The Blow-Up Rate for a Non-Scaling Invariant Semilinear Heat Equation

Abstract

We consider the semilinear heat equation

$$\begin{aligned} \partial _t u -\Delta u =f(u), \quad (x,t)\in {\mathbb {R}}^N\times [0,T),\qquad \qquad \qquad \qquad \qquad (1) \end{aligned}$$

with $f(u)=|u|^{p-1}u\log ^a (2+u^2)$, where $p>1$ is Sobolev subcritical and $a\in {\mathbb {R}}$. We first show an upper bound for any blow-up solution of (1). Then, using this estimate and the logarithmic property, we prove that the exact blow-up rate of any singular solution of (1) is given by the ODE solution associated with (1), namely $u' =|u|^{p-1}u\log ^a (2+u^2)$. In other words, all blow-up solutions in the Sobolev subcritical range are Type I solutions. To the best of our knowledge, this is the first determination of the blow-up rate for a semilinear heat equation where the main nonlinear term is not homogeneous.

Appendices

Appendix

We recall the interpolation result from Cazenave and Lions [1] and the interior regularity theorem in [6].

Lemma A.1

(Interpolation technique, Cazenave and Lions [1]) Let $t_0>0$. Assume that

$$\begin{aligned} v \in L^\alpha \left( [t_0,t_0+1]; L^\beta ({\mathbf {B}}_R) \right) , \; \partial _tv \in L^\gamma \left( [t_0,t_0+1]; L^\delta ({\mathbf {B}}_R) \right) \end{aligned}$$

for some $1< \alpha , \beta , \gamma , \delta < \infty$. Then

$$\begin{aligned} v \in {\mathscr {C}}\left( [t_0,t_0+1]; L^\lambda ({\mathbf {B}}_R) \right) \end{aligned}$$

for all $\lambda < \lambda _0 = \frac{(\alpha + \gamma ')\beta \delta }{\gamma '\beta + \alpha \delta }$ with $\gamma ' = \frac{\gamma }{\gamma - 1}$, and satisfies

$$\begin{aligned} \sup _{t \in [t_0,t_0+1]} \Vert v(t) \Vert _{L^\lambda ({\mathbf {B}}_R)} \leqq C \int _{t_0}^{t_0+1} \left( \Vert v(\tau )\Vert _{L^\beta ({\mathbf {B}}_R)}^\alpha + \Vert \partial _\tau v(\tau )\Vert _{L^\delta ({\mathbf {B}}_R)}^\gamma \right) {\mathrm {d}}\tau \end{aligned}$$

for $\lambda < \lambda _0$. The positive constant C depends only on $\alpha , \beta , \gamma , \delta , N$ and R.

The second one is an interior regularity result for a nonlinear parabolic equation:

Lemma A.2

(Interior regularity) Let $v(x,t)\in L^\infty \big ((0,+\infty ), L^2({\mathbf {B}}_R)\big ) \cap L^2\big ((0,+\infty ), H^1({\mathbf {B}}_R)\big )$ which satisfies

$$\begin{aligned} v_t - \Delta v + b. \nabla v = H,\quad (x,t) \in Q_R = {\mathbf {B}}_R \times (0, +\infty ), \end{aligned}$$

(A.1)

where $R > 0$, $|b(x,t)| \leqq \mu _1$ in $Q_R$ and $|H(x,t,v)| \leqq g(x,t)(|v| + 1)$ with

$$\begin{aligned} \int _{t}^{t +1} \left\| g(\tau )\right\| ^{\beta '}_{L^{\alpha '}({\mathbf {B}}_R)}d\tau \leqq \mu _2, \quad \forall t \in (0, +\infty ), \end{aligned}$$

(A.2)

and $\frac{1}{\beta '} + \frac{N}{2\alpha '} < 1$, and $\alpha ' \geqq 1$. If

$$\begin{aligned} \int _{t}^{t +1} \Vert v(\tau )\Vert ^2_{L^2({\mathbf {B}}_R)}d\tau \leqq \mu _3,\quad \forall t \in (0, +\infty ), \end{aligned}$$

(A.3)

and $\mu _1$, $\mu _2$ and $\mu _3$ are uniformly bounded in t, then there exists a positive constant C depending only on $\mu _1$, $\mu _2$, $\mu _3$, $\alpha '$, $\beta '$, N, R and $\tau \in (0,1)$ such that

$$\begin{aligned} |v(x,t)| \leqq C,\quad \forall (x,t) \in {\mathbf {B}}_{R/4} \times (\tau , +\infty ). \end{aligned}$$

Some Elementary Lemmas

Let f, F, $F_2$ be the functions defined in (1.2), (1.25) and (2.14). Clearly, we have

Lemma B.1

Let $q>1$,

$$\begin{aligned} \int _0^u|v|^{q-1}v\log ^{{a}}(2+v^2 ){\mathrm {d}}v\sim&\frac{| u|^{q+1}}{q+1}\log ^{{a}}(2+u^2 ),\quad \text { as } \;\; |u| \rightarrow \infty , \end{aligned}$$

(B.1)

$$\begin{aligned} F(u) \sim&\frac{uf(u)}{p+1} \quad \text { as } \;\; |u| \rightarrow \infty , \end{aligned}$$

(B.2)

$$\begin{aligned} F_2(u)\sim&\frac{Cuf(u)}{\log ^2(2+u^2)}\quad \text { as } \;\; |u| \rightarrow \infty . \end{aligned}$$

(B.3)

Proof

See Lemma A.1 in [15]. $\square$

Thanks to (B.1), (B.2) and (B.3), we will give the first and the second order terms in the expansion of the nonlinearity F(x) defined in (1.25), when |x| is large enough. More precisely, we now state the following estimates:

Lemma B.2

For all $s \geqq 1$, for all $z\in {\mathbb {R}}$,

$$\begin{aligned} C^{-1} \phi (s)z f(\phi (s)z))\leqq C+F\left( \phi (s)z)\leqq C (1+\phi (s)z f(\phi (s)z)\right) , \end{aligned}$$

(B.4)

$$\begin{aligned} F_1(\phi (s)z)\leqq C+C\frac{ \phi (s)z}{s}f(\phi (s)z),\quad \quad \end{aligned}$$

(B.5)

$$\begin{aligned} F_2(\phi (s)z)\leqq C+C \frac{ \phi (s)z}{s^2}f(\phi (s)z),\quad \quad \end{aligned}$$

(B.6)

$$\begin{aligned} e^{-\frac{ps}{p-1}}s^{\frac{a}{p-1}} |f(\phi (s)z)|\leqq C (\varepsilon ) + C |z|^{p+\varepsilon }, \quad \quad \forall \varepsilon \in (0,p-1), \end{aligned}$$

(B.7)

$$\begin{aligned} |z|^{p-\varepsilon }\leqq C e^{-\frac{ps}{p-1}}s^{\frac{a}{p-1}} | f(\phi (s)z)|+C (\varepsilon ), \quad \quad \forall \varepsilon \in (0,p-1), \end{aligned}$$

(B.8)

$$\begin{aligned} e^{-\frac{(p+1)s}{p-1}}s^{\frac{2a}{p-1}} F(\phi (s)z)\leqq C (\varepsilon ) + C |z|^{p+ \varepsilon +1}, \quad \quad \forall \varepsilon \in (0,p-1), \end{aligned}$$

(B.9)

$$\begin{aligned} |z|^{p-\varepsilon +1}\leqq e^{-\frac{(p+1)s}{p-1}}s^{\frac{2a}{p-1}} F(\phi (s)z) +C (\varepsilon ),\quad \quad \forall \varepsilon \in (0,p-1), \end{aligned}$$

(B.10)

where $\phi$, F, $F_1$ and $F_2$ are given in (1.20), (1.25), (2.13) and (2.14).

Proof

Note that (B.4) obviously follows from (B.2). In order to derive estimates (B.5) and (B.6), considering the first case $z^2\phi (s)\geqq 4$, then the case $z^2\phi (s)\leqq 4$, we would obtain (B.5) and (B.6) by using (B.1), (B.2) and(B.3). Similarly, by taking into account the inequality $\log ^a (2+u^2)\leqq C(\varepsilon )+ C(\varepsilon ) |u|^{\varepsilon }$ , we conclude easily (B.7), (B.8), (B.9) and (B.10). This ends the proof of Lemma B.2. $\square$

Proof of Proposition 2.6

Let us first derive the upper bound for ${\mathscr {E}}_\psi$.

Proof

(Proof of the upper bound for ${\mathscr {E}}_\psi$) Multiplying (1.18) by $\partial _{s} w \psi ^2\rho (y)$ and integrating over ${\mathbb {R}}^N,$ we obtain

$$\begin{aligned} \frac{d}{ds}{\mathscr {E}}_\psi (w(s),s)=&- \int _{{\mathbb {R}}^N}(\partial _{s}w)^2\psi ^2\rho (y){\mathrm {d}}y- 2\int _{{\mathbb {R}}^N}\partial _{s}w\nabla w. \nabla \psi \psi \rho (y){\mathrm {d}}y\nonumber \\&+\underbrace{\frac{a}{(p-1)s}\int _{{\mathbb {R}}^N}w\partial _{s}w\psi ^2\rho (y){\mathrm {d}}y}_{\Sigma ^1_{2}(s)} \nonumber \\&+\underbrace{\frac{p+1}{p-1} e^{-\frac{(p+1)s}{p-1}}s^{\frac{2a}{p-1}}\int _{{\mathbb {R}}^N}\big ( F(\phi w)-\frac{\phi wf(\phi w)}{p+1} \big )\psi ^2\rho (y){\mathrm {d}}y}_{\Sigma ^2_{2}(s)}\nonumber \\&\underbrace{-\frac{2a}{p-1}e^{-\frac{(p+1)s}{p-1}}s^{\frac{2a}{p-1}-1}\int _{{\mathbb {R}}^N}\big ( F(\phi w)-\frac{\phi wf(\phi w)}{2}\big )\psi ^2 \rho (y){\mathrm {d}}y}_{\Sigma ^3_{2}(s)}. \end{aligned}$$

(C.1)

Proceeding similarly as for the terms $\Sigma _{1}^{1}(s)$, $\Sigma _{1}^{2}(s)$ and $\Sigma _{1}^{3}(s)$ defined in (2.10), we get

$$\begin{aligned} \frac{d}{ds}{\mathscr {E}}_\psi (w(s),s)\leqq&- \frac{1}{2}\int _{{\mathbb {R}}^N}\psi ^2 (\partial _{s}w)^2\rho (y){\mathrm {d}}y- 2\int _{{\mathbb {R}}^N}\partial _{s}w\psi \nabla \psi . \nabla w\rho (y){\mathrm {d}}y\nonumber \\&+ \frac{C}{s^{a+1}}\int _{{\mathbb {R}}^N}\psi ^2|w|^{p+1}\log ^a(2+\phi ^2w^2)\rho (y){\mathrm {d}}y\nonumber \\&+\frac{C}{s^2}\int _{{\mathbb {R}}^N}\psi ^2 w^2\rho (y){\mathrm {d}}y+C e^{-s}. \end{aligned}$$

(C.2)

Using the fact that $2ab \leqq \frac{a^2}{4} + 4b^2$, we obtain

$$\begin{aligned} -2 \partial _{s}w\psi \nabla \psi . \nabla w \leqq \frac{1}{4}\psi ^2 (\partial _sw)^2 + 4 |\nabla \psi |^2 |\nabla w|^2, \end{aligned}$$

which implies, for all $s\geqq \max (-\log T,1)$,

$$\begin{aligned} \frac{d}{ds}{\mathscr {E}}_\psi (w(s),s)\leqq&\ \ C \int _{{\mathbb {R}}^N}|\nabla w|^2\rho (y){\mathrm {d}}y+ \frac{C}{s^{a+1}}\int _{{\mathbb {R}}^N}|w|^{p+1}\log ^a(2+\phi ^2w^2)\rho (y){\mathrm {d}}y\nonumber \\&+\frac{C}{s^2}\int _{{\mathbb {R}}^N}w^2\rho (y){\mathrm {d}}y+C e^{-s}, \end{aligned}$$

(C.3)

where $C = C(a, p, N, \Vert \psi \Vert _{L^\infty }, \Vert \nabla \psi \Vert _{L^\infty })$.

By combining (C.3), (2.40) and (2.41), we infer for all $s\geqq \max (-\log T,S_2)$

$$\begin{aligned} \int _s^{s+1} \frac{d}{ds}{\mathscr {E}}_\psi (w(\tau ),\tau ){\mathrm {d}}\tau \leqq Q_1 s^{b+1}. \end{aligned}$$

(C.4)

From the definition of ${\mathscr {E}}_\psi$ given in (2.48), using the fact that, $F(\phi w)\geqq 0,$ we have

$$\begin{aligned} {\mathscr {E}}_\psi (w(s),s)&\leqq \Vert \psi \Vert ^2_{L^\infty } \int _{\mathbb {R}^N}\left( \frac{1}{2}|\nabla w|^2 + \frac{1}{2(p-1)}|w|^2 \right) \rho (y){\mathrm {d}}y. \end{aligned}$$

By the definition of $H_{m}(w(s),s)$ given in (2.5), exploiting (2.39), we write for all $s\geqq \max (-\log T,S_2)$

$$\begin{aligned} {\mathscr {E}}_\psi (w(s),s)&\leqq C \left\{ H_{m_0}(w(s),s)+\frac{m_0}{2s} \int _{\mathbb {R}^N}w^2\rho (y){\mathrm {d}}y+e^{-\frac{(p+1)s}{p-1}}s^{\frac{2a}{p-1}} \int _{\mathbb {R}^N} F(\phi w)\rho (y){\mathrm {d}}y\right\} \nonumber \\&\leqq Q_2 s^{b+1} +Ce^{-\frac{(p+1)s}{p-1}}s^{\frac{2a}{p-1}} \int _{\mathbb {R}^N} F(\phi w)\rho (y){\mathrm {d}}y. \end{aligned}$$

(C.5)

Integrating the inequality (C.5) from s to $s+1$ and using (2.17), (B.4) and (2.41) we get, for all $s\geqq \max (-\log T,S_2)$

$$\begin{aligned} \int _s^{s+1} {\mathscr {E}}_\psi (w(\tau ),\tau ) {\mathrm {d}}\tau \leqq Q_3s^{b+1}. \end{aligned}$$

By using the mean value theorem, we derive the existence of $\sigma (s)\in [s,s+1]$ such that

$$\begin{aligned} {\mathscr {E}}_\psi (w(\sigma (s)),\sigma (s)) =\int _{s}^{s+1} {\mathscr {E}}_\psi (w(\tau ),\tau ){\mathrm {d}}\tau . \end{aligned}$$

(C.6)

Let us write the identity, for all $s\geqq \max (-\log T,S_2)$

$$\begin{aligned} {\mathscr {E}}_\psi (w(s),s)=&{\mathscr {E}}_\psi (w(\sigma (s)),\sigma (s)) + \int _{\sigma (s)}^{s}\frac{d}{d \tau } {\mathscr {E}}_\psi (w(\tau ),\tau ){\mathrm {d}}\tau . \end{aligned}$$

(C.7)

By combining (C.6), (C.7) and (C.4), we infer, for all $s\geqq \max (-\log T,S_2)$

$$\begin{aligned} {\mathscr {E}}_\psi (w(s),s)=&\leqq Q_4s^{b+1}. \end{aligned}$$

(C.8)

This concludes the proof of the upper bound for ${\mathscr {E}}_\psi$. $\square$

It remains to prove the lower bound.

[Proof of the lower bound for ${\mathscr {E}}_\psi$]

Consider now, for all $s \geqq \max (-\log T,1)$,

$$\begin{aligned} {\mathscr {I}}_\psi (w(s),s)=\frac{1}{s^{b+1}}\int _{{\mathbb {R}}^N}w^2 \psi ^2 \rho (y){\mathrm {d}}y. \end{aligned}$$

Multiplying equation (1.18) with $\psi ^2 w,$ integrating on $\mathbb {R}^N$ and using the same argument as in the proof of Lemma 2.3 yields

$$\begin{aligned} \frac{d}{ds}{\mathscr {I}}_\psi (w(s),s)\ \geqq&\ -\frac{p+3}{s^{b+1}}{\mathscr {E}}_{\psi }(w(s),s) + \frac{1}{2s^{b+1}}(1-\frac{C_4}{s}) \int _{{\mathbb {R}}^N}w^2 \psi ^2\rho (y){\mathrm {d}}y\nonumber \\&\ +\frac{p-1}{(p+1)s^{a+b+1}}(1-\frac{C_4}{s})\int _{{\mathbb {R}}^N}|w|^{p+1}\log ^a(2+\phi ^2 w^2) \psi ^2 \rho (y){\mathrm {d}}y\nonumber \\&- \frac{4}{s^{b+1}} \int _{\mathbb {R}^N} w \nabla w.\nabla \psi \psi \rho (y){\mathrm {d}}y. \end{aligned}$$

(C.9)

Therefore, there exists ${\tilde{S}}_2>S_2$ large enough, such that for all $s \geqq \max (-\log T, {\tilde{S}}_2)$, we have

$$\begin{aligned} \frac{d}{ds}{\mathscr {I}}_\psi (w(s),s)\ \geqq&\ \frac{p-1}{2(p+1)s^{a+b+1}}\int _{{\mathbb {R}}^N}|w|^{p+1}\log ^a(2+\phi ^2 w^2) \psi ^2\rho (y){\mathrm {d}}y\nonumber \\&\ -\frac{p+3}{s^{b+1}}{\mathscr {E}}_{\psi }(w(s),s)- \frac{4}{s^{b+1}} \int _{\mathbb {R}^N} w \nabla w.\nabla \psi \psi \rho (y){\mathrm {d}}y. \end{aligned}$$

(C.10)

Furthermore, after some integration by parts, we write

$$\begin{aligned}&-4 \int _{\mathbb {R}^N} w \nabla w.\nabla \psi \psi \rho (y){\mathrm {d}}y=2\int _{\mathbb {R}^N} w^2\, \text{ div }\, (\psi \rho (y)\nabla \psi ) {\mathrm {d}}y\nonumber \\&=2\int _{\mathbb {R}^N} w^2 |\nabla \psi |^2 \rho (y){\mathrm {d}}y+ 2 \int _{\mathbb {R}^N} w^2 \psi \Delta \psi \rho (y){\mathrm {d}}y- \int _{\mathbb {R}^N} w^2\psi y.\nabla \psi \rho (y){\mathrm {d}}y. \end{aligned}$$

(C.11)

Thanks to the estimates $\Vert \psi \Vert ^2_{L^\infty } +\Vert \Delta \psi \Vert ^2_{L^\infty } + \Vert \nabla \psi \Vert ^2_{L^\infty }+ \Vert y.\nabla \psi \Vert ^2_{L^\infty }\leqq C$, (C.11) and (2.42), we have for all $s \geqq \max (-\log T, {\tilde{S}}_2)$,

$$\begin{aligned} \Big |-4 \int _{\mathbb {R}^N} w \nabla w.\nabla \psi \psi \rho (y){\mathrm {d}}y\Big | \leqq C \int _{\mathbb {R}^N} w^2\rho (y){\mathrm {d}}y\leqq Q_5s^{b+1}. \end{aligned}$$

(C.12)

Using (C.10) and (C.12), we obtain for all $s \geqq \max (-\log T, {\tilde{S}}_2),$

$$\begin{aligned} \frac{d}{ds}{\mathscr {I}}_\psi (w(s),s)\ \geqq&\ \frac{p-1}{2(p+1)s^{a+b+1}}\int _{{\mathbb {R}}^N}|w|^{p+1}\log ^a(2+\phi ^2 w^2) \psi ^2\rho (y){\mathrm {d}}y\nonumber \\&\ -\frac{p+3}{s^{b+1}}{\mathscr {E}}_{\psi }(w(s),s)-Q_5. \end{aligned}$$

(C.13)

Let us define the following functional:

$$\begin{aligned} {\mathscr {G}}_{\psi }(w(s),s) = \frac{ p+3}{s^{b+1}}{\mathscr {E}}_{\psi }(w(s),s)+ Q_5, \end{aligned}$$

(C.14)

where ${\mathscr {G}}_{\psi }(w(s),s)$ is defined in (2.48).

We claim that the function of ${\mathscr {G}}_{\psi }(w(s),s)$ is bounded from below by some constant M, where M is a sufficiently large constant that will be determined later. Arguing by contradiction, we suppose that there exists a time $s^* \geqq \max (-\log T, {\tilde{S}}_2)$ such that ${\mathscr {G}}_{\psi }(w(s^*),s^*) \leqq - Q$, for some $Q>0$. Then, we write

$$\begin{aligned} {\mathscr {G}}_{\psi }(w(s),s) \leqq -Q + \int _{s^*}^s\frac{d}{d\tau } {\mathscr {G}}_{\psi }(w(\tau ),\tau ) {\mathrm {d}}\tau , \qquad \forall s \geqq s^*. \end{aligned}$$

(C.15)

If we now compute the time derivative of ${\mathscr {G}}_{\psi }(w(s),s)$ we get for all $s \geqq s^*,$

$$\begin{aligned} \frac{d}{ds} {\mathscr {G}}_{\psi }(w(s),s) =&\frac{ p+3}{s^{b+1}}\ \frac{d}{ds}{\mathscr {E}}_{\psi }(w(s),s)-\frac{(b+1) (p+3)}{s^{b+2}}{\mathscr {E}}_{\psi }(w(s),s). \end{aligned}$$

(C.16)

From the definition of ${\mathscr {E}}_\psi$ given in (2.48), using (B.4) and (2.17) we have for all $s \geqq s^*,$

$$\begin{aligned} -\frac{(b+1) (p+3)}{s^{b+2}}{\mathscr {E}}_{\psi }(w(s),s) \leqq \frac{C}{s^{a+b+2}}\int _{{\mathbb {R}}^N}|w|^{p+1}\log ^a(2+\phi ^2 w^2) \psi ^2 \rho (y){\mathrm {d}}y+Ce^{-s}. \end{aligned}$$

(C.17)

Thanks to (C.4) we conclude for all $s \geqq s^*,$

$$\begin{aligned} \int _{s^*}^{s} \frac{1}{\tau ^{b+1}}\frac{d}{ds}{\mathscr {E}}_\psi (w(\tau ),\tau ){\mathrm {d}}\tau \leqq Q_6 (s-s^*). \end{aligned}$$

(C.18)

Moreover, from (2.41), we obtain for all $s \geqq s^*,$

$$\begin{aligned} \int _{s^*}^{s}\frac{1}{ \tau ^{a+b+2}} \int _{{\mathbb {R}}^N}|w|^{p+1}\log ^a(2+\phi ^2 w^2) \psi ^2 \rho (y){\mathrm {d}}y{\mathrm {d}}\tau \leqq Q_7(s-s^*). \end{aligned}$$

(C.19)

Integrating the identity (C.16) over $[s^*,s]$ and combining (C.17), (C.18) and (C.19) we deduce that

$$\begin{aligned} \ \int _{s^*}^s\frac{d}{d\tau } {\mathscr {G}}_{\psi }(w(\tau ),\tau ) {\mathrm {d}}\tau \leqq Q_8(s - s^*), \qquad \forall s \geqq s^*. \end{aligned}$$

(C.20)

Combining (C.13), (C.15) and (C.20) we infer for all $s\geqq s^*,$

$$\begin{aligned} \frac{d}{ds}{\mathscr {I}}_\psi (w(s),s)\ \geqq&\ Q - Q_8(s - s^*) +\frac{C}{s^{a+b+1}}\int _{{\mathbb {R}}^N}|w|^{p+1}\log ^a(2+\phi ^2 w^2) \psi ^2\rho (y){\mathrm {d}}y. \end{aligned}$$

(C.21)

Thanks to (B.4) and (B.10), we have for all $s\geqq s^*,$ that

$$\begin{aligned} \frac{1}{s^a}\int _{{\mathbb {R}}^N} {| w|^{p+1}}\log ^{{a}}(2+\phi ^2 w^2 ) \psi ^2 \rho (y){\mathrm {d}}y\geqq C\int _{{\mathbb {R}}^N} {| w|^{\frac{p+3}{2}}} \psi ^2\rho (y){\mathrm {d}}y- C_5.\nonumber \\ \end{aligned}$$

(C.22)

Due to Jensen inequality, (C.21) and (C.22) we find for all $s\geqq s^*,$

$$\begin{aligned} \frac{d}{ds}{\mathscr {I}}_\psi (w(s),s)\ \geqq&\ {\tilde{Q}} - Q_{9}(s - s^*) +C_6\Big ({\mathscr {I}}_\psi (w(s),s)\Big )^{\frac{p+3}{4}}, \end{aligned}$$

(C.23)

where ${\tilde{Q}}=Q-C_5$.

It is interesting to denote that we easily prove that the solution of the differential inequality

$$\begin{aligned} \left\{ \begin{array}{l} h' (s)\geqq 1 + C_6h^{\frac{p+3}{4}}(s),\qquad s>s^*,\\ \\ h(s^*) \geqq 0 \end{array} \right. \end{aligned}$$

blows up in finite time before

$$\begin{aligned} s = s^* + \int _{0}^{+\infty }\frac{d\xi }{1 + C_6\xi ^\frac{p+3}{4}} = s^* + T^*. \end{aligned}$$

Now, we choose $Q= Q_{9}T^* +C_5+ 1$ to get ${\tilde{Q}} - Q_{9}(s-s^*) \geqq 1$ for all $s \in [s^*, s^* + T^*]$.

Therefore, ${\mathscr {I}}_\psi (w(s),s)$ blows up in some finite time before $s^* + T^*$. But this contradicts with the global existence of w. This implies (2.50), and we complete the proof of Proposition 2.6. $\square$