), then problem (1.1) admits a unique global solution *φ*(*t*) ∈ *C*([0, ∞);*H͠*) and *φ*(*t*) ∈ *W*_{δ} for *δ* ∈ [*δ*_{1}, *δ*_{2}], 0 ≤ *t* < ∞.

#### Proof

From Theorem 4.1 and Theorem 3.1 it follows that it is enough to prove *I*(*φ*_{0}) > 0. In fact, if it is false, then there exists a *δ̄* ∈ [*a*, *δ*_{2}] such that *I*_{δ̄} (*φ*_{0}) = 0 and *J*(*φ*_{0}) ≥ *d*(*δ̄*), which contradicts (3.1). □

#### Corollary 4.3

Let *p* satisfy (A), *φ*_{0} ∈ *H*, *a* < *δ*_{0} < $\begin{array}{}\frac{p+1}{2}\end{array}$. Assume that *E*(*φ*_{0}) ≤ *d*(*δ*_{0}) and *I*_{δ0}(*φ*_{0}) > 0. Then problem (1.1) admits a unique global solution *φ*(*t*) ∈ *C*([0, ∞); *H͠*) and *φ*(*t*) ∈ *W*_{δ0} for 0 ≤ *t* < ∞.

#### Proof

First we have *E*(*φ*_{0}) ≤ *d*(*δ*_{0}) < *d*(*a*) = *d* and

$$\begin{array}{}{\displaystyle E({\phi}_{0})=\frac{1}{2}\parallel |x|{\phi}_{0}{\parallel}^{2}+\left(\frac{1}{2}-\frac{{\delta}_{0}}{p+1}\right)\parallel \mathrm{\nabla}{\phi}_{0}{\parallel}^{2}+\frac{1}{p+1}{I}_{{\delta}_{0}}({\phi}_{0})>0.}\end{array}$$

If *δ*_{1} < *δ*_{2} are two roots of equation *d*(*δ*) = *E*(*φ*_{0}), then we have *δ*_{2} ≥ *δ*_{0}. Hence *I*_{δ0}(*φ*_{0}) > 0 gives *I*_{δ2}(*φ*_{0}) > 0. So by Corollary 4.2, problem (1.1) admits a unique global solution *φ*(*t*) ∈ *C*([0, ∞); *H͠*) and *φ*(*t*) ∈ *W*_{δ} for *δ* ∈ [*δ*_{1}, *δ*_{2}]. Since *δ*_{0} ∈ (*a*, *δ*_{2}] ⊂ [*δ*_{1}, *δ*_{2}] we obtain *φ*(*t*) ∈ *W*_{δ0} for 0 ≤ *t* < ∞. □

In the following theorem we give two results on global existence of solutions for problem (1.1) regarding ∥∇*φ*_{0}∥.

#### Theorem 4.4

*If in Corollary 4.2 the assumption* “*I*_{δ2}(*φ*_{0}) > 0” *is replaced by* “∥∇*φ*_{0}∥ < *r*(*δ*_{2})”, *then problem (1.1) admits a unique global solution φ*(*t*) ∈ *C*([0, ∞); *H͠*) *satisfying*

$$\begin{array}{}{\displaystyle \parallel \mathrm{\nabla}\phi {\parallel}^{2}\le \frac{E({\phi}_{0})}{a({\delta}_{1})},\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\parallel |x|\phi {\parallel}^{2}\le 2E({\phi}_{0}),\phantom{\rule{thinmathspace}{0ex}}0\le t<\mathrm{\infty},}\end{array}$$(4.1)

*where r*(*δ*) *and a*(*δ*) *are defined in Lemma 2.2 and Lemma 2.4 respectively*.

#### Proof

Firstly *E*(*φ*_{0}) > 0 gives ∥*φ*_{0}∥ ≠ 0 and (by Lemma 2.1) ∥∇*φ*_{0}∥ ≠ 0. Furthermore from 0 < ∥∇*φ*_{0}∥ < *r*(*δ*_{2}) we get *I*_{δ2}(*φ*_{0}) > 0. Hence from Corollary 4.2 it follows that problem (1.1) admits a unique global solution *φ*(*t*) ∈ *C*([0, ∞); *H͠*) and *φ*(*t*) ∈ *W*_{δ} for *δ* ∈ (*δ*_{1}, *δ*_{2}), 0 ≤ *t* < ∞. Finally in

$$\begin{array}{}{\displaystyle \frac{1}{2}\parallel |x|\phi {\parallel}^{2}+a(\delta )\parallel \mathrm{\nabla}\phi {\parallel}^{2}+\frac{1}{p+1}{I}_{\delta}(\phi )=E({\phi}_{0})}\end{array}$$

letting *δ* = *δ*_{1} we obtain (4.1). □

#### Theorem 4.5

*Let p satisfy (A)*, *φ*_{0} ∈ *H*. *Assume that E*(*φ*_{0}) = *d*, ∥∇*φ*_{0}∥ < *r*(*a*). *Then problem (1.1) admits a unique global solution φ*(*t*) ∈ *C*([0, ∞); *H͠*) *satisfying*

$$\begin{array}{}{\displaystyle \parallel \mathrm{\nabla}\phi {\parallel}^{2}\le \frac{2n(p-1)}{n(p-1)-4}d,\phantom{\rule{thinmathspace}{0ex}}\parallel |x|\phi {\parallel}^{2}\le 2d,\phantom{\rule{thinmathspace}{0ex}}0\le t<\mathrm{\infty}.}\end{array}$$(4.2)

#### Proof

Firstly by *E*(*φ*_{0}) = *d* > 0 we have ∥*φ*_{0}∥ ≠ 0. Hence by Lemma 2.2 we get *I*(*φ*_{0}) > 0. From Theorem 4.1 it follows that problem (1.1) admits a unique global solution *φ*(*t*) ∈ *C*([0, ∞); *H͠*) and *φ*(*t*) ∈ *W* for 0 ≤ *t* < ∞. Inequality (4.2) follows from

$$\begin{array}{}{\displaystyle \frac{1}{2}\parallel |x|\phi {\parallel}^{2}+\frac{n(p-1)-4}{2n(p-1)}\parallel \mathrm{\nabla}\phi {\parallel}^{2}+\frac{1}{p+1}I(\phi )=E({\phi}_{0})=d,\phantom{\rule{thinmathspace}{0ex}}0\le t<\mathrm{\infty}.}\end{array}$$

□

Next we discuss the finite time blow of solution for problem (1.1).

#### Theorem 4.6

*Let p satisfy (A)*, *φ*_{0} ∈ *H*. *Assume that E*(*φ*_{0}) < *d and I*(*φ*_{0}) < 0. *Then the solution of problem (1.1) blows up in finite time*.

#### Proof

First Proposition 1.1 gives the existence of unique local solution *φ* ∈ *C*([0, *T*); *H̃*), where *T* is the existence time of *φ*. Let us prove *T* < ∞. Arguing by contradiction, suppose *T* = ∞. Let

$$\begin{array}{}{\displaystyle F(t)=\int |x{|}^{2}|\phi {|}^{2}\mathrm{d}x.}\end{array}$$

Then by Proposition 1.2 we have

$$\begin{array}{}\begin{array}{rl}{F}^{\u2033}(t)& =8\int \left(|\mathrm{\nabla}\phi {|}^{2}-|x{|}^{2}|\phi {|}^{2}-\frac{n(p-1)}{2(p+1)}|\phi {|}^{p+1}\right)\mathrm{d}x\\ & \le 8\int \left(|\mathrm{\nabla}\phi {|}^{2}-\frac{n(p-1)}{2(p+1)}|\phi {|}^{p+1}\right)\mathrm{d}x=\frac{8}{a}{I}_{a}(\phi ).\end{array}\end{array}$$(4.3)

In order to finish this proof we consider the following two cases:

0 < *E*(*φ*_{0}) < *d*.

In this case from Theorem 3.1 we have *φ* ∈ *V*_{δ} for *δ*_{1} < *δ* < *δ*_{2} and 0 ≤ *t* < ∞, where *δ*_{1} < *δ*_{2} are two roots of equation *d*(*δ*) = *E*(*φ*_{0}). Clearly we have *δ*_{2} > *a* > 1. Hence we have *I*_{δ} (*φ*) < 0 and ∥∇*φ*∥ > *r*(*δ*) for *a* < *δ* < *δ*_{2}, 0 ≤ *t* < ∞. And *I*_{δ2}(*φ*) ≤ 0, ∥∇*φ*∥ ≥ *r*(*δ*_{2}) for 0 ≤ *t* < ∞. Thus from (4.3) we get

$$\begin{array}{}\begin{array}{rl}{F}^{\u2033}(t)& \le \frac{8}{a}{I}_{a}(\phi )=\frac{8}{a}\left((a-{\delta}_{2})\parallel \mathrm{\nabla}\phi {\parallel}^{2}+{I}_{{\delta}_{2}}(\phi )\right)\\ & \le \frac{8}{a}(a-{\delta}_{2})\parallel \mathrm{\nabla}\phi {\parallel}^{2}\le \frac{8}{a}(a-{\delta}_{2}){r}^{2}({\delta}_{2})=-C({\delta}_{2})<0,\end{array}\end{array}$$

$$\begin{array}{}{\displaystyle {F}^{\prime}(t)\le -C({\delta}_{2})t+{F}^{\prime}(0),\phantom{\rule{thinmathspace}{0ex}}0\le t<\mathrm{\infty},}\end{array}$$

Hence there exists a *t*_{0} ≥ 0 such that *F*′(*t*) < *F*′(*t*_{0}) < 0 for *t*_{0} < *t* < ∞ and

$$\begin{array}{}{\displaystyle F(t)\le {F}^{\prime}({t}_{0})(t-{t}_{0})+F({t}_{0}),\phantom{\rule{thinmathspace}{0ex}}{t}_{0}\le t<\mathrm{\infty}.}\end{array}$$(4.4)

Since *I*(*φ*_{0}) < 0 implies *F*(0) > 0 from (4.4) it follows that there exists a *T*_{1} > 0 such that *F*(*t*) > 0 for 0 ≤ *t* < *T*_{1} and

$$\begin{array}{}{\displaystyle \underset{t\to {T}_{1}}{lim}F(t)=0,}\end{array}$$

which together with

$$\begin{array}{}{\displaystyle \parallel {\phi}_{0}{\parallel}^{2}=\parallel \phi {\parallel}^{2}\le \frac{2}{n}\parallel \mathrm{\nabla}\phi \parallel {F}^{\frac{1}{2}}(t)}\end{array}$$

gives

$$\begin{array}{}{\displaystyle \underset{t\to {T}_{1}}{lim}sup\parallel \mathrm{\nabla}\phi \parallel =+\mathrm{\infty}.}\end{array}$$

This contradicts *T* = +∞.

*E*(*φ*_{0}) ≤ 0.

Since *I*(*φ*_{0}) < 0 implies ∥*φ*_{0}∥ ≠ 0, from Theorem 3.3 we have *φ* ∈ *V*_{δ} for 0 < *δ* < $\begin{array}{}\frac{p+1}{2}\end{array}$, 0 ≤ *t* < ∞. If in the proof of part (i) *δ*_{2} is replaced by $\begin{array}{}\frac{p+1}{2}\end{array}$, then we also obtain *T* < ∞.

Finally from Proposition 1.1 we get

$$\begin{array}{}{\displaystyle \underset{t\to T}{lim}\parallel \phi {\parallel}_{H}=+\mathrm{\infty}.}\end{array}$$

Theorem 4.6 is proved. □

#### Corollary 4.8

Let *p* satisfy (A), *φ*_{0} ∈ *H* and *a* < *δ* < $\begin{array}{}\frac{p+1}{2}\end{array}$. Assume that *E*(*φ*_{0}) ≤ *d*(*δ*) and *I*_{δ} (*φ*_{0}) < 0. Then solution of problem (1.1) blows up in finite time.

#### Proof

On the one hand, from *E*(*φ*_{0}) ≤ *d*(*δ*) and Corollary 2.5 we get *E*(*φ*_{0}) < *d*. On the other hand, from *I*_{δ} (*φ*_{0}) < 0 we get *I* (*φ*_{0}) < 0. So by Theorem 4.6 solution of problem (1.1) blows up in finite time. □

#### Corollary 4.9

Let *p* satisfy (A), *φ*_{0} ∈ *H*. Assume that *E*(*φ*_{0}) < 0 or *E*(*φ*_{0}) = 0, *φ*_{0} ≠ 0. Then solution of problem (1.1) blows up in finite time.

Next we discuss the blow up of solutions for problem (1.1) with *E*(*φ*_{0}) = *d*.

#### Theorem 4.10

*Let p satisfy (A)*, *φ*_{0} ∈ *H*. *Assume that E*(*φ*_{0}) = *d*, *I*(*φ*_{0}) < 0 *and F*′(0) ≤ 0. *Then solution of problem (1.1) blows up in finite time*.

#### Proof

Let *φ*(*t*) be any solution of problem (1.1) with *E*(*φ*_{0}) = *d*, *I*(*φ*_{0}) < 0 and *F*′(0) ≤ 0. Let us prove *T* < ∞. Arguing by contradiction, again suppose *T* = +∞. From Theorem 3.2 we have *φ*(*t*) ∈ *V* for 0 ≤ *t* < ∞. Hence we get

$$\begin{array}{}{\displaystyle {F}^{\u2033}(t)\le \frac{8}{a}I(\phi )<0,\phantom{\rule{thinmathspace}{0ex}}0\le t<\mathrm{\infty}.}\end{array}$$

From this and *F*′(0) ≤ 0 it follows that for any *t*_{0} > 0 we have *F*′(*t*_{0}) < 0 and

$$\begin{array}{}{\displaystyle F(t)<{F}^{\prime}({t}_{0})(t-{t}_{0})+F({t}_{0}),\phantom{\rule{thinmathspace}{0ex}}{t}_{0}<t<\mathrm{\infty}.}\end{array}$$

Hence there exists a *T*_{1} > 0 such that *F*(*t*) > 0 for 0 ≤ *t* < *T*_{1} and

$$\begin{array}{}{\displaystyle \underset{t\to {T}_{1}}{lim}F(t)=0.}\end{array}$$

The remainder of this proof is same as the proof of Theorem 4.6. □

From Theorem 4.1 and Theorem 4.6 we can obtain a sharp condition for global existence of solutions for problem (1.1) with *E*(*φ*_{0}) < *d* as follows:

#### Theorem 4.11

*Let p satisfy (A)*, *φ*_{0} ∈ *H*. *Assume that E*(*φ*_{0}) < *d*. *Then when I*(*φ*_{0}) > 0, *solution of problem (1.1) exists globally; and when I*(*φ*_{0}) < 0, *solution of problem (1.1) blows up in finite time*.

From Corollary 4.3 and Corollary 4.8 we can obtain a serious of sharp conditions for global existence and finite time blow up of solution for problem (1.1) in the following theorem.

#### Theorem 4.12

*Let p satisfy (A)*, *φ*_{0} ∈ *H*, *a* < *δ* < $\begin{array}{}\frac{p+1}{2}\end{array}$. *Assume that E*(*φ*_{0}) ≤ *d*(*δ*). *Then when I*_{δ} (*φ*_{0}) > 0, *the solution of problem (1.1) exists globally; and when I*_{δ} (*φ*_{0}) < 0, *solution of problem (1.1) blows up in finite time*.

From Theorem 4.1 and Theorem 4.10 we can obtain the following sharp condition for global existence of solution for problem (1.1) with *E*(*φ*_{0}) = *d*.

#### Theorem 4.13

*Let p satisfy (A)*, *φ*_{0} ∈ *H*. *Assume that E*(*φ*_{0}) = *d*, *F*′(0) ≤ 0. *Then when I*(*φ*_{0}) > 0, *the solution of problem (1.1) exists globally; and when I*(*φ*_{0}) < 0, *solution of problem (1.1) blows up in finite time*.

Note that from Lemma 2.4 we have

$$\begin{array}{}{\displaystyle d=d(a)\ge \frac{n(p-1)-4}{2n(p-1)}{r}^{2}(a)={d}_{0}.}\end{array}$$

From this we get the following sharp condition that only depends on ∥∇*φ*_{0}∥.

#### Corollary 4.14

Let *p* satisfy (A), *φ*_{0} ∈ *H*. Assume that *E*(*φ*_{0}) < *d*_{0}. Then when ∥∇*φ*_{0}∥ < *r*(*a*), solution of problem (1.1) exists globally. When ∥∇*φ*_{0}∥ ≥ *r*(*a*), solution of problem (1.1) blows up in finite time.

#### Proof

If ∥∇*φ*_{0}∥ < *r*(*a*), then 0 < ∥∇*φ*_{0}∥ < *r*(*a*) or ∥∇*φ*_{0}∥ = 0, i.e. *I*(*φ*_{0}) > 0 or ∥∇*φ*_{0}∥ = 0. Then Theorem 4.1 gives the existence of unique global solution *φ*(*t*) ∈ *C*([0, ∞);*H͠*). If ∥∇*φ*_{0}∥ ≥ *r*(*a*), then by

$$\begin{array}{}\begin{array}{rl}& \frac{1}{2}\parallel |x|{\phi}_{0}{\parallel}^{2}+\frac{n(p-1)-4}{2n(p-1)}\parallel \mathrm{\nabla}{\phi}_{0}{\parallel}^{2}+\frac{1}{p+1}I({\phi}_{0})\\ =& E({\phi}_{0})<{d}_{0}=\frac{n(p-1)-4}{2n(p-1)}{r}^{2}(a),\end{array}\end{array}$$

we get *I*(*φ*_{0}) < 0. Hence from Theorem 4.6 it follows that solution of problem (1.1) blows up in finite time. □

#### Corollary 4.15

Let *p* satisfy (A), *φ*_{0} ∈ *H*. Assume that *E*(*φ*_{0}) > 0 and

$$\begin{array}{}{\displaystyle \frac{n(p-1)}{2(p+1)}\parallel {\phi}_{0}{\parallel}^{p+1-\frac{n(p-1)}{2}}{\left(\frac{2n(p-1)}{n(p-1)-4}E({\phi}_{0})\right)}^{\frac{n(p-1)-4}{4}}<{C}_{\ast}^{-1}.}\end{array}$$(4.5)

Then when

$$\begin{array}{}{\displaystyle \frac{n(p-1)}{2(p+1)}\parallel {\phi}_{0}{\parallel}^{p+1-\frac{n(p-1)}{2}}\parallel \mathrm{\nabla}{\phi}_{0}{\parallel}^{\frac{n(p-1)-4}{2}}<{C}_{\ast}^{-1},}\end{array}$$(4.6)

the solution of problem (1.1) exists globally. And when

$$\begin{array}{}{\displaystyle \frac{n(p-1)}{2(p+1)}\parallel {\phi}_{0}{\parallel}^{p+1-\frac{n(p-1)}{2}}\parallel \mathrm{\nabla}{\phi}_{0}{\parallel}^{\frac{n(p-1)-4}{2}}\ge {C}_{\ast}^{-1},}\end{array}$$(4.7)

solution of problem (1.1) blows up in finite time.

#### Proof

This corollary follows from Corollary 4.14 and the fact that (4.5), (4.6) and (4.7) are equivalent to *E*(*φ*_{0}) < *d*_{0}, ∥∇*φ*_{0}∥ < *r*(*a*) and ∥∇*φ*_{0}∥ ≥ *r*(*a*) respectively. □

From *E*(*φ*_{0}) < $\begin{array}{}\frac{1}{2}\end{array}$(∥|*x*|*φ*_{0}∥^{2} + ∥∇*φ*_{0}∥^{2}) for *φ*_{0} ≠ 0 and Corollary 4.15 we can obtain the following corollary.

#### Corollary 4.16

Let *p* satisfy (A), *φ*_{0} ∈ *H*. Assume that *E*(*φ*_{0}) > 0 and

$$\begin{array}{}{\displaystyle \frac{n(p-1)}{2(p+1)}\parallel {\phi}_{0}{\parallel}^{p+1-\frac{n(p-1)}{2}}{\left(\frac{n(p-1)}{n(p-1)-4}\left(\parallel |x|{\phi}_{0}{\parallel}^{2}+\parallel \mathrm{\nabla}{\phi}_{0}{\parallel}^{2}\right)\right)}^{\frac{n(p-1)-4}{4}}\le {C}_{\ast}^{-1},}\end{array}$$(4.8)

then solution of problem (1.1) exists globally.

#### Proof

One can see that (4.8) gives both (4.5) and (4.6), which proves this corollary. □

Finally we give another series of sharp conditions for globally existence and finite time blow up of solution for problem (1.1) as follows:

#### Corollary 4.17

Let *p* satisfy (A), *φ*_{0} ∈ *H*, *a* < *δ* < $\begin{array}{}\frac{p+1}{2}\end{array}$. Assume that *E*(*φ*_{0}) ≤ *a*(*δ*) *r*^{2}(*δ*), where *r*(*δ*) and *a*(*δ*) are defined in Lemma 2.2 and Lemma 2.4. Then when ∥∇*φ*_{0}∥ < *r*(*δ*) the solution of problem (1.1) exists globally; and when ∥∇*φ*_{0}∥ ≥ *r*(*δ*) the solution of problem
(1.1) blows up in finite time.

#### Proof

If ∥∇*φ*_{0}∥ < *r*(*δ*), then we have 0 < ∥∇*φ*_{0}∥ < *r*(*δ*) or ∥∇*φ*_{0}∥ = 0, which gives *I*_{δ}(*φ*_{0}) > 0 or ∥*φ*_{0}∥ = 0. Hence by Theorem 4.12 and Theorem 4.1 the solution of problem (1.1) exists globally.

If ∥∇*φ*_{0}∥ ≥ *r*(*δ*), then from ∥|*x*|*φ*_{0}∥ > 0 and

$$\begin{array}{}{\displaystyle \frac{1}{2}\parallel |x|{\phi}_{0}{\parallel}^{2}+a(\delta )\parallel \mathrm{\nabla}{\phi}_{0}{\parallel}^{2}+\frac{1}{p+1}{I}_{\delta}({\phi}_{0})=E({\phi}_{0})\le a(\delta ){r}^{2}(\delta ),}\end{array}$$

we get *I*_{δ}(*φ*_{0}) < 0. Again by Theorem 4.12 solution of problem (1.1) blows up in finite time. □

## Comments (0)

General note:By using the comment function on degruyter.com you agree to our Privacy Statement. A respectful treatment of one another is important to us. Therefore we would like to draw your attention to our House Rules.