Let us establish the Fujita type theorems.
Definition 3.1
We call u the blow-up solution to equation (1) if there exists some \(0< T_{*}<+\infty\) such that
$$\begin{aligned} \lim_{t\to T_{*}^{-}} \bigl\Vert u(\cdot,t)\bigr\Vert _{L^{\infty}(\mathbb {R}^{n})} =\lim_{t\to T_{*}^{-}}\sup_{x\in\mathbb {R}^{n}}u(x,t)=+ \infty. \end{aligned}$$
Theorem 3.1
Assume that
\(1< p< p_{c}=q+(q+1+\mu_{2})/(n+\mu_{1})\)
and
\(0\le u_{0}\in C_{0}(\mathbb {R}^{n})\)
is nontrivial. Then the problem (1), (2) admits a blow-up solution.
Proof
Due to Proposition 2.1, we may assume \(u_{0}(0)>0\). By the comparison principle, we only need to prove the conclusion for radial and nonincreasing \(u_{0}(x)\), i.e.,
$$\begin{aligned} u_{0}(x)=h_{0}\bigl(\vert x\vert \bigr),\quad x\in \mathbb {R}^{n}, \end{aligned}$$
where \(h_{0}\in C^{1}_{0}([0,+\infty))\) satisfies \(h'_{0}(0)=0\) and \(h'_{0}(r)\le0\) for \(r>0\). With such initial data, the solution u is also radial, namely
$$\begin{aligned} u(x,t)=h\bigl(\vert x\vert ,t\bigr), \quad x\in\mathbb {R}^{n}, t\ge0. \end{aligned}$$
(6)
If \(\mu_{1}=\mu_{2}\), it is easy to know that u is also nonincreasing by a standard regularization argument and the maximum principle. However, this method is invalid if \(\mu_{1}<\mu_{2}\). In the following discussion, we will first of all consider a nonincreasing u, namely \(h(r,t)\) is nonincreasing with respect to \(r\in[0,+\infty)\) for any \(t\ge0\), and we treat the general case finally.
Let
$$\begin{aligned} \psi(x)=\left \{ \textstyle\begin{array}{@{}l@{\quad}l} 1,&0\le \vert x\vert \le1,\\ f(\vert x\vert -1), &1< \vert x\vert < 2,\\ 0,&\vert x\vert \ge2, \end{array}\displaystyle \right . \end{aligned}$$
where f is the principal eigenfunction of −Δ in the unit ball \(B_{1}\) of \(\mathbb {R}^{n}\) with homogeneous initial-boundary condition, normalized by \(\Vert f\Vert _{L^{\infty}(B_{1})}=1\). For \(l>1\), define
$$\begin{aligned} \psi_{l}(x)=\psi(x/l), \quad x\in\mathbb {R}^{n}. \end{aligned}$$
Then
$$\begin{aligned} \vert \nabla\psi_{l}\vert \le\frac{M_{0}}{l}, \qquad \vert \Delta \psi_{l}\vert \le\frac{M_{0}}{l^{2}}, \qquad\frac{\vert \Delta\psi_{l}\vert }{\psi_{l}}\le \frac {M_{0}}{l^{2}}, \quad x\in B_{2l}\setminus B_{l}, \end{aligned}$$
with \(M_{0}>0\) independent of l. Set
$$\begin{aligned} \eta_{l}(t)= \int_{\mathbb {R}^{n}}\bigl(\vert x\vert +1\bigr)^{\mu_{1}} u \psi_{l} \,dx,\quad t\ge0. \end{aligned}$$
(7)
Definition 2.1 gives
$$\begin{aligned} \frac{d\eta_{l}}{dt} =- \int_{B_{2l}}\vert \nabla u\vert ^{q-1}\nabla u\cdot \nabla \psi_{l} \,dx+ \int_{\mathbb {R}^{n}}\bigl(\vert x\vert +1\bigr)^{\mu_{2}}u^{p} \psi_{l} \,dx. \end{aligned}$$
For radial and nonincreasing \(u(x,t)\), one has
$$\begin{aligned} \int_{B_{2l}}\vert \nabla u\vert ^{q-1}\nabla u\cdot \nabla \psi_{l} \,dx &= \int_{B_{2l}}\vert \nabla u\vert ^{q}\vert \nabla \psi_{l}\vert \,dx \\ &\le \biggl( \int_{B_{2l}}\vert \nabla u\vert \cdot \vert \nabla \psi_{l}\vert \,dx \biggr)^{q} \biggl( \int_{B_{2l}}\vert \nabla\psi_{l}\vert \,dx \biggr)^{1-q} \\ &= \biggl( \int_{B_{2l}}\nabla u\cdot\nabla\psi_{l}\,dx \biggr)^{q} \biggl( \int_{B_{2l}}\vert \nabla\psi_{l}\vert \,dx \biggr)^{1-q} \\ &\le M_{0}l^{(n-1)(1-q)} \biggl( \int_{B_{2l}}\nabla u\cdot\nabla\psi_{l}\,dx \biggr)^{q} \end{aligned}$$
and
$$\begin{aligned} 0\le \int_{B_{2l}}\nabla u\cdot\nabla\psi_{l}\,dx = \int_{\partial B_{2l}}u\nabla\psi_{l}\cdot\boldsymbol {\nu}\,d\sigma- \int_{B_{2l}} u\Delta\psi_{l} \,dx \le- \int_{B_{2l}} u\Delta\psi_{l} \,dx, \end{aligned}$$
where ν is the unit outer normal to \(\partial B_{2l}\). Hence
$$\begin{aligned} \frac{d\eta_{l}}{dt}\ge-M_{0}l^{(1-q)(n-1)} \biggl\vert \int_{B_{2l}} u\Delta\psi_{l} \,dx \biggr\vert ^{q} + \int_{\mathbb {R}^{n}}\bigl(\vert x\vert +1\bigr)^{\mu_{2}}u^{p} \psi_{l} \,dx. \end{aligned}$$
(8)
The Hölder inequality yields
$$\begin{aligned} \biggl\vert \int_{B_{2l}} u\Delta\psi_{l} \,dx \biggr\vert ^{q} \le{}& \biggl( \int_{B_{2l}\setminus B_{l}} u\vert \Delta\psi_{l}\vert \,dx \biggr)^{q} \\ \le{}& \biggl( \int_{B_{2l}\setminus B_{l}}\bigl(\vert x\vert +1\bigr)^{-\mu_{2}/(p-1)}\vert \Delta\psi_{l}\vert ^{p/(p-1)} \psi_{l}^{-1/(p-1)}\,dx \biggr)^{q(p-1)/p} \\ &{} \cdot \biggl( \int_{B_{2l}\setminus B_{l}}\bigl(\vert x\vert +1\bigr)^{\mu_{2}}u^{p} \psi _{l} \,dx \biggr)^{q/p} \\ \le{}& M_{1}l^{q(n-2)-q(n+\mu_{2})/p} \biggl( \int_{\mathbb {R}^{n}}\bigl(\vert x\vert +1\bigr)^{\mu_{2}}u^{p} \psi_{l} \,dx \biggr)^{q/p} \end{aligned}$$
with \(M_{1}>0\) independent of l, which, together with (8), implies
$$\begin{aligned} \frac{d\eta_{l}}{dt} \ge{}& \biggl( \int_{\mathbb {R}^{n}}\bigl(\vert x\vert +1\bigr)^{\mu_{2}}u^{p} \psi_{l} \,dx \biggr)^{q/p} \\ &{} \cdot \biggl\{ -M_{0}M_{1}l^{n-q-1-q(n+\mu_{2})/p}+ \biggl( \int_{\mathbb {R}^{n}}\bigl(\vert x\vert +1\bigr)^{\mu_{2}}u^{p} \psi_{l} \,dx \biggr)^{(p-q)/p} \biggr\} . \end{aligned}$$
(9)
By the Hölder inequality,
$$\begin{aligned} \int_{\mathbb {R}^{n}}\bigl(\vert x\vert +1\bigr)^{\mu_{1}}u \psi_{l}\,dx \le{}& \biggl( \int_{B_{2l}}\bigl(\vert x\vert +1\bigr)^{(p\mu_{1}-\mu_{2})/(p-1)} \psi_{l}\,dx \biggr)^{(p-1)/p} \\ &{} \cdot \biggl( \int_{\mathbb {R}^{n}}\bigl(\vert x\vert +1\bigr)^{\mu_{2}}u^{p} \psi_{l} \,dx \biggr)^{1/p} \\ \le{}& M_{2}l^{(n+\mu_{1})-(n+\mu_{2})/p} \biggl( \int_{\mathbb {R}^{n}}\bigl(\vert x\vert +1\bigr)^{\mu_{2}}u^{p} \psi_{l} \,dx \biggr)^{1/p}, \end{aligned}$$
and hence
$$\begin{aligned} \int_{\mathbb {R}^{n}}\bigl(\vert x\vert +1\bigr)^{\mu_{2}}u^{p} \psi_{l} \,dx \ge M_{2}^{-p}l^{-p(n+\mu_{1})+(n+\mu_{2})} \eta_{l}^{p} \end{aligned}$$
(10)
with \(M_{2}>0\) independent of l. Equations (9) and (10) show that
$$\begin{aligned} \frac{d\eta_{l}}{dt} \ge{}& \bigl(M_{2}^{-p}l^{-p(n+\mu_{1})+(n+\mu_{2})} \bigr)^{q/p}\eta_{l}^{q} \\ &{} \cdot \bigl\{ -M_{0}M_{1}l^{n-q-1-q(n+\mu_{2})/p} + M_{2}^{q-p}l^{[-p(n+\mu_{1})+(n+\mu_{2})](p-q)/p}\eta_{l}^{p-q} \bigr\} . \end{aligned}$$
(11)
We mention that the above discussion holds provided that \(p>1\).
If \(p< p_{c}=q+(q+1+\mu_{2})/(n+\mu_{1})\), then
$$\begin{aligned} n-q-1-(n+\mu_{2})q/p< \bigl[-p(n+\mu_{1})+(n+ \mu_{2}) \bigr](p-q)/p. \end{aligned}$$
Notice that \(\eta_{l}\) is nondecreasing with respect to \(l\in(1,+\infty)\) and \(\sup\{\eta_{l}(0):l\in(1,+\infty)\}>0\), and from (11) one shows that, for \(l>1\) large enough, there exists a constant \(\delta>0\) such that
$$\begin{aligned} \frac{d\eta_{l}}{dt} \ge{}& \bigl(M_{2}^{-p}l^{-p(n+\mu_{1})+(n+\mu_{2})} \bigr)^{q/p}\eta_{l}^{q} \\ &{} \cdot \biggl(\frac{1}{2}M_{2}^{q-p}l^{[-p(n+\mu_{1})+(n+\mu _{2})](p-q)/p} \eta_{l}^{p-q} \biggr) \\ \ge{}&\delta\eta_{l}^{p}. \end{aligned}$$
So there exists some \(0< T_{*}<+\infty\) such that
$$\begin{aligned} \lim_{t\to T_{*}^{-}}\eta_{l}(t)=+\infty. \end{aligned}$$
Due to \(\operatorname{supp} \psi_{l}=B_{2l}\), we obtain
$$\begin{aligned} \lim_{t\to T_{*}^{-}}\bigl\Vert u(\cdot,t)\bigr\Vert _{L^{\infty}(\mathbb {R}^{n})}=+\infty. \end{aligned}$$
Next, for the general case without the assumption that \(u(x,t)\) is nonincreasing, define
$$\begin{aligned} \underline{u}(x,t)=\min_{0\le r\le \vert x\vert } h(r,t),\quad x \in\mathbb {R}^{n}, t\ge0. \end{aligned}$$
(12)
Then \(\underline{u}(x,t)\) is nonincreasing,
$$\begin{aligned} 0\le\underline{u}(x,t)\le u(x,t), \quad x\in\mathbb {R}^{n}, t\ge0, \end{aligned}$$
(13)
and
$$\begin{aligned} \frac{d}{dt} \int_{\mathbb {R}^{n}}\bigl(\vert x\vert +1\bigr)^{\mu_{1}} \underline{u} \psi_{l} \,dx \ge- \int_{B_{2l}}\vert \nabla\underline{u}\vert ^{q-1} \nabla\underline {u}\cdot\nabla\psi_{l} \,dx + \int_{\mathbb {R}^{n}}\bigl(\vert x\vert +1\bigr)^{\mu_{2}} \underline{u}^{p} \psi_{l} \,dx. \end{aligned}$$
(14)
From the above argument, we get
$$\begin{aligned} \lim_{t\to\tilde{T}_{*}^{-}}\bigl\Vert \underline{u}(\cdot,t)\bigr\Vert _{L^{\infty}(\mathbb {R}^{n})}=+\infty \end{aligned}$$
for some \(0<\tilde{T}_{*}<+\infty\), and (13) ensures that u is a blow-up solution. □
Let us turn to the case \(p>p_{c}\). Suppose that
$$\begin{aligned} U(x,t)={(t+1)^{-\alpha}}V \bigl((t+1)^{-\beta}\bigl( \vert x\vert +1\bigr) \bigr), \end{aligned}$$
(15)
where
$$\begin{aligned} \alpha=\frac{q+1+\mu_{2}}{\mu_{1}(p-q)+\mu _{2}(q-1)+(q+1)(p-1)},\qquad \beta=\frac{p-q}{q+1+\mu_{2}}\alpha, \end{aligned}$$
(16)
is a self-similar solution to (1). It is easy to show that \(V(r)\) solves
$$\begin{aligned} \bigl(\bigl\vert V'\bigr\vert ^{q-1}V' \bigr)'+\frac{n-1}{r}\bigl\vert V'\bigr\vert ^{q-1}V'+ \beta r^{\mu_{1}+1}V'+\alpha r^{\mu_{1}}V+r^{\mu_{2}}V^{p}=0, \quad r>0. \end{aligned}$$
(17)
Lemma 3.1
Assume that
\(p>p_{c}=q+(q+1+\mu_{2})/(n+\mu_{1})\). Then for
\(\varepsilon>0\)
small enough, the function
$$\begin{aligned} V(r)=\varepsilon \bigl(1+\rho(\varepsilon)r^{\lambda}\bigr)^{-\gamma}, \quad r>0, \end{aligned}$$
where
$$\begin{aligned} \lambda=1+\frac{\mu_{1}+1}{q},\qquad \gamma=\frac{q}{1-q}, \qquad\rho (\varepsilon)= \frac{1}{\gamma\lambda}\beta^{1/q}\varepsilon^{(1-q)/q}, \end{aligned}$$
is a supersolution to equation (17), i.e.
$$\begin{aligned} \bigl(\bigl\vert V'\bigr\vert ^{q-1}V' \bigr)'+ \frac{n-1}{r}\bigl\vert V'\bigr\vert ^{q-1}V'+\beta r^{\mu_{1}+1}V'+\alpha r^{\mu_{1}}V+r^{\mu_{2}}V^{p} \le0, \quad r>0. \end{aligned}$$
Proof
It is not hard to show that it suffices to verify
$$\begin{aligned} &q(\gamma+1)\rho(\varepsilon)\lambda \bigl[\varepsilon\gamma \lambda\rho( \varepsilon) \bigr]^{q} \bigl[1+\rho(\varepsilon)r^{\lambda}\bigr]^{-q(\gamma+1)-1}r^{(q+1)(\lambda-1)} \\ &\quad{} -q(\lambda-1) \bigl[\varepsilon\gamma\lambda\rho (\varepsilon) \bigr]^{q} \bigl[1+\rho(\varepsilon)r^{\lambda}\bigr]^{-q(\gamma+1)}r^{q(\lambda-1)-1} \\ &\quad{} -(n-1) \bigl[\varepsilon\gamma\lambda\rho(\varepsilon) \bigr]^{q} \bigl[1+\rho(\varepsilon)r^{\lambda}\bigr]^{-q(\gamma+1)}r^{q(\lambda-1)-1} \\ &\quad{} -\beta\varepsilon\gamma\lambda\rho(\varepsilon) \bigl[1+\rho( \varepsilon)r^{\lambda}\bigr]^{-(\gamma+1)}r^{\mu_{1}+\lambda} +\alpha \varepsilon \bigl[1+\rho(\varepsilon)r^{\lambda}\bigr]^{-\gamma }r^{\mu_{1}} \\ &\quad{} +\varepsilon^{p} \bigl[1+\rho(\varepsilon)r^{\lambda}\bigr]^{-p\gamma }r^{\mu_{2}} \le0,\quad r>0, \end{aligned}$$
namely
$$\begin{aligned} & \bigl\{ \bigl[\varepsilon\gamma\lambda\rho (\varepsilon) \bigr]^{q}-\beta\varepsilon \bigr\} \gamma\lambda\rho(\varepsilon) \bigl[1+\rho(\varepsilon)r^{\lambda}\bigr]^{-(\gamma+1)}r^{\mu _{1}+\lambda} \\ &\quad{} - \bigl\{ \bigl[q(\lambda-1)+n-1 \bigr]\varepsilon ^{-1} \bigl[ \varepsilon\gamma\lambda\rho(\varepsilon) \bigr]^{q} -\alpha- \varepsilon^{p-1} \bigl[1+\rho(\varepsilon)r^{\lambda}\bigr]^{-(p-1)\gamma}r^{\mu_{2}-\mu_{1}} \bigr\} \\ &\quad{} \cdot\varepsilon \bigl[1+\rho(\varepsilon)r^{\lambda}\bigr]^{-\gamma }r^{\mu_{1}} \le0, \quad r>0. \end{aligned}$$
(18)
From the definition of \([\varepsilon\gamma\lambda\rho(\varepsilon )]^{q}=\beta\varepsilon\), we have
$$\begin{aligned} & \bigl[q(\lambda-1)+n-1 \bigr]\varepsilon^{-1} \bigl[\varepsilon \gamma \lambda\rho(\varepsilon) \bigr]^{q} -\alpha-\varepsilon^{p-1} \bigl[1+\rho(\varepsilon)r^{\lambda}\bigr]^{-(p-1)\gamma}r^{\mu _{2}-\mu_{1}} \\ &\quad= (n+\mu_{1})\beta-\alpha-\varepsilon^{p-1} \bigl[1+ \rho(\varepsilon)r^{\lambda}\bigr]^{-(p-1)\gamma}r^{\mu_{2}-\mu_{1}} \\ &\quad\ge(n+\mu_{1})\beta-\alpha-\varepsilon^{p-1} \bigl[ \rho(\varepsilon) \bigr]^{-(p-1)\gamma}r^{\mu_{2}-\mu_{1}-\lambda (p-1)\gamma}. \end{aligned}$$
Due to \(p>p_{c}=q+(q+1+\mu_{2})/(n+\mu_{1})\) and \((n-1)/q-(n+1)<\mu_{1}\le\mu_{2}\),
$$\begin{aligned} (n+\mu_{1})\beta>\alpha, \quad\lambda(p-1)\gamma\ge\mu_{2}- \mu_{1}\ge0. \end{aligned}$$
Hence,
$$\begin{aligned} (n+\mu_{1})\beta-\alpha-\varepsilon^{p-1} \bigl[\rho( \varepsilon) \bigr]^{-(p-1)\gamma}r^{\mu_{2}-\mu_{1}-\lambda (p-1)\gamma}>0,\quad r>0 \end{aligned}$$
holds for sufficiently small \(\varepsilon>0\), and (18) is obtained. □
Theorem 3.2
Assume that
\(p>p_{c}=q+(q+1+\mu_{2})/(n+\mu_{1})\). Then the solution to the problem (1), (2) exists globally with small
\(u_{0}\), or blows up with large
\(u_{0}\).
Proof
Let \(V(r)\) be a supersolution to equation (17) in Lemma 3.1. From
$$\begin{aligned} V'(r)< 0, \quad r>0, \end{aligned}$$
one can show that \(U(x,t)\) given in (15) is a supersolution to equation (1) with α and β given in (16). Therefore, the comparison principle implies the problem (1), (2) has a nontrivial global solution with small \(u_{0}\).
Let us turn to the case of large \(u_{0}\). Denote by the radial u a solution to the problem (1), (2). Temporarily suppose u is nonincreasing. Then (11) holds with \(\eta_{l}\) defined by (7). If \(u_{0}\) is so large that
$$\begin{aligned} M_{0}M_{1}l_{0}^{n-q-1-(n+\mu_{2})q/p} < \frac{1}{2}M_{2}^{q-p}l_{0}^{[-p(n+\mu_{1})+(n+\mu_{2})](p-q)/p} \eta_{l_{0}}^{p-q}(0) \end{aligned}$$
holds for some \(l_{0}>1\), then from (11), we get
$$\begin{aligned} \eta_{l_{0}}(t)\ge\eta_{l_{0}}(0),\quad t>0, \end{aligned}$$
and
$$\begin{aligned} \frac{d\eta_{l_{0}}}{dt} \ge{}& \bigl(M_{2}^{-p}l_{0}^{-p(n+\mu _{1})+(n+\mu_{2})} \bigr)^{q/p}\eta_{l_{0}}^{q} \\ &{} \cdot \biggl(\frac{1}{2} M_{2}^{q-p}l_{0}^{[-p(n+\mu_{1})+(n+\mu_{2})](p-q)/p} \eta_{l_{0}}^{p-q} \biggr) \\ ={}&\delta_{0}\eta_{l_{0}}^{p} \end{aligned}$$
for some \(\delta_{0}>0\). Therefore, u is a blow-up solution.
For the general case without the assumption that \(u(x,t)\) is nonincreasing, considering a new function just as in the proof of Theorem 3.1, one can also see that u is a blow-up solution. □