1 Introduction

In the paper, we investigate the blow-up theorems of Fujita type for the following Cauchy problem:

$$\begin{aligned} &\bigl( \vert x \vert +1\bigr)^{\lambda_{1}}\frac{\partial u}{\partial t}= \Delta u +b\bigl( \vert x \vert \bigr)x\cdot\nabla u+\bigl( \vert x \vert +1\bigr)^{\lambda_{2}}u^{p}, \quad x\in\mathbb {R}^{n}, t>0, \end{aligned}$$
(1.1)
$$\begin{aligned} &u(x,0)=u_{0}(x),\quad x\in\mathbb {R}^{n}, \end{aligned}$$
(1.2)

where \(p>1\), \(-2<\lambda_{1}\leq\lambda_{2}\), \(0\leq u_{0}\in C_{0}(\mathbb {R}^{n})\) and \(b\in C^{1}([0,+\infty))\) satisfies

$$\begin{aligned} \lim_{s\to+\infty}s^{2}b(s)=\kappa\quad(- \infty\le\kappa\le +\infty), \end{aligned}$$
(1.3)

and in the case that \(-n-\lambda_{1}<\kappa\leq+\infty\), b also satisfies

$$\begin{aligned} \kappa_{0}=\inf\bigl\{ s(s+1)b(s): s>0\bigr\} >-n- \lambda_{1}. \end{aligned}$$
(1.4)

The critical exponents for nonlinear diffusion equations have attached extensive attention since 1966, when Fujita [1] proved that, for the Cauchy problem of Eq. (1.1) with \(b\equiv0\) and \(\lambda_{1}=\lambda_{2}=0\), the nontrivial nonnegative solution blows up in a finite time if \(1< p< p_{c}=1+2/n\), whereas it exists globally for small initial data and blows up in a finite time for large ones if \(p>p_{c}=1+2/n\). This result reveals that the exponent p of the nonlinear reaction plays a remarkable role in affecting the properties of solutions. We call \(p_{c}\) with the above properties the critical Fujita exponent and the similar result a blow-up theorem of Fujita type. There have been many kinds of extensions of Fujita’s results since then, such as different types of parabolic equations and systems with or without degeneracies or singularities, various geometries of domains, different nonlinear reactions or nonhomogeneous boundary sources, etc. One can see the survey papers [2, 3] and the references therein, and more recent work [4–18]. For the Cauchy problem of

$$\frac{\partial u}{\partial t}=\Delta u+\mathbf{b}_{0}\cdot\nabla u+u^{p}, \quad x\in\mathbb {R}^{n}, t>0, $$

with \(\mathbf{b}_{0}\) being a nonzero constant vector, Aguirre and Escobedo [19] showed that

$$p_{c}=1+2/(n+1) $$

is its critical Fujita exponent. Wang and Zheng [11] considered the Cauchy problem of Eq. (1.1) with \(b\equiv0\), and showed that the critical Fujita exponent is

$$p_{c}=1+(2+\lambda_{2})/(n+\lambda_{1}). $$

Recently, the Cauchy problem of Eq. (1.1) with \(\lambda _{1}=\lambda_{2}=0\) was studied in [18] and it was shown that

$$p_{c}= \textstyle\begin{cases} 1,&\kappa=+\infty, \\ 1+2/(n+\kappa),&-n< \kappa< +\infty, \\ +\infty,&-\infty\le\kappa\le-n. \end{cases} $$

As to Neumann exterior problems, Levine and Zhang [20] investigated the critical Fujita exponent of the homogeneous Neumann exterior problem of (1.1) with \(b\equiv0\) and \(\lambda_{1}=\lambda_{2}=0\), and proved that \(p_{c}\) is still \(1+2/n\). In [21], Zheng and Wang concerned the homogeneous Neumann exterior problem of (1.1) with

$$b(s)=\frac{\kappa}{s^{2}},\quad s>0\ (-\infty< \kappa< +\infty), $$

and formulated the critical Fujita exponent as

$$p_{c}= \textstyle\begin{cases} 1+(2+\lambda_{2})/(n+\kappa+\lambda_{1}),&\kappa>-n-\lambda_{1}, \\ +\infty,&\kappa\le-n-\lambda_{1}. \end{cases} $$

Moreover, the general case of b is considered in [8] if \(0\le \lambda_{1}\leq\lambda_{2}\le p\lambda_{1}+(p-1)n\) and \(\kappa\ge0\).

In this paper, we investigate the blow-up theorems of Fujita type for the Cauchy problem (1.1), (1.2). It is proved that the critical Fujita exponent to the problem is

$$\begin{aligned} p_{c}= \textstyle\begin{cases} 1,&\kappa=+\infty, \\ 1+(2+\lambda_{2})/(n+\kappa+\lambda_{1}),&-n-\lambda_{1}< \kappa< +\infty, \\ +\infty,&-\infty\le\kappa\leq-n-\lambda_{1}. \end{cases}\displaystyle \end{aligned}$$
(1.5)

That is to say, if \(1< p< p_{c}\), there does not exist any nontrivial nonnegative global solution, whereas if \(p>p_{c}\), there exist both nontrivial nonnegative global and blow-up solutions. The technique used in this paper is mainly inspired by [11, 18, 21, 22]. To prove the blow-up of solutions, we use precise energy integral estimates instead of constructing subsolutions. For the global existence of nontrivial solutions, we construct a nontrivial global supersolution. It should be noted that we have to seek a complicated supersolution and do some precise calculations in order to overcome the difficulty from the non-self-similar construction of (1.1). Furthermore, the properties of such models which will be proved in the paper provide theoretical foundation for the numerical simulation which involved difference schemes.

The paper is organized as follows. Some preliminaries and main results are introduced in Sect. 2, such as the local well-posedness of the problem (1.1), (1.2) and some auxiliary lemmas to be used later, as well as the blow-up theorems of Fujita type. The main results are proved in Sect. 3.

2 Preliminaries and main results

The solutions to the problem (1.1), (1.2) are defined as follows.

Definition 2.1

A nonnegative function u is called a solution to the problem (1.1), (1.2) in \((0,T)\) with \(0< T\le+\infty\), if

$$\begin{aligned} &u\in C\bigl([0,T\bigr),L_{\mathrm{loc}}^{1}\bigl(\mathbb {R}^{n}\bigr))\cap L^{\infty}_{\mathrm{loc}}\bigl(0,T;L^{\infty}\bigl(\mathbb {R}^{n}\bigr)\bigr), \\ & \int_{0}^{T} \int_{\mathbb {R}^{n}}\bigl( \vert x \vert +1\bigr)^{\lambda _{1}}u(x,t) \frac{\partial\varphi}{\partial t}(x,t)\,\mathrm{d}x\,\mathrm{d}t\\ &\quad {} + \int_{0}^{T} \int_{\mathbb {R}^{n}}u(x,t) \bigl(\Delta\varphi(x,t) -\operatorname{div} \bigl(b\bigl( \vert x \vert \bigr)\varphi (x,t)x\bigr)\bigr)\,\mathrm{d}x \,\mathrm{d}t \\ &\quad{}+ \int_{0}^{T} \int_{\mathbb {R}^{n}}\bigl( \vert x \vert +1\bigr)^{\lambda_{2}}u^{p}(x,t) \varphi(x,t)\,\mathrm{d}x\,\mathrm{d}t=0,\quad \varphi\in C_{0}^{\infty}\bigl(\mathbb {R}^{n}\times(0,T)\bigr), \end{aligned}$$

and

$$\lim_{t\to0} \int_{\mathbb{R}^{n}}u(x,t)\psi(x)\,\mathrm{d}x= \int _{\mathbb{R}^{n}}u_{0}(x)\psi(x)\,\mathrm{d}x, \quad\psi\in C_{0}^{\infty}\bigl(\mathbb{R}^{n}\bigr). $$

Definition 2.2

A solution u to the problem (1.1), (1.2) is called a blow-up solution if there exists some \(T_{*}\in(0,+\infty)\), which is called blow-up time, such that

$$\bigl\Vert u(\cdot,t) \bigr\Vert _{L^{\infty}(\mathbb {R}^{n})}\to +\infty\quad\mbox{as } t \to T_{*}^{-}. $$

Otherwise, u is called a global solution.

For \(0\le u_{0}\in C_{0}(\mathbb {R}^{n})\) and \(b\in C^{1}([0,+\infty))\) and \(p>1\), one can establish the existence, uniqueness and the comparison principle for solutions to the problem (1.1), (1.2) locally in time by use of the classical theory on parabolic equations (see, e.g., [23]).

The blow-up theorems of Fujita type for the problem (1.1), (1.2) are stated as follows.

Theorem 2.1

Assume that \(b\in C^{1}([0,+\infty))\) satisfies (1.3) and (1.4) with \(-\infty\le\kappa<+\infty\). If \(1< p< p_{c}\) with \(p_{c}\) given by (1.5), then, for any nontrivial \(0\le u_{0}\in C_{0}(\mathbb {R}^{n})\), the solution to the problem (1.1), (1.2) must blow up in a finite time.

Theorem 2.2

Assume that \(b\in C^{1}([0,+\infty))\) satisfies (1.3) and (1.4) with \(-n<\kappa\le+\infty\). If \(p>p_{c}\) with \(p_{c}\) given by (1.5), then there exist both nontrivial nonnegative global and blow-up solutions to the problem (1.1), (1.2).

3 Proofs of main results

To prove Theorem 2.1, the following auxiliary lemma is needed. We omit the proof and a similar one may be found in [18, 21].

Lemma 3.1

Assume that \(b\in C^{1}([0,+\infty))\) satisfies (1.3) and (1.4) with \(-\infty\le\kappa<+\infty\), u is a solution to the problem (1.1), (1.2), and

$$\eta_{R}(r)= \textstyle\begin{cases} h(r),&0\leq r\leq R, \\ \frac{1}{2}h(r) (1+\cos\frac{(r-R)\pi}{(\delta-1)R} ),&R< r< \delta R, \\ 0,&r\geq\delta R, \end{cases} $$

with

$$h(r)=\exp\biggl\{ \int_{0}^{r} sb(s)\,\mathrm{d}s\biggr\} ,\quad r\ge0. $$

Then there exist three numbers \(R_{0}>0\), \(\delta>1\) and \(M_{0}>0\) depending only on n and b, such that, for any \(R>R_{0}\),

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t} \int_{\mathbb {R}^{n}}\bigl( \vert x \vert +1\bigr)^{\lambda_{1}}u(x,t) \eta_{R}\bigl( \vert x \vert \bigr)\,\mathrm{d}x & \ge-M_{0}R^{-2} \int_{B_{\delta R}\setminus B_{R}} u(x,t)\eta_{R}\bigl( \vert x \vert \bigr)\,\mathrm{d}x \\ &\quad{}+ \int_{\mathbb {R}^{n}}\bigl( \vert x \vert +1\bigr)^{\lambda _{2}}u^{p}(x,t) \eta_{R}\bigl( \vert x \vert \bigr)\,\mathrm{d}x,\quad t>0, \end{aligned}$$
(3.1)

in the distribution sense, where \(B_{r}\) denotes the open ball in \(\mathbb {R}^{n}\) with radius r and centered at the origin.

Remark 3.1

For the case \(\kappa=+\infty\), one can prove that (3.1) holds for each fixed \(R>0\), but \(\delta>1\) and \(M_{0}>0\) depend also on R.

Proof of Theorem 2.1

Let \(\eta_{R}\), h, \(R_{0}\), δ and \(M_{0}\) be introduced in Lemma 3.1. It follows from \(-\infty\le\kappa<+\infty\) and \(1< p< p_{c}\) that

$$n+\kappa+\lambda_{1}-\frac{\lambda_{2}}{p-1}< \frac{2}{p-1}. $$

Fix \(\tilde{\kappa}>\kappa\) to satisfy

$$\begin{aligned} -\frac{\lambda_{1}}{p-1}< n+\tilde{\kappa}+\lambda_{1}- \frac{\lambda _{2}}{p-1}< \frac{2}{p-1}, \end{aligned}$$
(3.2)

which, together with (1.3), shows that there exists \(R_{1}>1\) such that

$$s^{2}b(s)< \tilde{\kappa},\quad s>R_{1}. $$

For any \(R>R_{1}\), one can get

$$\int_{0}^{r}sb(s)\,\mathrm{d}s\leq \textstyle\begin{cases} K_{0},&0\leq r\leq R_{1}, \\ K_{0}+\ln r^{\tilde{\kappa}},& r>R_{1}, \end{cases} $$

and

$$h(r)=\exp\biggl\{ \int_{0}^{r}sb(s)\,\mathrm{d}s\biggr\} \leq \textstyle\begin{cases} \mathrm{e}^{K_{0}},&0\leq r\leq R_{1}, \\ \mathrm{e}^{K_{0}}r^{\tilde{\kappa}},& r>R_{1}, \end{cases}\displaystyle \leq K(r+1)^{\tilde{\kappa}},\quad r\geq0, $$

where

$$K=\max\biggl\{ \sup_{0\leq r\leq R_{1}}\frac{\mathrm{e}^{K_{0}}}{(r+1)^{\tilde{\kappa}}}, \sup _{r>R_{1}}\frac{\mathrm{e}^{K_{0}}r^{\tilde{\kappa}}}{(r+1)^{\tilde{\kappa}}}\biggr\} ,\quad K_{0}= \vert \tilde{\kappa} \vert \ln R_{1}+\sup_{0\leq r\leq R_{1}} \int_{0}^{r}sb(s)\,\mathrm{d}s. $$

Therefore,

$$\begin{aligned} 0\le\eta_{R}\bigl( \vert x \vert \bigr)&\le h \bigl( \vert x \vert \bigr)\chi_{[0,\delta R]}\bigl( \vert x \vert \bigr) =K\bigl( \vert x \vert +1\bigr)^{\tilde{\kappa}}\chi_{[0,\delta R]}\bigl( \vert x \vert \bigr),\quad x\in\mathbb{R}, \end{aligned}$$
(3.3)

where \(\chi_{[0,\delta R]}\) is the characteristic function of the interval \([0,\delta R]\), while \(K>0\) depends only on n, b, \(R_{1}\), δ and κ̃. Let u be the solution to the problem (1.1), (1.2), and denote

$$w_{R}(t)= \int_{\mathbb {R}^{n}}\bigl( \vert x \vert +1\bigr)^{\lambda _{1}}u(x,t) \eta_{R}(x)\,\mathrm{d}x,\quad t\ge0. $$

For any \(R>\max\{R_{0}, R_{1}\}\), Lemma 3.1 implies

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}w_{R}(t) &\geq-M_{0}R^{-2} \int_{B_{\delta R}\setminus B_{R}} u(x,t)\eta_{R}\bigl( \vert x \vert \bigr)\,\mathrm{d}x \\ &\quad {} + \int_{\mathbb {R}^{n}}\bigl( \vert x \vert +1\bigr)^{\lambda _{2}}u^{p}(x,t) \eta_{R}\bigl( \vert x \vert \bigr)\,\mathrm{d}x,\quad t>0. \end{aligned}$$
(3.4)

The Hölder inequality and (3.3) yield

$$\begin{aligned} & \int_{B_{\delta R}\setminus B_{R}}u(x,t)\eta_{R}\bigl( \vert x \vert \bigr)\,\mathrm{d}x \\ &\quad \leq \biggl( \int_{B_{\delta R}\setminus B_{R}} \bigl( \vert x \vert +1\bigr)^{-\lambda_{2}/(p-1)} \eta_{R}\bigl( \vert x \vert \bigr)\,\mathrm{d}x\biggr)^{(p-1)/p} \\ &\qquad {}\times\biggl( \int_{B_{\delta R}\setminus B_{R}} \bigl( \vert x \vert +1\bigr)^{\lambda_{2}}u^{p}(x,t) \eta_{R}\bigl( \vert x \vert \bigr)\,\mathrm{d}x\biggr)^{1/p} \\ &\quad \leq \biggl(K \int_{B_{\delta R}\setminus B_{R}} \bigl( \vert x \vert +1\bigr)^{\tilde{\kappa}-\lambda _{2}/(p-1)} \,\mathrm{d}x\biggr)^{(p-1)/p} \biggl( \int_{B_{\delta R}\setminus B_{R}}\bigl( \vert x \vert +1\bigr)^{\lambda_{2}} u^{p}(x,t)\eta_{R}\bigl( \vert x \vert \bigr)\,\mathrm{d}x \biggr)^{1/p} \\ &\quad \leq \biggl(K\omega_{n} \int_{R}^{\delta R}(r+1)^{n+\tilde{\kappa}-1-\lambda_{2}/(p-1)} \,\mathrm{d}r \biggr)^{(p-1)/p} \biggl( \int_{B_{\delta R}\setminus B_{R}}\bigl( \vert x \vert +1\bigr)^{\lambda_{2}} u^{p}(x,t)\eta_{R}\bigl( \vert x \vert \bigr)\,\mathrm{d}x \biggr)^{1/p} \\ &\quad \le M_{1}^{(p-1)/p}R^{n+\tilde{\kappa}-(n+\tilde{\kappa}+\lambda_{2})/p} \biggl( \int_{\mathbb{R}^{n}}\bigl( \vert x \vert +1\bigr)^{\lambda _{2}}u^{p}(x,t) \eta_{R}\bigl( \vert x \vert \bigr)\,\mathrm{d}x\biggr)^{1/p}, \quad t>0, \end{aligned}$$
(3.5)

where \(\omega_{n}\) is the volume of the unit ball in \(\mathbb {R}^{n}\), while \(M_{1}>0\) depends only on n, b, \(R_{1}\), δ and κ̃. Substituting (3.5) into (3.4) gives

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}w_{R}(t) &\geq-M_{0}M_{1}^{(p-1)/p}R^{n+\tilde{\kappa}-2-(n+\tilde{\kappa}+\lambda_{2})/p} \biggl( \int_{\mathbb {R}^{n}}\bigl( \vert x \vert +1\bigr)^{\lambda _{2}}u^{p}(x,t) \eta_{R}\bigl( \vert x \vert \bigr)\,\mathrm{d}x\biggr)^{1/p} \\ &\quad{}+ \int_{\mathbb {R}^{n}}\bigl( \vert x \vert +1\bigr)^{\lambda _{2}}u^{p}(x,t) \eta_{R}\bigl( \vert x \vert \bigr)\,\mathrm{d}x,\quad t>0. \end{aligned}$$
(3.6)

It follows from (3.2), (3.3) and the Hölder inequality that

$$\begin{aligned} w_{R}(t)&\leq\biggl( \int_{\mathbb {R}^{n}}\bigl( \vert x \vert +1\bigr)^{(p\lambda_{1}-\lambda_{2})/(p-1)} \eta_{R}\bigl( \vert x \vert \bigr)\,\mathrm{d}x\biggr)^{(p-1)/p} \\ &\quad {}\times \biggl( \int_{\mathbb {R}^{n}}\bigl( \vert x \vert +1\bigr)^{\lambda _{2}}u^{p}(x,t) \eta_{R}\bigl( \vert x \vert \bigr)\,\mathrm{d}x\biggr)^{1/p} \\ &\leq\biggl(K \int_{B_{\delta R}} \bigl( \vert x \vert +1\bigr)^{\tilde{\kappa}+\lambda_{1}+(\lambda _{1}-\lambda_{2})/(p-1)} \,\mathrm{d}x\biggr)^{(p-1)/p} \\ &\quad {}\times \biggl( \int_{\mathbb {R}^{n}}\bigl( \vert x \vert +1\bigr)^{\lambda _{2}}u^{p}(x,t) \eta_{R}\bigl( \vert x \vert \bigr)\,\mathrm{d}x\biggr)^{1/p} \\ &\leq\biggl(K\omega_{n} \int_{0}^{\delta R} (r+1)^{n+\tilde{\kappa}+\lambda_{1}-1+(\lambda_{1}-\lambda _{2})/(p-1)}\,\mathrm{d}r \biggr)^{(p-1)/p} \\ &\quad {}\times \biggl( \int_{\mathbb {R}^{n}}\bigl( \vert x \vert +1\bigr)^{\lambda _{2}}u^{p}(x,t) \eta_{R}\bigl( \vert x \vert \bigr)\,\mathrm{d}x\biggr)^{1/p} \\ &\leq M_{2}^{(p-1)/p}R^{n+\tilde{\kappa}+\lambda_{1}-(n+\tilde{\kappa}+\lambda_{2})/p} \biggl( \int_{\mathbb {R}^{n}}\bigl( \vert x \vert +1\bigr)^{\lambda _{2}}u^{p}(x,t) \eta_{R}\bigl( \vert x \vert \bigr)\,\mathrm{d}x\biggr)^{1/p}, \quad t>0, \end{aligned}$$

with \(M_{2}>0\) depending only on n, b, \(R_{1}\), δ and κ̃, and

$$\begin{aligned} \int_{\mathbb {R}^{n}}\bigl( \vert x \vert +1\bigr)^{\lambda _{2}}u^{p}(x,t) \eta_{R}\bigl( \vert x \vert \bigr)\,\mathrm{d}x \ge M_{2}^{-(p-1)}R^{-(p-1)(n+\tilde{\kappa}+\lambda_{1})-\lambda _{1}+\lambda_{2}}w_{R}^{p}(t), \quad t>0. \end{aligned}$$
(3.7)

Substituting (3.7) into (3.6), one gets, for any \(R>\max\{R_{0},R_{1}\}\),

$$\begin{aligned} &\frac{\mathrm{d}}{\mathrm{d}t}w_{R}(t) \\ &\quad \geq-M_{0} \biggl(\frac{M_{1}}{M_{2}} \biggr)^{(p-1)/p}R^{-2-\lambda_{1}}w_{R}(t) +M_{2}^{-(p-1)}R^{-(p-1)(n+\tilde{\kappa}+\lambda_{1})-\lambda_{1}+\lambda _{2}}w_{R}^{p}(t) \\ &\quad =w_{R}(t) \biggl( -M_{0} \biggl(\frac{M_{1}}{M_{2}} \biggr)^{(p-1)/p}R^{-2-\lambda_{1}} +M_{2}^{-(p-1)}R^{-(p-1)(n+\tilde{\kappa}+\lambda_{1})-\lambda_{1}+\lambda _{2}}w_{R}^{p-1}(t) \biggr), \\ & \quad t>0. \end{aligned}$$
(3.8)

Note that (3.2) implies

$$-2-\lambda_{1}< -(p-1) (n+\tilde{\kappa}+\lambda_{1})- \lambda_{1}+\lambda_{2}, $$

while \(w_{R}(0)\) is nondecreasing with respect to \(R\in(0,+\infty)\) and

$$\sup\bigl\{ w_{R}(0):R>0\bigr\} >0. $$

Therefore, there exists \(R_{2}>0\) such that, for any \(R>R_{2}\),

$$\begin{aligned} M_{0} \biggl(\frac{M_{1}}{M_{2}} \biggr)^{(p-1)/p}R^{-2-\lambda_{1}} \leq\frac{1}{2}M_{2}^{-(p-1)}R^{-(p-1)(n+\tilde{\kappa}+\lambda _{1})-\lambda_{1}+\lambda_{2}}w_{R}^{p-1}(0). \end{aligned}$$
(3.9)

Fix \(R>\max\{R_{0},R_{1},R_{2}\}\). (3.8) and (3.9) yield

$$\frac{\mathrm{d}}{\mathrm{d}t}w_{R}(t) \geq\frac{1}{2}M_{2}^{-(p-1)}R^{-(p-1)(n+\tilde{\kappa}+\lambda_{1}) -\lambda_{1}+\lambda_{2}}w_{R}^{p}(t), \quad t>0. $$

Since \(p>1\), there exists \(T_{*}>0\) such that

$$w_{R}(t)= \int_{\mathbb {R}^{n}}\bigl( \vert x \vert +1\bigr)^{\lambda _{1}}u(x,t) \eta_{R}\bigl( \vert x \vert \bigr)\,\mathrm{d}x \to+\infty \quad \mbox{as } t\to T_{*}^{-}. $$

It follows from \(\operatorname{supp}\eta_{R}( \vert x \vert )=\overline{B}_{\delta R}\) that

$$\bigl\Vert u(\cdot,t) \bigr\Vert _{L^{\infty}(\mathbb {R}^{n})}\to +\infty\quad\mbox{as } t \to T_{*}^{-}, $$

i.e., u blows up in a finite time. □

Now, let us prove Theorem 2.2. Firstly, we study self-similar supersolutions of (1.1) of the form

$$\begin{aligned} u(x,t)=(t+\tau)^{-\alpha}U\bigl((t+\tau)^{-\beta} \bigl( \vert x \vert +1\bigr)\bigr),\quad x\in\mathbb {R}^{n}, t\ge0, \end{aligned}$$
(3.10)

with

$$\alpha=\frac{2+\lambda_{2}}{(2+\lambda_{1})(p-1)},\qquad \beta=\frac{1}{2+\lambda_{1}}, $$

and \(\tau>0\) will be determined. If \(U\in C^{1,1}([0,+\infty))\) with \(U'\leq0\) in \((0,+\infty)\) satisfies

$$\begin{aligned} &U''(r)+\frac{n-1}{r}U'(r)+(t+ \tau)^{\beta}\bigl((t+\tau)^{\beta}r-1\bigr)b\bigl((t+ \tau)^{\beta}r-1\bigr)U'(r) +\beta r^{1+\lambda_{1}}U'(r) \\ &\quad {}+\alpha r^{\lambda_{1}}U(r)+r^{\lambda_{2}}U^{p}(r)\leq0,\quad r>(t+ \tau )^{-\beta}, \end{aligned}$$

then u given by (3.10) is a supersolution to (1.1).

Lemma 3.2

Assume that \(b\in C^{1}([0,+\infty))\) satisfies (1.3) and (1.4) with \(-n-\lambda_{1}<\kappa\leq+\infty\), \(p>p_{c}\),

$$\begin{aligned} U(r)=\varepsilon\mathrm{e}^{-A(r)},\quad&r\ge0, \end{aligned}$$
(3.11)

with \(A\in C^{1,1}([0,+\infty))\) satisfies \(A(0)=0\) and

$$A'(r)= \textstyle\begin{cases} A_{1}r^{1+\lambda_{1}}, &0\leq r\leq l^{2}, \\ (A_{2}+(A_{1}-A_{2})\frac{l^{2(n+\kappa_{2}+\lambda_{1})}}{r^{n+\kappa _{2}+\lambda_{1}}})r^{1+\lambda_{1}}, & l^{2}< r< l, \\ (A_{2}+(A_{1}-A_{2})l^{n+\kappa_{2}+\lambda_{1}})r^{1+\lambda_{1}}, & r\geq l, \end{cases} $$

where \(0< l<1\) will be determined,

$$A_{1}=\frac{2(2+\lambda_{2})}{(2+\lambda_{1})(n+\kappa_{1}+\lambda _{1})(p+p_{c}-2)},\qquad A_{2}=\frac{2(2+\lambda_{2})}{(2+\lambda_{1})(n+\kappa_{2}+\lambda_{1})(p+p_{c}-2)}, $$

with \(\kappa_{1}\), \(\kappa_{2}\) satisfying

$$\kappa_{1}< \kappa_{0},\quad -n-\lambda_{1}< \kappa_{1}< \frac{2(2+\lambda_{2})}{p+p_{c}-2}-n-\lambda_{1} < \kappa_{2}< \kappa. $$

Then there exist \(\varepsilon>0\), \(0< l<1\) and \(\tau>0\) such that u given by (3.10) and (3.11) is a supersolution to (1.1).

Proof

The choice of \(\kappa_{1}\), \(\kappa_{2}\) leads to \(A_{2}<\beta<A_{1}\). Fix

$$\begin{aligned} 0< l< &\min\biggl\{ 1, \biggl(\frac{\kappa_{0}-\kappa_{1}}{A_{1}} \biggr)^{1/(4+2\lambda_{1})}, \biggl( \frac{(2+\lambda_{2})(p-p_{c})}{2A_{1}^{2}(2+\lambda _{1})(p+p_{c}-2)(p-1)} \biggr)^{1/(2+\lambda_{1})}, \\ &{} \biggl(\frac{\beta-A_{2}}{A_{1}-A_{2}} \biggr)^{1/(n+\kappa_{2}+\lambda_{1})}\biggr\} . \end{aligned}$$
(3.12)

Additionally, (1.3) allows us to choose \(\tau>0\) sufficiently large such that

$$\begin{aligned} (t+\tau)^{\beta}\bigl((t+\tau)^{\beta}r-1\bigr)b \bigl((t+\tau)^{\beta}r-1\bigr)\geq\frac {\kappa_{2}}{r},\quad r>l^{2}, t>0. \end{aligned}$$
(3.13)

For \(0< r< l^{2}\) and \(t>0\), we have from (1.3) and (3.12)

$$\begin{aligned} &U''(r)+\frac{n-1}{r}U'(r)+(t+ \tau)^{\beta}\bigl((t+\tau)^{\beta}r-1\bigr)b\bigl((t+ \tau)^{\beta}r-1\bigr)U'(r) \\ &\qquad{}+\beta r^{1+\lambda_{1}}U'(r)+\alpha r^{\lambda_{1}}U(r) \\ &\quad = \bigl(-(n+\lambda_{1})A_{1} -A_{1}(t+ \tau)^{\beta}r\bigl((t+\tau)^{\beta}r-1\bigr)b\bigl((t+ \tau)^{\beta}r-1\bigr) +\alpha \\ &\qquad{}+A_{1}(A_{1}-\beta)r^{2+\lambda_{1}} \bigr)r^{\lambda_{1}}U(r) \\ &\quad \leq \bigl(-(n+\kappa_{0}+\lambda_{1})A_{1}+ \alpha +A_{1}^{2}l^{4+2\lambda_{1}} \bigr)r^{\lambda_{1}}U(r) \\ &\quad \leq \biggl(-(\kappa_{0}-\kappa_{1})A_{1} -\frac{(2+\lambda_{2})(p-p_{c})}{(2+\lambda _{1})(p+p_{c}-2)(p-1)}+A_{1}^{2}l^{4+2\lambda_{1}} \biggr)r^{\lambda_{1}}U(r) \\ &\quad \leq -\frac{(2+\lambda_{2})(p-p_{c})}{2(2+\lambda_{1})(p+p_{c}-2)(p-1)} r^{\lambda_{1}}U(r). \end{aligned}$$
(3.14)

Then, for \(l^{2}< r< l\) and \(t>0\), (3.12) and (3.13) ensure that

$$\begin{aligned} & \begin{aligned} &U''(r)+\frac{n-1}{r}U'(r)+(t+ \tau)^{\beta}\bigl((t+\tau)^{\beta}r-1\bigr)b\bigl((t+ \tau)^{\beta}r-1\bigr)U'(r) \\ &\qquad{}+\beta r^{1+\lambda_{1}}U'(r)+\alpha r^{\lambda_{1}}U(r) \\ &\quad \leq U''(r)+\frac{n+\kappa_{2}-1}{r}U'(r)+ \beta r^{1+\lambda_{1}}U'(r) +\alpha r^{\lambda_{1}}U(r) \end{aligned} \\ &\quad = \biggl(\bigl(A'(r)\bigr)^{2}-A''(r)- \frac{n+\kappa_{2}-1}{r}A'(r) -\beta r^{1+\lambda_{1}}+\alpha r^{\lambda_{1}} \biggr)U(r) \\ &\quad = \biggl(\biggl(A_{2}+(A_{1}-A_{2}) \frac{l^{2(n+\kappa_{2}+\lambda_{1})}}{r^{n+\kappa _{2}+\lambda_{1}}}\biggr) \biggl(A_{2}+(A_{1}-A_{2}) \frac{l^{2(n+\kappa_{2}+\lambda_{1})}}{r^{n+\kappa _{2}+\lambda_{1}}}-\beta\biggr) r^{2+\lambda_{1}} \\ &{}\qquad-\frac{(2+\lambda_{2})(p-p_{c})}{(2+\lambda_{1})(p+p_{c}-2)(p-1)}\biggr) r^{\lambda_{1}}U(r) \\ &\quad \leq \biggl(-\frac{(2+\lambda_{2})(p-p_{c})}{(2+\lambda_{1})(p+p_{c}-2)(p-1)} +A_{1}^{2}l^{2+\lambda_{1}} \biggr)r^{\lambda_{1}}U(r) \\ &\quad \leq -\frac{(2+\lambda_{2})(p-p_{c})}{2(2+\lambda _{1})(p+p_{c}-2)(p-1)}r^{\lambda_{1}}U(r). \end{aligned}$$
(3.15)

Finally, for \(r>l\) and \(t>0\), (3.12) and (3.13) guarantee that

$$\begin{aligned} &U''(r)+\frac{n-1}{r}U'(r)+(t+ \tau)^{\beta}\bigl((t+\tau)^{\beta}r-1\bigr)b\bigl((t+ \tau)^{\beta}r-1\bigr)U'(r) \\ &\qquad{}+\beta r^{1+\lambda_{1}}U'(r)+\alpha r^{\lambda_{1}}U(r) \\ &\quad \leq U''(r)+\frac{n+\kappa_{2}-1}{r}U'(r)+ \beta r^{1+\lambda_{1}}U'(r) +\alpha r^{\lambda_{1}}U(r) \\ &\quad = \bigl(A_{2}+(A_{1}-A_{2})l^{n+\kappa_{2}+\lambda_{1}} \bigr) \bigl(A_{2}+(A_{1}-A_{2})l^{n+\kappa_{2}+\lambda_{1}}- \beta \bigr)r^{2+2\lambda_{1}}U(r) \\ &\qquad{}+ \bigl(\alpha-(n+\kappa_{2}+\lambda_{1}) \bigl(A_{2}+(A_{1}-A_{2})l^{n+\kappa_{2}+\lambda_{1}} \bigr) \bigr)r^{\lambda_{1}}U(r) \\ &\quad \leq \bigl(\alpha-(n+\kappa_{2}+\lambda_{1})A_{2} \bigr)r^{\lambda_{1}}U(r) \\ &\quad \leq -\frac{(2+\lambda_{2})(p-p_{c})}{2(2+\lambda _{1})(p+p_{c}-2)(p-1)}r^{\lambda_{1}}U(r). \end{aligned}$$
(3.16)

Due to \(-2<\lambda_{1}\leq\lambda_{2}\), \(p>1\) and the definition of the function \(A(r)\),

$$0< \sigma_{0}=\sup_{r>0}r^{\lambda_{2}-\lambda_{1}} \mathrm{e}^{-(p-1)A(r)}< +\infty. $$

Let \(\varepsilon>0\) be sufficiently small such that

$$\varepsilon^{p-1}\le\frac{(2+\lambda_{2})(p-p_{c})}{2\sigma _{0}(2+\lambda_{1})(p+p_{c}-2)(p-1)}, $$

then (3.14)–(3.16) imply that

$$\begin{aligned} & \begin{aligned} &U''(r)+\frac{n-1}{r}U'(r)+ (t+ \tau)^{\beta}\bigl((t+\tau)^{\beta}r-1\bigr)b\bigl((t+ \tau)^{\beta}r-1\bigr)U'(r) \\ &\qquad{}+\beta r^{1+\lambda_{1}}U'(r)+\alpha r^{\lambda _{1}}U(r)+r^{\lambda_{2}}U^{p}(r) \\ &\quad \leq r^{\lambda_{1}}U(r) \biggl(-\frac{(2+\lambda _{2})(p-p_{c})}{2(2+\lambda_{1})(p+p_{c}-2)(p-1)} + \varepsilon^{p-1}r^{\lambda_{2}-\lambda_{1}}\mathrm{e}^{-(p-1)A(r)} \biggr) \end{aligned} \\ &\quad \leq r^{\lambda_{1}}U(r) \biggl(-\frac{(2+\lambda _{2})(p-p_{c})}{2(2+\lambda_{1})(p+p_{c}-2)(p-1)} + \varepsilon^{p-1}\sigma_{0} \biggr) \\ &\quad \leq 0,\quad r\in\bigl(0,l^{2}\bigr)\cup\bigl(l^{2},l \bigr)\cup(l,+\infty), t>0. \end{aligned}$$

Therefore, u given by (3.10) and (3.11) is a supersolution to (1.1). □

Proof of Theorem 2.2

The comparison principle and Lemma 3.2 show that there exists a nontrivial global solution to the problem (1.1), (1.2). We will show that the problem also admits a blow-up solutions. Fix \(R>R_{0}\). Assume that u is a solution to the problem (1.1), (1.2). Lemma 3.1 and Remark 3.1 imply that

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}w_{R}(t) \ge-M_{0}R^{-2}w_{R}(t) + \int_{\mathbb {R}^{n}}\bigl( \vert x \vert +1\bigr)^{\lambda _{2}}u^{p}(x,t) \eta_{R}\bigl( \vert x \vert \bigr)\,\mathrm{d}x,\quad t>0, \end{aligned}$$
(3.17)

where \(\eta_{R}\), \(R_{0}\), δ, \(M_{0}\) and \(w_{R}(t)\) are given in Lemma 3.1, Remark 3.1 and the proof of Theorem 2.1. The Hölder inequality yields

$$\begin{aligned} w_{R}(t)&\leq\biggl( \int_{\mathbb {R}^{n}} \bigl( \vert x \vert +1\bigr)^{(p\lambda_{1}-\lambda_{2})/(p-1)}\eta _{R}\bigl( \vert x \vert \bigr)\,\mathrm{d}x\biggr)^{(p-1)/p}\\ &\quad {}\times\biggl( \int_{\mathbb {R}^{n}}\bigl( \vert x \vert +1\bigr)^{\lambda _{2}}u^{p}(x,t) \eta_{R}\bigl( \vert x \vert \bigr)\,\mathrm{d}x\biggr)^{1/p} \\ &\leq\biggl( \int_{\mathbb {R}^{n}}\bigl( \vert x \vert +1\bigr)^{p\lambda _{1}/(p-1)} \eta_{R}\bigl( \vert x \vert \bigr)\,\mathrm{d}x\biggr)^{(p-1)/p}\\ &\quad {}\times \biggl( \int_{\mathbb {R}^{n}}\bigl( \vert x \vert +1\bigr)^{\lambda _{2}}u^{p}(x,t) \eta_{R}\bigl( \vert x \vert \bigr)\,\mathrm{d}x\biggr)^{1/p}, \quad t>0, \end{aligned}$$

which implies

$$\begin{aligned} &\int_{\mathbb {R}^{n}}\bigl( \vert x \vert +1\bigr)^{\lambda _{2}}u^{p}(x,t) \eta_{R}\bigl( \vert x \vert \bigr)\,\mathrm{d}x \\ &\quad \geq\biggl( \int_{\mathbb {R}^{n}}\bigl( \vert x \vert +1\bigr)^{p\lambda_{1}/(p-1)} \eta_{R}\bigl( \vert x \vert \bigr)\,\mathrm{d}x\biggr)^{1-p}w_{R}^{p}(t), \quad t>0. \end{aligned}$$
(3.18)

Substituting (3.18) into (3.17) we get

$$\begin{aligned} &\frac{\mathrm{d}}{\mathrm{d}t}w_{R}(t) \\ &\quad \ge w_{R}(t) \biggl(-M_{0}R^{-2}+\biggl( \int_{\mathbb {R}^{n}} \bigl( \vert x \vert +1\bigr)^{p\lambda_{1}/(p-1)} \eta_{R}\bigl( \vert x \vert \bigr)\,\mathrm{d}x\biggr)^{1-p}w_{R}^{p-1}(t) \biggr), \quad t>0. \end{aligned}$$
(3.19)

If \(u_{0}\) is so large that

$$M_{0}R^{-2}\leq\frac{1}{2}\biggl( \int_{\mathbb {R}^{n}} \bigl( \vert x \vert +1\bigr)^{p\lambda_{1}/(p-1)} \eta_{R}\bigl( \vert x \vert \bigr)\,\mathrm{d}x\biggr)^{1-p}w_{R}^{p-1}(0), $$

then (3.19) leads to

$$\frac{\mathrm{d}}{\mathrm{d}t}w_{R}(t) \geq\frac{1}{2}\biggl( \int_{\mathbb {R}^{n}} \bigl( \vert x \vert +1\bigr)^{p\lambda_{1}/(p-1)} \eta_{R}\bigl( \vert x \vert \bigr)\,\mathrm{d}x\biggr)^{1-p}w_{R}^{p}(t), \quad t>0. $$

The same argument as in the proof of Theorem 2.1 shows that u must blow up in a finite time. □

Remark 3.2

For the critical case \(p=p_{c}\) with \(-n-\lambda_{1}<\kappa<+\infty\). we need an additional condition (see [18]) that

$$\begin{aligned} -\infty\le \int_{1}^{+\infty}\frac{s^{2}b(s)-\kappa}{s}\,\mathrm{d}s< +\infty \quad\mbox{if }-n-\lambda_{1}< \kappa< +\infty. \end{aligned}$$
(3.20)

Similar to the proof in critical case in [18, 21], one can show the blow-up of the solutions to the problem (1.1), (1.2) for the critical case \(p=p_{c}\) with \(-n-\lambda _{1}<\kappa<+\infty\) if (3.20) holds.