In this section, we prove Theorem 1.2 with four lemmas. Firstly, we define the set of initial data with a fixed constant

$\epsilon \in (0,1)$,

${\mathbb{V}}_{0}=\{({u}_{0},{v}_{0})|\mathrm{\Delta}{u}_{0}-\epsilon {u}_{0}^{\alpha}{e}^{p{v}_{0}}\ge 0,\mathrm{\Delta}{v}_{0}-\epsilon {u}_{0}^{q}{e}^{\beta {v}_{0}}\ge 0,x\in {B}_{R}\}.$

**Lemma 3.1** *For any*$({u}_{0},{v}_{0})\in {\mathbb{V}}_{0}$,

*there must be*$\begin{array}{r}{u}_{t}(x,t)\ge \epsilon \left({u}^{\alpha}{e}^{pv}\right)(x,t)\\ {v}_{t}(x,t)\ge \epsilon \left({u}^{q}{e}^{\beta v}\right)(x,t)\end{array}\phantom{\rule{1em}{0ex}}(x,t)\in {B}_{R}\times [0,T).$

(3.1)

*Proof* Set

$J={u}_{t}-\epsilon {u}^{\alpha}{e}^{pv},\phantom{\rule{2em}{0ex}}K={v}_{t}-\epsilon {u}^{q}{e}^{\beta v},\phantom{\rule{1em}{0ex}}(x,t)\in {B}_{R}\times [0,T).$

By computations, we can check that

By the comparison principle, $J(x,t),K(x,t)\ge 0$, for $(x,t)\in {B}_{R}\times [0,T)$. □

**Lemma 3.2**
*For any*
$t\in [0,T)$
*Proof* First, we prove (3.2). From (3.1), we get

${u}_{t}(0,t)\ge \epsilon {u}^{\alpha}(0,t){e}^{pv(0,t)},$

then

${u}_{t}(0,t)\ge \epsilon {u}^{\alpha}(0,t){e}^{p{v}_{0}(0,t)}.$

(3.4)

Integrating (3.4) from

*t* to

*T*,

${\int}_{t}^{T}\frac{{u}_{t}(0,t)\phantom{\rule{0.2em}{0ex}}dt}{{u}^{\alpha}(0,t)}\ge \epsilon {e}^{p{v}_{0}(0)}(T-t),$

thus

$-{u}^{-\alpha +1}(0,t)\ge \epsilon {e}^{p{v}_{0}(0)}(T-t)(-\alpha +1),$

then

$u(0,t)\le {[(\alpha -1)\epsilon {e}^{p{v}_{0}(0)}]}^{-\frac{1}{\alpha -1}}{(T-t)}^{-\frac{1}{\alpha -1}}.$

Similarly, we can also prove (3.3) from (3.1),

${v}_{t}(0,t)\ge \epsilon {u}^{q}(0,t){e}^{\beta v(0,t)}\ge \epsilon {u}_{0}^{q}(0){e}^{\beta v(0,t)}.$

Integrating the above inequality from

*t* to

*T*, then

□

The following lemma proves the sufficient and necessary condition on the existence of *u* blowing up alone.

**Lemma 3.3** *There exist suitable initial data such that* *u* *blows up while* *v* *remains bounded if and only if*$\alpha >q+1$.

*Proof* Firstly, we prove the sufficiency.

Let

$\mathrm{\Gamma}(x,y,t,\tau )=\frac{1}{{[4\pi (t-\tau )]}^{N/2}}\cdot exp\left\{\frac{-{|x-y|}^{2}}{4(t-\tau )}\right\}$

be the fundamental solution of the heat equation. Assume

$({\tilde{u}}_{0},{\tilde{v}}_{0})$ is a pair of initial data such that the solution of (1.1) blows up. Fix radially symmetric

${v}_{0}$ (

$\ge {\tilde{v}}_{0}$) in

${B}_{R}$ and take

${M}_{0}>{v}_{0}(0)$. Let the minimum of

${u}_{0}$ (

$\ge {\tilde{u}}_{0}$) be large such that

*T* is small and satisfies

${M}_{0}\ge {v}_{0}(0)+\frac{\alpha -1}{\alpha -q-1}{[(\alpha -1)\epsilon {e}^{p{v}_{0}(0)}]}^{-\frac{q}{\alpha -1}}{e}^{\beta {M}_{0}}\cdot {T}^{\frac{\alpha -q-1}{\alpha -1}}.$

Consider the auxiliary problem

$\{\begin{array}{c}{\overline{v}}_{t}=\mathrm{\Delta}\overline{v},\phantom{\rule{1em}{0ex}}(x,t)\in {B}_{R}\times (0,T),\hfill \\ \frac{\partial \overline{v}}{\partial \eta}={[(\alpha -1)\epsilon {e}^{p{v}_{0}(0)}]}^{-\frac{q}{\alpha -1}}{e}^{\beta {M}_{0}}{(T-t)}^{-\frac{q}{\alpha -1}},\phantom{\rule{1em}{0ex}}(x,t)\in \partial {B}_{R}\times (0,T),\hfill \\ \overline{v}(x,0)={v}_{0}(x),\phantom{\rule{1em}{0ex}}x\in {B}_{R}.\hfill \end{array}$

For

$\alpha >q+1$ and by Green’s identity [

5], we have

$\begin{array}{rcl}\overline{v}(x,t)& =& {\int}_{{B}_{R}}\mathrm{\Gamma}(x,y,t,0)\cdot {v}_{0}(y)\phantom{\rule{0.2em}{0ex}}dy+{\int}_{0}^{t}{\int}_{\partial {B}_{R}}\mathrm{\Gamma}(x,y,t,\tau )\cdot \frac{\partial \overline{v}}{\partial \eta}\cdot d{S}_{y}\cdot d\tau \\ =& {\int}_{{B}_{R}}\mathrm{\Gamma}(x,y,t,0)\cdot {v}_{0}(y)\phantom{\rule{0.2em}{0ex}}dy\\ +{\int}_{0}^{t}{\int}_{\partial {B}_{R}}\mathrm{\Gamma}(x,y,t,\tau )\cdot {[(\alpha -1)\epsilon {e}^{p{v}_{0}(0)}]}^{-\frac{q}{\alpha -1}}{e}^{\beta {M}_{0}}{(T-\tau )}^{-\frac{q}{\alpha -1}}\cdot d{S}_{y}\cdot d\tau \\ \le & {v}_{0}(0)+\frac{\alpha -1}{\alpha -q-1}{[(\alpha -1)\epsilon {e}^{p{v}_{0}(0)}]}^{-\frac{q}{\alpha -1}}{e}^{\beta {M}_{0}}{T}^{\frac{\alpha -q-1}{\alpha -1}}\\ \le & {M}_{0},\end{array}$

thus,

${M}_{0}\ge \overline{v}(x,t)$, for any

$(x,t)\in {B}_{R}\times (0,T)$. So

$\overline{v}$ satisfies

$\{\begin{array}{c}{\overline{v}}_{t}=\mathrm{\Delta}\overline{v},\phantom{\rule{1em}{0ex}}(x,t)\in {B}_{R}\times (0,T),\hfill \\ \frac{\partial \overline{v}}{\partial \eta}\ge {[(\alpha -1)\epsilon {e}^{p{v}_{0}(0)}]}^{-\frac{q}{\alpha -1}}{e}^{\beta \overline{v}}{(T-t)}^{-\frac{q}{\alpha -1}},\phantom{\rule{1em}{0ex}}(x,t)\in \partial {B}_{R}\times (0,T),\hfill \\ \overline{v}(x,0)={v}_{0}(x),\phantom{\rule{1em}{0ex}}x\in {B}_{R}.\hfill \end{array}$

Combining the radial symmetry and the monotonicity of the initial data with the estimate (3.2), we have

${u}^{q}(|x|,t)\le {u}^{q}(0,t)\le {[(\alpha -1)\epsilon {e}^{p{v}_{0}(0)}]}^{-\frac{q}{\alpha -1}}{(T-t)}^{-\frac{q}{\alpha -1}}\phantom{\rule{1em}{0ex}}(x,t)\in {B}_{R}\times (0,T).$

So,

*v* satisfies that

$\{\begin{array}{c}{v}_{t}=\mathrm{\Delta}v,\phantom{\rule{1em}{0ex}}(x,t)\in {B}_{R}\times (0,T),\hfill \\ \frac{\partial v}{\partial \eta}\le {[(\alpha -1)\epsilon {e}^{p{v}_{0}(0)}]}^{-\frac{q}{\alpha -1}}{e}^{\beta v}{(T-t)}^{-\frac{q}{\alpha -1}},\phantom{\rule{1em}{0ex}}(x,t)\in \partial {B}_{R}\times (0,T),\hfill \\ v(x,0)={v}_{0}(x),\phantom{\rule{1em}{0ex}}x\in {B}_{R}.\hfill \end{array}$

By the comparison principle, $v\le \overline{v}\le {M}_{0}$, so *v* remains bounded up to time *T*. Since $({u}_{0},{v}_{0})\ge ({\tilde{u}}_{0},{\tilde{v}}_{0})$, $(u,v)$ blows up, hence only *u* blows up at time *T*.

Secondly, we prove the necessity. Assume *u* blows up while *v* remains bounded, say $v\le C$.

By Green’s identity, we have

$u(0,t)\le u(0,z)+C{u}^{\alpha}(0,t)(T-z),$

for any

$z\in (0,T)$, take

*t* such that

$u(0,t)=2u(0,z)$, then

$u(0,z)\le C{u}^{\alpha}(0,z)(T-z),$

hence,

$u(0,t)\ge C{(T-t)}^{-\frac{1}{\alpha -1}}\phantom{\rule{1em}{0ex}}t\in (0,T).$

For some

${t}_{1}\in (0,T)$, we can find a suitable

${\epsilon}_{1}\in (0,1)$, such that

${u}_{t}(x,{t}_{1})-{\epsilon}_{1}\left({u}_{0}^{\alpha}{e}^{p{v}_{0}}\right)(x,{t}_{1})\ge 0.$

Similarly to Lemma 3.1, we can prove there must be

$\begin{array}{r}{u}_{t}(x,t)\ge {\epsilon}_{1}\left({u}^{\alpha}{e}^{pv}\right)(x,t)\\ {v}_{t}(x,t)\ge {\epsilon}_{1}\left({u}^{q}{e}^{\beta v}\right)(x,t)\end{array}\phantom{\rule{1em}{0ex}}(x,t)\in {B}_{R}\times [{t}_{1},T).$

(3.5)

Then

${v}_{t}(0,t)\ge {\epsilon}_{1}{e}^{\beta {v}_{0}(0)}{C}^{q}{(T-t)}^{-\frac{q}{\alpha -1}},\phantom{\rule{1em}{0ex}}t\in [{t}_{1},T).$

Integrating the above inequality from

${t}_{1}$ to

*t*, we have

$v(0,t)\ge {\epsilon}_{1}{e}^{\beta {v}_{0}(0)}{C}^{q}{\int}_{{t}_{1}}^{t}{(T-\tau )}^{-\frac{q}{\alpha -1}}\phantom{\rule{0.2em}{0ex}}d\tau +v(0,{t}_{1}).$

The boundedness of *v* requires that $\alpha >q+1$. □

The following lemma proves the sufficient and necessary condition on the existence of *v* blowing up alone.

**Lemma 3.4** *There exist suitable initial data such that* *v* *blows up while* *u* *remains bounded if and only if*$\beta >p$.

*Proof* Firstly, we prove the sufficiency. Assume

$({\tilde{u}}_{0},{\tilde{v}}_{0})$ is a pair of initial data such that the solution of (1.1) blows up. Fix radially symmetric

${u}_{0}$ (

$\ge {\tilde{u}}_{0}$) in

${B}_{R}$ and take

${M}_{1}>{u}_{0}(0)$. Let the minimum of

${v}_{0}$ (

$\ge {\tilde{v}}_{0}$) be large such that

*T* is small and satisfies

${M}_{1}\ge {u}_{0}(0)+\frac{\beta}{\beta -p}{[\beta \epsilon {u}_{0}^{q}(0)]}^{-\frac{p}{\beta}}{M}_{1}^{\alpha}{T}^{\frac{\beta -p}{\beta}}.$

Consider the auxiliary problem

$\{\begin{array}{c}{\overline{u}}_{t}=\mathrm{\Delta}\overline{u},\phantom{\rule{1em}{0ex}}(x,t)\in {B}_{R}\times (0,T),\hfill \\ \frac{\partial \overline{u}}{\partial \eta}={[\beta \epsilon {u}_{0}^{q}(0)]}^{-\frac{p}{\beta}}{M}_{1}^{\alpha}{(T-t)}^{-\frac{p}{\beta}},\phantom{\rule{1em}{0ex}}(x,t)\in \partial {B}_{R}\times (0,T),\hfill \\ \overline{u}(x,0)={u}_{0}(x),\phantom{\rule{1em}{0ex}}x\in {B}_{R}.\hfill \end{array}$

For

$\beta >p$, and by Green’s identity, we have

$\overline{u}(x,t)\le {u}_{0}(0)+\frac{\beta}{\beta -p}{[\beta \epsilon {u}_{0}^{q}(0)]}^{-\frac{p}{\beta}}{M}_{1}^{\alpha}{T}^{\frac{\beta -p}{\beta}}\le {M}_{1}.$

So

$\overline{u}$ satisfies

$\frac{\partial \overline{u}}{\partial \eta}\ge {[\beta \epsilon {u}_{0}^{q}(0)]}^{-\frac{p}{\beta}}{\overline{u}}^{\alpha}{(T-t)}^{-\frac{p}{\beta}},\phantom{\rule{1em}{0ex}}(x,t)\in \partial {B}_{R}\times (0,T).$

From (3.3), we have

$\frac{\partial u}{\partial \eta}\le {[\beta \epsilon {u}_{0}^{q}(0)]}^{-\frac{p}{\beta}}{u}^{\alpha}{(T-t)}^{-\frac{p}{\beta}},\phantom{\rule{1em}{0ex}}(x,t)\in \partial {B}_{R}\times (0,T).$

By the comparison principle, $u\le \overline{u}\le {M}_{1}$. Since $({u}_{0},{v}_{0})\ge ({\tilde{u}}_{0},{\tilde{v}}_{0})$, $(u,v)$ blows up, hence only *v* blows up at time *T*.

Secondly, we prove the necessity. Assume *v* blows up while *u* remains bounded, say $u\le C$.

By Green’s identity, we have

$v(0,t)\le v(0,z)+C{e}^{\beta v(0,t)}(T-z).$

For any

$z\in (0,T)$, take

*t* such that

$v(0,t)=v(0,z)+1$, then

$C{e}^{\beta v(0,z)}(T-z)\ge 1,$

thus

$v(0,t)\ge ln{[C(T-t)]}^{-\frac{1}{\beta}},\phantom{\rule{1em}{0ex}}t\in (0,T).$

(3.6)

From (3.5) and (3.6), we have

${u}_{t}(0,t)\ge {\epsilon}_{1}{u}_{0}^{\alpha}(0){C}^{-\frac{p}{\beta}}{(T-t)}^{-\frac{p}{\beta}},\phantom{\rule{1em}{0ex}}t\in ({t}_{1},T).$

(3.7)

Integrating (3.7) from

${t}_{1}$ to

*t*, we obtain that

$u(0,t)\ge u(0,{t}_{1})+{\epsilon}_{1}{u}_{0}^{\alpha}(0){C}^{-\frac{p}{\beta}}{\int}_{{t}_{1}}^{t}{(T-\tau )}^{-\frac{p}{\beta}}\phantom{\rule{0.2em}{0ex}}d\tau .$

The boundedness of *u* requires that $\beta >p$. □