1 Introduction

In this paper, we study the following Cauchy problem of fast diffusion parabolic equation with a nonlinear nonlocal source:

$$\begin{aligned} \left \{ \textstyle\begin{array}{l@{\quad}l} u_{t}=\Delta u^{m}+ (\int_{\mathbb{R}^{n}}u^{q}(x,t)\,dx )^{\frac{p-1}{q}}u^{r+1}, &(x,t)\in\mathbb{R}^{n}\times(0,T), \\ u(x,0)=u_{0}(x), & x\in \mathbb{R}^{n}, \end{array}\displaystyle \right . \end{aligned}$$
(1.1)

where the spatial dimension \(n\ge1\), the coefficients m, p, q, r satisfy \(\max\{0,1-\frac{2}{n}+r\}< m<1\), \(p> 1\), \(q\ge1\), \(0\le r<\frac{2}{n}\), and the initial data \(u_{0}(x)\) is a nontrivial nonnegative continuous function.

The quasilinear parabolic equations involving a nonlocal term originate in the phenomena of diffusion about concentration of some Newtonian fluids or the density of some biological species and heat transfer in a special medium with nonlocal source (see [2, 3] and the references therein). In the past three decades, various nonlocal mathematical models were established to describe many physical phenomena (see [1, 4,5,6,7,8,9] and references therein). At the same time, many important results have appeared on the blow-up problem for a nonlinear parabolic equation with nonlocal source (see [2, 6, 8,9,10,11] and references therein), and for nonlocal nonlinear diffusion equations [12, 13]. However, most of efforts have been devoted in bounded domains, there were few researches for the Cauchy problems (see [1, 14, 15]).

It is well known that the classical Cauchy problem

$$\begin{aligned} u_{t}=\Delta u+u^{p} \quad\text{in } \mathbb{R}^{n}\times(0,T) \end{aligned}$$
(1.2)

possesses the critical exponent \(1+\frac{2}{n}\) [16,17,18,19], that is to say, any nontrivial solution blows up in finite time if \(1< p\le1+\frac {2}{n}\), whereas global and non-global solutions coexist if \(p>1+\frac{2}{n}\), depending on the size of initial data. From then on, the Fujita phenomenon has been observed for many nonlinear PDEs (see surveys [20, 21] and references therein).

The study for the Cauchy problem of nonlocal nonlinear parabolic equation was proposed by Galaktionov et al. [1], in which it was proved that the Cauchy problem (1.1) with \(m=1\) has a critical Fujita exponent, and Wang et al. [15] obtained similar results by other methods. Recently, Zhou [14] considered the global and non-global existence of solutions for (1.1) with \(m>1\).

The present paper investigates a fast diffusion parabolic equation (1.1) (\(\max\{0,1-\frac{2}{n}\}< m<1\)) with a nonlocal source, and establishes the critical Fujita exponent \(p_{c}=m+\frac{2q-n(1-m)-nqr}{n(q-1)}\). Comparing with the known result for the parallel problem with a local source

$$\begin{aligned} u_{t}=\Delta u^{m}+u^{p} \quad \mbox{in } \mathbb{R}^{n}\times(0,T), \end{aligned}$$

the critical Fujita exponent was obtained in [22, 23] and shown to be \(p_{c}=1+\frac{2m}{n}\).

In the rest of the paper, we always let u be a solution to (1.1), and \(p_{c}=m+\frac{2q-n(1-m)-nqr}{n(q-1)}\). The main results are stated in the following theorems.

Theorem 1.1

For \(1< p\leq p_{c}\), there are no global nontrivial solutions to (1.1).

Theorem 1.2

For \(p>p_{c}\), there are both global and non-global solutions to (1.1).

This paper is organized as follows. Section 2 concerns the non-global solution to prove Theorem 1.1. Section 3 deals with the global existence to prove Theorem 1.2. And Sect. 4 shows in what ways the parameter q of the nonlocal source affects the behavior of solutions in the fast diffusion problem (1.1).

2 Non-global solutions

This section mainly applies the test function method (refer to [15, 22]) to prove that any solution of (1.1) must blow up in finite time for \(1< p\le p_{c}\). Introducing the test function

$$\begin{aligned} \varphi_{k}(x)= \biggl(\frac{k}{\pi} \biggr)^{\frac{n}{2}}\mathrm{e}^{-k|x|^{2}} \end{aligned}$$
(2.1)

for some \(k>0\), we can simply verify that

$$\begin{aligned} \int_{\mathbb{R}^{n}}\varphi_{k}(x)\,dx=1,\qquad \bigl\Vert \varphi_{k}(x) \bigr\Vert _{L^{\infty}}= \biggl(\frac{k}{\pi} \biggr)^{\frac {n}{2}},\qquad \Delta\varphi_{k}(x)\ge-2kn \varphi_{k}(x). \end{aligned}$$

Define

$$\begin{aligned} F(t)= \int_{\mathbb{R}^{n}}u(x,t)\varphi_{k}(x)\,dx. \end{aligned}$$

It is sufficient to show that \(F(t)\) blows up in finite time as \(1< p\le p_{c}\) to deal with Theorem 1.1.

Proof of Theorem 1.1

Firstly, we consider the case of \(1< p< p_{c}\). Multiplying equation (1.1) by \(\varphi_{k}(x)\) and integrating by parts in \(\mathbb{R}^{n}\), we get

$$\begin{aligned} F'(t)={}& \int_{\mathbb{R}^{n}}u_{t}\varphi_{k}\,dx \\ ={}& \int_{\mathbb{R}^{n}}\Delta u^{m}\varphi_{k}\,dx+ \biggl( \int_{\mathbb {R}^{n}}u^{q}\,dx \biggr)^{\frac{p-1}{q}} \int_{\mathbb{R}^{n}} u^{r+1}\varphi _{k}\,dx \\ \ge{}& {-}2kn \int_{\mathbb{R}^{n}} u^{m}\varphi_{k}\,dx+ \Vert \varphi_{k} \Vert _{L^{\infty}}^{-\frac{p-1}{q}} \biggl( \int_{\mathbb{R}^{n}}u^{q}\varphi_{k}\,dx \biggr)^{\frac {p-1}{q}} \int_{\mathbb{R}^{n}} u^{r+1}\varphi_{k}\,dx. \end{aligned}$$

Using Jensen’s inequality for \(m<1\), \(q>1\), and \(r>0\),

$$\begin{aligned} F'(t)\ge {}&{-}2knF^{m}(t)+ \biggl(\frac{k}{\pi} \biggr)^{-\frac{n(p-1)}{2q}}F^{p+r}(t) \\ ={}&F^{p+r}(t) \biggl( \biggl(\frac{k}{\pi} \biggr)^{-\frac {n(p-1)}{2q}}-2knF^{-(p+r-m)}(t) \biggr). \end{aligned}$$

Assuming

$$\begin{aligned} F(t)> \bigl(\pi^{\frac{n(p-1)}{2q}}(4n)^{-1} \bigr)^{-\frac{1}{p+r-m}}k^{\frac {2q+n(p-1)}{2q(p+r-m)}}, \end{aligned}$$
(2.2)

we obtain

$$\begin{aligned} \biggl(\frac{k}{\pi} \biggr)^{-\frac{n(p-1)}{2q}}>4knF^{-(p+r-m)}(t), \end{aligned}$$

and

$$\begin{aligned} F'(t)\ge \frac{1}{2} \biggl(\frac{k}{\pi} \biggr)^{-\frac{n(p-1)}{2q}}F^{p+r}(t). \end{aligned}$$
(2.3)

This implies

$$F(t)\geq \biggl(F^{-(p+r-1)}(0)-\frac{p+r-1}{2} \biggl(\frac{k}{\pi} \biggr)^{-\frac{n(p-1)}{2q}}t \biggr)^{-\frac{1}{p+r-1}}. $$

Obviously, \(F(t)\) blows up for any nonnegative initial data as \(t\to T=\frac{2F^{-(p+r-1)}(0)}{p+r-1} (\frac{k}{\pi} )^{\frac{n(p-1)}{2q}}\). In the following, we show that

$$\begin{aligned} F(0)> \bigl(\pi^{\frac{n(p-1)}{2q}}(4n)^{-1} \bigr)^{-\frac {1}{p+r-m}}k^{\frac{2q+n(p-1)}{2q(p+r-m)}} \end{aligned}$$
(2.4)

is a sufficient condition to prove condition (2.2). If not, there exists some τ, such that

$$\begin{aligned} F(\tau)= \bigl(\pi^{\frac{n(p-1)}{2q}}(4n)^{-1} \bigr)^{-\frac{1}{p+r-m}}k^{\frac {2q+n(p-1)}{2q(p+r-m)}}, \end{aligned}$$

and

$$\begin{aligned} F(t)> \bigl(\pi^{\frac{n(p-1)}{2q}}(4n)^{-1} \bigr)^{-\frac{1}{p+r-m}}k^{\frac {2q+n(p-1)}{2q(p+r-m)}},\quad t\in[0,\tau). \end{aligned}$$

This implies \(F'(\tau_{0})<0\) for some \(\tau_{0}\in(0,\tau)\), which contradicts \(F'(t)\ge0\), \(t\in(0,\tau)\). Thereby, to prove that a solution of (1.1) blows up in finite time, we only show (2.4) is true for any nonnegative nontrivial initial data \(u_{0}(x)\). Since \(\frac{n}{2}<\frac{2q+n(p-1)}{2q(p+r-m)}\) which was derived by \(1< p< p_{c}\), there exists a \(k>0\) small enough, such that

$$F(0)= \biggl(\frac{k}{\pi} \biggr)^{\frac{n}{2}} \int_{\mathbb{R}^{n}}\mathrm {e}^{-k|x|^{2}}u_{0}(x)\,dx> \bigl(\pi^{\frac{n(p-1)}{2q}}(4n)^{-1} \bigr)^{-\frac{1}{p+r-m}}k^{\frac{2q+n(p-1)}{2q(p+r-m)}}. $$

Next, we consider the case of \(p=p_{c}\). Supposing a solution of (1.1) is global for any \(t\ge0\), it holds that

$$\begin{aligned} F(t)= \biggl(\frac{k}{\pi} \biggr)^{\frac{n}{2}} \int_{\mathbb{R}^{n}}{\mathrm{ e}}^{-k|x|^{2}}u(x,t)\,dx\le \bigl( \pi^{\frac{n(p-1)}{2q}}(4n)^{-1} \bigr)^{-\frac{1}{p+r-m}}k^{\frac{2q+n(p-1)}{2q(p+r-m)}}. \end{aligned}$$
(2.5)

That is, if (2.5) is not true, namely \(F(t_{1})> (\pi^{\frac{n(p-1)}{2q}}(4n)^{-1} )^{-\frac {1}{p+r-m}}k^{\frac{2q+n(p-1)}{2q(p+r-m)}}\) for some \(t_{1}>0\), then the solution \(u(x,t)\) must blow up in finite time by the above proof. The condition \(p=p_{c}\) means \(\frac{n}{2}=\frac{2q+n(p-1)}{2q(p+r-m)}\), and (2.5) can be rewritten as

$$\begin{aligned} \int_{\mathbb{R}^{n}}{\mathrm{ e}}^{-k|x|^{2}}u(x,t)\,dx\le\pi^{\frac{n}{2}} \bigl(\pi^{\frac {n(p-1)}{2q}}(4n)^{-1} \bigr)^{-\frac{1}{p+r-m}}\quad \text{for } t>0. \end{aligned}$$
(2.6)

Without loss of generality, assuming \(u_{0}(x)\) has compact support in \(\mathbb{R}^{n}\), we get that \(u(x,t)\in L(\mathbb{R}^{n})\) for any fixed \(t>0\) (see [24]). By Lebesgue Dominated Convergence Theorem, as \(k\to0\) in (2.6),

$$\begin{aligned} \int_{\mathbb{R}^{n}}u(x,t)\,dx\le\pi^{\frac{n}{2}} \bigl( \pi^{\frac {n(p-1)}{2q}}(4n)^{-1} \bigr)^{-\frac{1}{p+r-m}}. \end{aligned}$$
(2.7)

Integrating equation (1.1) on \(\mathbb{R}^{n}\times[0,t]\), we have

$$\begin{aligned} \int_{\mathbb{R}^{n}}u(x,t)\,dx- \int_{\mathbb{R}^{n}}u_{0}(x)\,dx= \int_{0}^{t} \biggl( \int_{\mathbb{R}^{n}}u^{q}\,dx \biggr)^{\frac{p-1}{q}} \int_{\mathbb {R}^{n}}u^{r+1}\,dx\,dt. \end{aligned}$$

And then

$$\begin{aligned} \int_{0}^{t} \biggl( \int_{\mathbb{R}^{n}}u^{q}\,dx \biggr)^{\frac{p-1}{q}} \int _{\mathbb{R}^{n}}u^{r+1}\,dx\,dt\le \int_{\mathbb{R}^{n}}u(x,t)\,dt\le \pi^{\frac{n}{2}} \bigl( \pi^{\frac{n(p-1)}{2q}}(4n)^{-1} \bigr)^{-\frac {1}{p+r-m}} \end{aligned}$$

as \(u_{0}(x,t)\ge0\). This implies that

$$\begin{aligned} \int_{0}^{\infty}\biggl( \int_{\mathbb{R}^{n}}u^{q}\,dx \biggr)^{\frac{p-1}{q}} \int _{\mathbb{R}^{n}}u^{r+1}\,dx\,dt< +\infty. \end{aligned}$$
(2.8)

On the other hand, from [25] we know that there exists \(\delta>0\) such that the solution of (1.1) satisfies

$$\begin{aligned} u(x,\tau)>\delta \bigl(1+B \vert x \vert ^{2} \bigr)^{-\frac{1}{1-m}} \end{aligned}$$

for \(B=\frac{(1-m)\alpha\delta^{1-m}}{2mn}\) and some \(\tau>0\). Setting

$$\underline{u}(x,t)=\delta(1+t)^{-\alpha} \bigl(1+B \vert x \vert ^{2}(1+t)^{-\frac {2\alpha}{n}} \bigr)^{-\frac{1}{1-m}} $$

with \(\alpha=\frac{n}{2-n(1-m)}\), it is simple to verify

$$\begin{aligned} \underline{u}(x,t)\le u(x,t+\tau) \quad\text{for } x\in \mathbb{R}^{n}, t>0. \end{aligned}$$

And we have

$$\begin{aligned}& \int_{0}^{\infty}\biggl( \int_{\mathbb{R}^{n}}u^{q}(x,t)\,dx \biggr)^{\frac {p-1}{q}} \int_{\mathbb{R}^{n}}u^{r+1}(x,t)\,dx\,dt \\& \quad\ge \int_{0}^{\infty}\biggl( \int_{\mathbb{R}^{n}}u^{q}(x,t+\tau)\,dx \biggr)^{\frac{p-1}{q}} \int_{\mathbb{R}^{n}}u^{r+1}(x,t+\tau)\,dx\,dt \\& \quad\ge \int_{0}^{\infty}\biggl( \int_{\mathbb{R}^{n}}\underline{u}^{q}(x,t)\,dx \biggr)^{\frac{p-1}{q}} \int_{\mathbb{R}^{n}}\underline{u}^{r+1}(x,t)\,dx\,dt \\& \quad= B^{-\frac{p+q-1}{2q}}\delta^{p+r} \biggl( \int_{\mathbb{R}^{n}}\bigl(1+ \vert \xi \vert ^{2} \bigr)^{-\frac{q}{1-m}}\,d\xi \biggr)^{\frac{p-1}{q}} \int_{\mathbb{R}^{n}}\bigl(1+ \vert \xi \vert ^{2} \bigr)^{-\frac{r+1}{1-m}}\,d\xi \int_{0}^{\infty}(1+t)^{-1}\,dt \\& \quad=+\infty, \end{aligned}$$

since \(-\alpha(p+r-1)+\frac{\alpha(p-1)}{q}=-1\) for \(p=p_{c}\) and \(\xi=\sqrt{B}x(1+t)^{-\frac{\alpha}{n}}\). This contradicts (2.8), and so our assumption that the solution of (1.1) globally exist for \(t>0\) is not true, which proves Theorem 1.1 with \(p=p_{c}\). □

3 Coexistence of global and non-global solutions

This section mainly deals with the global solution for the case of \(p>p_{c}\) to derive Theorem 1.2.

Proof of Theorem 1.2

Firstly, we show that the solution of (1.1) must blow up in finite time for large initial data \(u_{0}(x)\). The proof of Theorem 1.1 means that \(u(x,t)\) does not exist globally, provided \(u_{0}\) satisfies

$$\begin{aligned} \biggl(\frac{k}{\pi} \biggr)^{\frac{n}{2}} \int_{\mathbb{R}^{n}}\mathrm{e}^{-k|x|^{2}} u_{0}(x)\,dx> \bigl(\pi^{\frac{n(p-1)}{2q}}(4n)^{-1} \bigr)^{-\frac {1}{p+r-m}}k^{\frac{2q+n(p+1)}{2q(p+r-m)}}. \end{aligned}$$
(3.1)

For any fixed \(k=k_{0}>0\), we can choose large \(u_{0}(x)\) to fulfil condition (3.1).

Next, we prove that the solution of (1.1) exists globally for any small initial data \(u_{0}(x)\). Let

$$\bar{u}=(t+1)^{-\beta} \bigl(D_{1}+D_{2} \vert x \vert ^{2}(t+1)^{-\beta(1-m)-1} \bigr)^{-\frac{1}{1-m}}, $$

where \(\beta=\frac{n(p-1)+2q}{2q(p+r-1)-n(1-m)(p-1)}\), and \(D_{1},D_{2}>0\) are to be determined. We demonstrate that ū is a global supersolution of (1.1) for suitable \(D_{1}\) and \(D_{2}\). Setting

$$Z=D_{1}+D_{2} \vert x \vert ^{2}(t+1)^{-\beta(1-m)-1}=:D_{1}+D_{2}z, $$

with \(z=|x|^{2}(t+1)^{-\beta(1-m)-1}\), we have

$$\begin{aligned}& \bar{u}_{t}-\Delta \bar{u}^{m}- \biggl( \int_{\mathbb{R}^{n}}\bar{u}^{q}\,dx \biggr)^{\frac {p-1}{q}}\bar{u}^{r+1} \\& \quad=(t+1)^{-\beta-1}Z^{-\frac{1}{1-m}-1} \biggl[-\beta Z+\frac{D_{2}(\beta-\beta m+1)}{1-m}z+ \frac{2mD_{2}n}{1-m}Z- \frac{4mD_{2}^{2}}{(1-m)^{2}}z \\& \qquad{}-(t+1)^{-\beta r+1} \biggl( \int_{\mathbb{R}^{n}}(t+1)^{-\beta q} \bigl(D_{1}+D_{2} \vert y \vert ^{2}(t+1)^{-1-\beta(1-m)} \bigr)^{-\frac{q}{1-m}}\,dy \biggr) ^{\frac{p-1}{q}}Z^{-\frac{r}{1-m}+1} \biggr] \\& \quad=:(t+1)^{-\beta-1}Z^{-\frac{1}{1-m}-1}G(Z). \end{aligned}$$
(3.2)

For \(\max\{0,1-\frac{2}{n}+r\}< m<1\), \(q\ge1\), \(r\ge0\) implying \(\frac{2q}{1-m}\ge\frac{2}{1-m}>n\), there exists a constant \(C>0\) such that

$$\begin{aligned}& \int_{\mathbb{R}^{n}}(t+1)^{-\beta q} \bigl(D_{1}+D_{2} \vert y \vert ^{2}(t+1)^{-1-\beta(1-m)} \bigr)^{-\frac{q}{1-m}}\,dy \\& \quad= \int_{\mathbb{R}^{n}}(t+1)^{-\beta q+\frac{n+n\beta(1-m)}{2}} \bigl(D_{1}+D_{2} \vert w \vert ^{2} \bigr)^{-\frac{q}{1-m}}\,dw \\& \quad\le C(t+1)^{-\beta q+\frac{n+n\beta(1-m)}{2}}. \end{aligned}$$

Substituting the above inequity into the expression of \(G(Z)\) in (3.2), and using \(D_{2}z=Z-D_{1}\), \(\beta=\frac{n(p-1)+2q}{2q(p+r-1)-n(1-m)(p-1)}\), we have

$$\begin{aligned} G(Z)\ge{}&{-}\beta Z+\frac{D_{2}(\beta-\beta m+1)}{1-m}z+\frac{2mD_{2}n}{1-m}Z- \frac{4mD_{2}^{2}}{(1-m)^{2}}z \\ &-C(t+1)^{-\beta(p+r-1)+\frac{n+n\beta(1-m)}{2q}(p-1)+1}Z^{-\frac {r}{1-m}+1} \\ ={}& \biggl(-\beta+\frac{\beta-\beta m+1}{1-m}+\frac{2mD_{2}n}{1-m}-\frac {4mD_{2}}{(1-m)^{2}} \biggr)Z \\ &- \biggl(\frac{\beta-\beta m+1}{1-m} -\frac{4mD_{2}}{(1-m)^{2}} \biggr)D_{1}-CZ^{-\frac{r}{1-m}+1} \\ =:{}&F(Z). \end{aligned}$$
(3.3)

To describe \(F(Z)\ge0\) for some \(D_{1}\) and \(D_{2}\), we have to show (i) \(F(D_{1})\ge0\) and (ii) \(F'(Z)\ge0\) for \(Z\ge D_{1}\).

(i) \(F(D_{1})= (-\beta+\frac{2mDn}{1-m} )D_{1}-CD_{1}^{-\frac {r}{1-m}+1}\ge 0\) is equivalent to

$$\begin{aligned}& D_{1}^{-\frac{r}{1-m}}\le\frac{1}{C} \biggl(-\beta+ \frac {2mD_{2}n}{1-m} \biggr), \end{aligned}$$
(3.4)
$$\begin{aligned}& D_{2}>\frac{\beta(1-m)}{2mn}. \end{aligned}$$
(3.5)

(ii) By simple computation, \(F'(Z)=-\beta+\frac{\beta-\beta m+1}{1-m}+\frac{2mD_{2}n}{1-m}- \frac{4mD_{2}}{(1-m)^{2}}-C (1-\frac{r}{1-m} )Z^{-\frac{r}{1-m}}\). If \(1-\frac{r}{1-m}\le0\), condition (ii) is ensured by

$$\begin{aligned} -\beta+\frac{\beta-\beta m+1}{1-m}+\frac{2mD_{2}n}{1-m}- \frac{4mD_{2}}{(1-m)^{2}}> 0. \end{aligned}$$
(3.6)

If \(1-\frac{r}{1-m}> 0\), condition (ii) is ensured by (3.6) and

$$\begin{aligned} D_{1}^{-\frac{r}{1-m}}\le\frac{1-m}{C(1-m-r)} \biggl(-\beta+\frac{\beta -\beta m+1}{1-m}+\frac{2mD_{2}n}{1-m}- \frac{4mD_{2}}{(1-m)^{2}} \biggr). \end{aligned}$$
(3.7)

Inequalities (3.5) and (3.6) require

$$\begin{aligned} \frac{\beta(1-m)}{2mn}< D_{2}< \frac{1-m}{2m(2-n(1-m))}. \end{aligned}$$
(3.8)

Due to \(\beta=\frac{n(p-1)+2q}{2q(p+r-1)-n(1-m)(p-1)}\) and \(p>p_{c}\), we can choose some \(D_{2}>0\) that fulfils (3.8). For such \(D_{2}\), choose \(D_{1}>0\) large enough to satisfy (3.4) and (3.7).

In conclusion, ū is a global supersolution to problem (1.1) with small initial data \(u_{0}(x)\le\bar{u}(x,0)=(D_{1}+D_{2}|x|^{2})^{-\frac{1}{1-m}}\). □

4 Conclusion

This paper shows that the model (1.1) possesses critical Fujita exponent \(p_{c}=m+\frac{2q-n(1-m)-nqr}{n(q-1)}\) in Theorems 1.1 and 1.2, and we find that the coefficient q of the nonlocal term affects the critical Fujita exponent. It’s easy to see that \(p_{c}\) is decreasing in q with \(\lim_{q\to\infty}p_{c}=m+\frac{2}{n}-r\) and \(\lim_{q\to1}p_{c}=\infty\). That is to say, the scope \(1< p\leq p_{c}\) for the blow-up of any nontrivial solutions will be enlarged as q is decreasing, and any nontrivial solution of (1.1) will blow up when \(p>1\) and \(q=1\). Refer to Fig. 1.

Figure 1
figure 1

Critical Fujita exponent curve in q–p plane

Full size image