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Critical Fujita exponents to a class of non-Newtonian filtration equations with fast diffusion
- Mingjun Zhou1,
- Huilai Li1,
- Wei Guo2Email authorView ORCID ID profile and
- Xu Zhou3
Received: 9 May 2016
Accepted: 29 July 2016
Published: 9 August 2016
Abstract
We consider the Cauchy problem to a class of fast-diffusion non-Newtonian filtration equations. Besides the usual degeneracy in the fast-diffusion non-Newtonian filtration, the equation is degenerate or singular at infinity, depending on the sign of the parameter related to the coefficient of diffusion. Fujita type theorems are established and the critical Fujita exponent is determined. Specially, we also prove that the nontrivial solution blows up in a finite time on the critical situation.
Keywords
critical Fujita exponentdegeneracysingularitynon-Newtonian filtration equationfast diffusion
MSC
35K5535B33
1 Introduction
The purpose of this paper is to investigate the critical Fujita exponent for the following initial value problem:
where \(p>1\), \(0< q<1\), \(\max\{-n,(n-1)/q-(n+1)\}<\mu_{1}\le\mu_{2}<p\mu_{1}+(p-1)n\), and \(0\le u_{0}\in C_{0}(\mathbb {R}^{n})\).
$$\begin{aligned} &\bigl(\vert x\vert +1\bigr)^{\mu_{1}}\frac{\partial u}{\partial t}= \operatorname{div} \bigl(\vert \nabla u\vert ^{q-1}\nabla u \bigr)+ \bigl(\vert x\vert +1\bigr)^{\mu_{2}}u^{p}, \quad x\in\mathbb {R}^{n}, t>0, \end{aligned}$$
(1)
$$\begin{aligned} &u(x,0)=u_{0}(x),\quad x\in\mathbb {R}^{n}, \end{aligned}$$
(2)
The study of critical exponents began in 1966 by Fujita in [1], where it was proved for the initial value problem of that the problem admits no nontrivial nonnegative global solution if \(1< p< p_{c}=1+2/n\), whereas if \(p>p_{c}\), it admits both global (with small data) and non-global (with large initial data) solutions. Later, in 1981, Weissler [2] proved that the critical case \(p=p_{c}\) is still a blow-up case.
$$\begin{aligned} \frac{\partial u}{\partial t}=\Delta u+u^{p}, \quad x\in\mathbb {R}^{n}, t>0 \end{aligned}$$
In Fujita’s work, the new phenomenon of nonlinear parabolic equations was discovered. From then on, there has been a lot of work on the critical Fujita exponents for various nonlinear evolution equations and systems (see, e.g., the survey papers [3, 4] and the references therein, and also [5–15]). Among those, the Fujita type theorems for the slow-diffusion non-Newtonian filtration equation was investigated by Galaktionov in [16, 17], where \(p, q>1\). He proved that \(p_{c}=q+(q+1)/n\) by blow-up subsolutions and global supersolutions. Recently, the same problem for an interesting variant of (3) is studied by the authors [13]. The non-Newtonian filtration equations with fast diffusion were considered by Qi and Wang in [18], where the critical Fujita exponent was determined for the Cauchy problem of the equation with \(p>1\), \((n-1)/(n+1)< q<1\), and \(\sigma>n(1-q)-q-1\). It is shown that \(p_{c}=q+(q+1+\sigma)/n\) by energy functions. Obviously, they did not cover the portion \(0< q\le(n-1)/(n+1)\) of the fast-diffusion range.
$$\begin{aligned} \frac{\partial u}{\partial t}=\operatorname{div} \bigl(\vert \nabla u\vert ^{q-1}\nabla u \bigr)+u^{p}, \quad x\in\mathbb {R}^{n}, t>0 \end{aligned}$$
(3)
$$\begin{aligned} \frac{\partial u}{\partial t} =\operatorname{div} \bigl(\vert \nabla u\vert ^{q-1}\nabla u \bigr)+\vert x\vert ^{\sigma}u^{p}, \quad x\in\mathbb {R}^{n}, t>0 \end{aligned}$$
(4)
In the present paper, we study the problem (1), (2) and formulate the critical Fujita exponent as and the critical situation \(p=p_{c}\) is still the blow-up case. The range of m considered in this paper is \(0< m<1\), the whole fast-diffusion range of (1). Like the non-Newtonian filtration equation with fast diffusion, (1) is singular at points where \(\vert \nabla u\vert =0\). In addition, (1) is degenerate at \(\vert x\vert =+\infty\) for \(\mu_{1}>0\) and singular for \(\mu_{1}<0\), different from both (3) and (4). Inspired by [11, 18, 19], to prove the solutions’ blow-up, we analyze the interaction between the nonlinear source and nonlinear diffusion via precise estimates through constructing energy functions by use of the normalized principal eigenfunction of −Δ in the unit ball \(B_{1}\) of \(\mathbb {R}^{n}\) with homogeneous initial-boundary condition, rather than constructing subsolutions as the author did in [16, 17]. This method for equation (1) and its special case (4) basically depends upon the nonincreasing properties in the spatial variant of solutions, which is trivial with \(\mu_{1}=\mu_{2}\), while it may be invalid if \(\mu_{1}<\mu_{2}\). For all these reasons, we have to overcome some technical difficulties.
$$\begin{aligned} p_{c}=q+(q+1+\mu_{2})/(n+\mu_{1}) \end{aligned}$$
This paper is arranged as follows. Some preliminaries are introduced in Section 2, including the local existence theorem, the comparison principle, and a property of solutions from propagation of disturbances. The Fujita type theorems are established in Section 3. Finally in Section 4, the critical case will be concerned.
2 Preliminaries
Throughout this paper, we use \(B_{r}\) to indicate the ball in \(\mathbb {R}^{n}\) with radius r and center at the origin. The solution considered here is taken in the following sense.
Definition 2.1
We call a solution to the problem (1), (2) in \((0,T)\) with \(0< T\le+\infty\) if holds for any \(\phi\in C^{\infty}_{0}(\mathbb {R}^{n}\times(0,T))\) and for any \(\zeta\in C^{\infty}_{0}(\mathbb {R}^{n})\).
$$\begin{aligned} 0\le u\in C \bigl([0,T );L^{\infty} \bigl(\mathbb {R}^{n} \bigr)\bigr) \cap L_{\mathrm{loc}}^{q+1} \bigl(0,T;W_{\mathrm{loc}}^{1,q+1} \bigl(\mathbb {R}^{n} \bigr) \bigr) \end{aligned}$$
$$\begin{aligned} \int_{0}^{T} \int_{\mathbb {R}^{n}} \biggl(\bigl(\vert x\vert +1 \bigr)^{\mu_{1}}u\frac{\partial\phi }{\partial t} -\vert \nabla u\vert ^{q-1} \nabla u\cdot\nabla\phi+\bigl(\vert x\vert +1\bigr)^{\mu_{2}}u^{p} \phi \biggr)\,dx\,dt=0 \end{aligned}$$
$$\begin{aligned} \lim_{t\to0^{+}} \int_{\mathbb {R}^{n}}u(x,t)\zeta(x)\,dx = \int_{\mathbb {R}^{n}}u_{0}(x)\zeta(x)\,dx \end{aligned}$$
Like the non-Newtonian filtration equation, it is not hard to prove the well-posedness to the problem (1), (2), one can see, e.g., [20].
Next, we will prove the following proposition on a property of solutions from propagation of disturbances.
Proposition 2.1
Assume that u is a solution to the problem (1), (2) with \(0\le u_{0}\in C_{0}(\mathbb {R}^{n})\) nontrivial, then \(u(0,t_{0})>0\) for some \(t_{0}>0\).
Proof
That \(u_{0}\) is nontrivial shows that there exists \(0\neq x_{0}\in\mathbb {R}^{n}\) and \(\kappa, \rho>0\) such that where \(s_{+}=\max\{s,0\}\). Let with \(R(t)=\kappa^{q-1}t+\rho^{q+1}\), and \(\xi>1\) independent of κ and ρ to be chosen later.
$$\begin{aligned} u_{0}(x)\ge\kappa \biggl(1- \biggl(\frac{\vert x-x_{0}\vert ^{q+1}}{\rho^{q+1}} \biggr)^{1/q} \biggr)_{+}^{2},\quad x\in\mathbb {R}^{n}, \end{aligned}$$
$$\begin{aligned} &\varPhi(x,t)=\frac{\kappa\rho^{(q+1)\xi}}{R^{\xi}(t)} \biggl(1- \biggl(\frac{\vert x-x_{0}\vert ^{q+1}}{R(t)} \biggr)^{1/q} \biggr)_{+}^{2},\quad x\in\mathbb {R}^{n}, t>0, \\ &D= \biggl\{ (x,t)\in\mathbb {R}^{n+1}\times\mathbb {R}_{+}: \vert x-x_{0}\vert < 2\vert x_{0}\vert , \vert x-x_{0}\vert ^{q+1}< R(t), 0< t< \frac{\kappa^{1-q}\rho^{q+1}}{\xi} \biggr\} , \end{aligned}$$
Denote A direct calculation within D shows Setting then Divide D into two sets with \(\delta>0\) satisfying where Then in \(D^{(1)}\), For the chosen \(\delta>0\), we have in \(D^{(2)}\)
where Due to we know So for fixed \(\delta>0\) satisfying (5) and \(\xi>1\) satisfying we obtain Clearly, the constant \(\xi>1\) is independent of κ and ρ. The comparison principle implies In particular, with \(t_{1}=\frac{\kappa^{1-q}\rho^{q+1}}{\xi}\) and If \(0\in\varGamma_{1}\), the proof is complete. Otherwise, where From the above argument, we have where with \(R_{1}(t)=\kappa_{1}^{q-1}(t-t_{1})+\rho_{1}^{q+1}\). In particular, with \(t_{2}=t_{1}+\frac{\kappa_{1}^{1-q}\rho_{1}^{q+1}}{\xi}\) and If \(0\in\varGamma_{2}\), the proof is complete. Otherwise, repeat the above procedure. We get the conclusion in finite steps. □
$$\begin{aligned} \Vert z\Vert =\frac{\vert x-x_{0}\vert ^{q+1}}{R(t)}, \qquad H=1-\Vert z\Vert ^{1/q}. \end{aligned}$$
$$\begin{aligned} &\frac{\partial\varPhi}{\partial t}=-\frac{\xi\kappa^{q}\rho ^{(q+1)\xi}}{R^{\xi+1}(t)}H^{2} +\frac{2}{q} \frac{\kappa^{q}\rho^{(q+1)\xi}}{R^{\xi+1}(t)}H\Vert z\Vert ^{1/q}, \\ &\operatorname{div} \bigl(\vert \nabla\varPhi \vert ^{q-1}\nabla \varPhi \bigr) =- \biggl(\frac{2(q+1)}{q} \biggr)^{q} \frac{1}{R(t)} \biggl( \frac{\kappa\rho^{(q+1)\xi}}{R^{\xi}(t)}H \biggr)^{q} \biggl(n-(q+1)\frac{\Vert z\Vert ^{1/q}}{H} \biggr). \end{aligned}$$
$$\mathscr{L}[\varPhi]=\frac{R^{\xi+1}(t)}{\kappa^{q}\rho^{(q+1)\xi }H} \biggl(\bigl(\vert x\vert +1 \bigr)^{\mu_{1}} \frac{\partial\varPhi}{\partial t}-\operatorname{div} \bigl(\vert \nabla \varPhi \vert ^{q-1}\nabla \varPhi \bigr) \biggr),\quad(x,t)\in D, $$
$$\begin{aligned} \mathscr{L}[\varPhi] ={}&{-}\bigl(\vert x\vert +1\bigr)^{\mu_{1}}\xi H+ \frac {2}{q}\bigl(\vert x\vert +1\bigr)^{\mu_{1}} \Vert z\Vert ^{1/q} \\ &{}+ \biggl(\frac{2(q+1)}{q} \biggr)^{q} \biggl(\frac{\rho ^{(q+1)\xi}}{R^{\xi}(t)}H \biggr)^{q-1} \biggl(n-(q+1)\frac{\Vert z\Vert ^{1/q}}{H} \biggr). \end{aligned}$$
$$\begin{aligned} D^{(1)}= \bigl\{ (x,t)\in D:H< \delta \bigr\} \quad\mbox{and}\quad D^{(2)}= \bigl\{ (x,t)\in D:H\ge\delta \bigr\} \end{aligned}$$
$$\begin{aligned} \delta^{q-1} \biggl((q+1) \biggl(\frac{1}{\delta}-1 \biggr)-n \biggr)\ge\frac{2}{q}\varLambda_{1}, \end{aligned}$$
(5)
$$\begin{aligned} \varLambda_{1}=\left \{ \textstyle\begin{array}{@{}l@{\quad}l} (3\vert x_{0}\vert +1)^{\mu_{1}}, &\mu_{1}\ge0,\\ 1, &\mu_{1}< 0. \end{array}\displaystyle \right . \end{aligned}$$
$$\begin{aligned} \mathscr{L}[\varPhi]&\le\frac{2}{q}{\varLambda_{1}}+ \biggl( \frac {2(q+1)}{q} \biggr)^{q} \delta^{q-1} \biggl(n-(q+1) \biggl(\frac{1}{\delta}-1 \biggr) \biggr) \\ &\le\frac{2}{q}{\varLambda_{1}}+\delta^{q-1} \biggl(n+q+1-\frac {q+1}{\delta}\biggr) \\ &\le0. \end{aligned}$$
$$\begin{aligned} \mathscr{L}[\varPhi]\le-\varLambda_{2}\xi\delta+\frac{2}{q}{ \varLambda_{1}}+n \biggl(\frac {2(q+1)}{q} \biggr)^{q} \biggl(\frac{\rho^{(q+1)\xi}}{R^{\xi}(t)}\delta \biggr)^{q-1}, \end{aligned}$$
$$\begin{aligned} \varLambda_{2}=\left \{ \textstyle\begin{array}{@{}l@{\quad}l} 1, &\mu_{1}\ge0,\\ (3\vert x_{0}\vert +1)^{\mu_{1}}, &\mu_{1}< 0. \end{array}\displaystyle \right . \end{aligned}$$
$$\begin{aligned} \biggl(\frac{\rho^{(q+1)\xi}}{R^{\xi}(t)}\delta \biggr)^{q-1}\le \biggl(1+ \frac{1}{\xi}\biggr)^{\xi(1-q)}\delta^{q-1} \le \biggl( \frac{\mathrm{e}}{\delta}\biggr)^{1-q}, \quad(x,t)\in D, \end{aligned}$$
$$\begin{aligned} \mathscr{L}[\varPhi] \le-\varLambda_{2}\xi\delta+\frac{2}{q}{ \varLambda_{1}}+n \biggl(\frac {2(q+1)}{q} \biggr)^{m} \biggl(\frac{\mathrm{e}}{\delta}\biggr)^{1-q},\quad(x,t)\in D. \end{aligned}$$
$$\begin{aligned} \varLambda_{2}\xi\delta\ge\frac{2}{q}{\varLambda_{1}}+n \biggl(\frac {2(q+1)}{q} \biggr)^{q} \biggl(\frac{\mathrm{e}}{\delta}\biggr)^{1-q}, \end{aligned}$$
$$\bigl(\vert x\vert +1\bigr)^{\mu_{1}}\frac{\partial\varPhi}{\partial t}-\operatorname {div} \bigl(\vert \nabla \varPhi \vert ^{q-1}\nabla\varPhi \bigr)\le0, \quad x\in\mathbb {R}^{n}, 0< t< \frac{\kappa^{1-q}\rho^{q+1}}{\xi}. $$
$$\begin{aligned} u(x,t)\ge\varPhi(x,t),\quad(x,t)\in D. \end{aligned}$$
$$\begin{aligned} u(x,t_{1})>0,\quad x\in\varGamma_{1} \end{aligned}$$
$$\begin{aligned} \varGamma_{1}= \biggl\{ x\in\mathbb {R}^{n}:\vert x-x_{0}\vert < 2\vert x_{0}\vert , \vert x-x_{0}\vert ^{q+1}< \frac{\xi+1}{\xi}\rho^{q+1} \biggr\} . \end{aligned}$$
$$\begin{aligned} u(x,t_{1})\ge\varPhi(x,t_{1})=\kappa_{1} \biggl(1- \biggl(\frac {\vert x-x_{0}\vert ^{q+1}}{\rho_{1}^{q+1}} \biggr)^{1/q} \biggr)_{+}^{2}, \quad x \in\mathbb {R}^{n}, \end{aligned}$$
$$\begin{aligned} \kappa_{1}=\kappa \biggl(\frac{\xi}{\xi+1} \biggr)^{\xi},\qquad \rho_{1}=\rho \biggl(\frac{\xi+1}{\xi}\biggr)^{1/(q+1)}. \end{aligned}$$
$$\begin{aligned} u(x,t)\ge\varPhi_{1}(x,t),\quad(x,t)\in D_{1}, \end{aligned}$$
$$\begin{aligned} &\varPhi_{1}(x,t)=\frac{\kappa_{1}\rho_{1}^{(q+1)\xi}}{R_{1}^{\xi}(t)} \biggl(1- \biggl( \frac{\vert x-x_{0}\vert ^{q+1}}{R_{1}(t)} \biggr)^{1/q} \biggr)_{+}^{2},\quad x\in \mathbb {R}^{n}, t>t_{1}, \\ &D_{1}= \biggl\{ (x,t)\in\mathbb {R}^{n}\times\mathbb {R}_{+}: \vert x-x_{0}\vert < 2\vert x_{0}\vert , \vert x-x_{0}\vert ^{q+1}< R_{1}(t), t_{1}< t< t_{1}+\frac{\kappa_{1}^{1-q}\rho_{1}^{q+1}}{\xi} \biggr\} , \end{aligned}$$
$$\begin{aligned} u(x,t_{2})>0,\quad x\in\varGamma_{2}, \end{aligned}$$
$$\begin{aligned} \varGamma_{2}= \biggl\{ x\in\mathbb {R}^{n}:\vert x-x_{0}\vert < 2\vert x_{0}\vert , \vert x-x_{0}\vert ^{q+1}< \frac{\xi+1}{\xi} \rho_{1}^{q+1} \biggr\} . \end{aligned}$$
3 Fujita type theorems
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., 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 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.
$$\begin{aligned} u_{0}(x)=h_{0}\bigl(\vert x\vert \bigr),\quad x\in \mathbb {R}^{n}, \end{aligned}$$
$$\begin{aligned} u(x,t)=h\bigl(\vert x\vert ,t\bigr), \quad x\in\mathbb {R}^{n}, t\ge0. \end{aligned}$$
(6)
Let 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 Then with \(M_{0}>0\) independent of l. Set Definition 2.1 gives For radial and nonincreasing \(u(x,t)\), one has and where ν is the unit outer normal to \(\partial B_{2l}\). Hence The Hölder inequality yields with \(M_{1}>0\) independent of l, which, together with (8), implies By the Hölder inequality, and hence with \(M_{2}>0\) independent of l. Equations (9) and (10) show that
$$\begin{aligned} \psi(x)=\left \{ \textstyle\begin{array}{@{}l@{\quad}l} 1,&0\le \vert x\vert \le1,\\ f(\vert x\vert -1), &1< \vert x\vert < 2,\\ 0,&\vert x\vert \ge2, \end{array}\displaystyle \right . \end{aligned}$$
$$\begin{aligned} \psi_{l}(x)=\psi(x/l), \quad x\in\mathbb {R}^{n}. \end{aligned}$$
$$\begin{aligned} \vert \nabla\psi_{l}\vert \le\frac{M_{0}}{l}, \qquad \vert \Delta \psi_{l}\vert \le\frac{M_{0}}{l^{2}}, \qquad\frac{\vert \Delta\psi_{l}\vert }{\psi_{l}}\le \frac {M_{0}}{l^{2}}, \quad x\in B_{2l}\setminus B_{l}, \end{aligned}$$
$$\begin{aligned} \eta_{l}(t)= \int_{\mathbb {R}^{n}}\bigl(\vert x\vert +1\bigr)^{\mu_{1}} u \psi_{l} \,dx,\quad t\ge0. \end{aligned}$$
(7)
$$\begin{aligned} \frac{d\eta_{l}}{dt} =- \int_{B_{2l}}\vert \nabla u\vert ^{q-1}\nabla u\cdot \nabla \psi_{l} \,dx+ \int_{\mathbb {R}^{n}}\bigl(\vert x\vert +1\bigr)^{\mu_{2}}u^{p} \psi_{l} \,dx. \end{aligned}$$
$$\begin{aligned} \int_{B_{2l}}\vert \nabla u\vert ^{q-1}\nabla u\cdot \nabla \psi_{l} \,dx &= \int_{B_{2l}}\vert \nabla u\vert ^{q}\vert \nabla \psi_{l}\vert \,dx \\ &\le \biggl( \int_{B_{2l}}\vert \nabla u\vert \cdot \vert \nabla \psi_{l}\vert \,dx \biggr)^{q} \biggl( \int_{B_{2l}}\vert \nabla\psi_{l}\vert \,dx \biggr)^{1-q} \\ &= \biggl( \int_{B_{2l}}\nabla u\cdot\nabla\psi_{l}\,dx \biggr)^{q} \biggl( \int_{B_{2l}}\vert \nabla\psi_{l}\vert \,dx \biggr)^{1-q} \\ &\le M_{0}l^{(n-1)(1-q)} \biggl( \int_{B_{2l}}\nabla u\cdot\nabla\psi_{l}\,dx \biggr)^{q} \end{aligned}$$
$$\begin{aligned} 0\le \int_{B_{2l}}\nabla u\cdot\nabla\psi_{l}\,dx = \int_{\partial B_{2l}}u\nabla\psi_{l}\cdot\boldsymbol {\nu}\,d\sigma- \int_{B_{2l}} u\Delta\psi_{l} \,dx \le- \int_{B_{2l}} u\Delta\psi_{l} \,dx, \end{aligned}$$
$$\begin{aligned} \frac{d\eta_{l}}{dt}\ge-M_{0}l^{(1-q)(n-1)} \biggl\vert \int_{B_{2l}} u\Delta\psi_{l} \,dx \biggr\vert ^{q} + \int_{\mathbb {R}^{n}}\bigl(\vert x\vert +1\bigr)^{\mu_{2}}u^{p} \psi_{l} \,dx. \end{aligned}$$
(8)
$$\begin{aligned} \biggl\vert \int_{B_{2l}} u\Delta\psi_{l} \,dx \biggr\vert ^{q} \le{}& \biggl( \int_{B_{2l}\setminus B_{l}} u\vert \Delta\psi_{l}\vert \,dx \biggr)^{q} \\ \le{}& \biggl( \int_{B_{2l}\setminus B_{l}}\bigl(\vert x\vert +1\bigr)^{-\mu_{2}/(p-1)}\vert \Delta\psi_{l}\vert ^{p/(p-1)} \psi_{l}^{-1/(p-1)}\,dx \biggr)^{q(p-1)/p} \\ &{} \cdot \biggl( \int_{B_{2l}\setminus B_{l}}\bigl(\vert x\vert +1\bigr)^{\mu_{2}}u^{p} \psi _{l} \,dx \biggr)^{q/p} \\ \le{}& M_{1}l^{q(n-2)-q(n+\mu_{2})/p} \biggl( \int_{\mathbb {R}^{n}}\bigl(\vert x\vert +1\bigr)^{\mu_{2}}u^{p} \psi_{l} \,dx \biggr)^{q/p} \end{aligned}$$
$$\begin{aligned} \frac{d\eta_{l}}{dt} \ge{}& \biggl( \int_{\mathbb {R}^{n}}\bigl(\vert x\vert +1\bigr)^{\mu_{2}}u^{p} \psi_{l} \,dx \biggr)^{q/p} \\ &{} \cdot \biggl\{ -M_{0}M_{1}l^{n-q-1-q(n+\mu_{2})/p}+ \biggl( \int_{\mathbb {R}^{n}}\bigl(\vert x\vert +1\bigr)^{\mu_{2}}u^{p} \psi_{l} \,dx \biggr)^{(p-q)/p} \biggr\} . \end{aligned}$$
(9)
$$\begin{aligned} \int_{\mathbb {R}^{n}}\bigl(\vert x\vert +1\bigr)^{\mu_{1}}u \psi_{l}\,dx \le{}& \biggl( \int_{B_{2l}}\bigl(\vert x\vert +1\bigr)^{(p\mu_{1}-\mu_{2})/(p-1)} \psi_{l}\,dx \biggr)^{(p-1)/p} \\ &{} \cdot \biggl( \int_{\mathbb {R}^{n}}\bigl(\vert x\vert +1\bigr)^{\mu_{2}}u^{p} \psi_{l} \,dx \biggr)^{1/p} \\ \le{}& M_{2}l^{(n+\mu_{1})-(n+\mu_{2})/p} \biggl( \int_{\mathbb {R}^{n}}\bigl(\vert x\vert +1\bigr)^{\mu_{2}}u^{p} \psi_{l} \,dx \biggr)^{1/p}, \end{aligned}$$
$$\begin{aligned} \int_{\mathbb {R}^{n}}\bigl(\vert x\vert +1\bigr)^{\mu_{2}}u^{p} \psi_{l} \,dx \ge M_{2}^{-p}l^{-p(n+\mu_{1})+(n+\mu_{2})} \eta_{l}^{p} \end{aligned}$$
(10)
$$\begin{aligned} \frac{d\eta_{l}}{dt} \ge{}& \bigl(M_{2}^{-p}l^{-p(n+\mu_{1})+(n+\mu_{2})} \bigr)^{q/p}\eta_{l}^{q} \\ &{} \cdot \bigl\{ -M_{0}M_{1}l^{n-q-1-q(n+\mu_{2})/p} + M_{2}^{q-p}l^{[-p(n+\mu_{1})+(n+\mu_{2})](p-q)/p}\eta_{l}^{p-q} \bigr\} . \end{aligned}$$
(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 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 So there exists some \(0< T_{*}<+\infty\) such that Due to \(\operatorname{supp} \psi_{l}=B_{2l}\), we obtain
$$\begin{aligned} n-q-1-(n+\mu_{2})q/p< \bigl[-p(n+\mu_{1})+(n+ \mu_{2}) \bigr](p-q)/p. \end{aligned}$$
$$\begin{aligned} \frac{d\eta_{l}}{dt} \ge{}& \bigl(M_{2}^{-p}l^{-p(n+\mu_{1})+(n+\mu_{2})} \bigr)^{q/p}\eta_{l}^{q} \\ &{} \cdot \biggl(\frac{1}{2}M_{2}^{q-p}l^{[-p(n+\mu_{1})+(n+\mu _{2})](p-q)/p} \eta_{l}^{p-q} \biggr) \\ \ge{}&\delta\eta_{l}^{p}. \end{aligned}$$
$$\begin{aligned} \lim_{t\to T_{*}^{-}}\eta_{l}(t)=+\infty. \end{aligned}$$
$$\begin{aligned} \lim_{t\to T_{*}^{-}}\bigl\Vert u(\cdot,t)\bigr\Vert _{L^{\infty}(\mathbb {R}^{n})}=+\infty. \end{aligned}$$
Next, for the general case without the assumption that \(u(x,t)\) is nonincreasing, define Then \(\underline{u}(x,t)\) is nonincreasing, and From the above argument, we get for some \(0<\tilde{T}_{*}<+\infty\), and (13) ensures that u is a blow-up solution. □
$$\begin{aligned} \underline{u}(x,t)=\min_{0\le r\le \vert x\vert } h(r,t),\quad x \in\mathbb {R}^{n}, t\ge0. \end{aligned}$$
(12)
$$\begin{aligned} 0\le\underline{u}(x,t)\le u(x,t), \quad x\in\mathbb {R}^{n}, t\ge0, \end{aligned}$$
(13)
$$\begin{aligned} \frac{d}{dt} \int_{\mathbb {R}^{n}}\bigl(\vert x\vert +1\bigr)^{\mu_{1}} \underline{u} \psi_{l} \,dx \ge- \int_{B_{2l}}\vert \nabla\underline{u}\vert ^{q-1} \nabla\underline {u}\cdot\nabla\psi_{l} \,dx + \int_{\mathbb {R}^{n}}\bigl(\vert x\vert +1\bigr)^{\mu_{2}} \underline{u}^{p} \psi_{l} \,dx. \end{aligned}$$
(14)
$$\begin{aligned} \lim_{t\to\tilde{T}_{*}^{-}}\bigl\Vert \underline{u}(\cdot,t)\bigr\Vert _{L^{\infty}(\mathbb {R}^{n})}=+\infty \end{aligned}$$
Let us turn to the case \(p>p_{c}\). Suppose that where is a self-similar solution to (1). It is easy to show that \(V(r)\) solves
$$\begin{aligned} U(x,t)={(t+1)^{-\alpha}}V \bigl((t+1)^{-\beta}\bigl( \vert x\vert +1\bigr) \bigr), \end{aligned}$$
(15)
$$\begin{aligned} \alpha=\frac{q+1+\mu_{2}}{\mu_{1}(p-q)+\mu _{2}(q-1)+(q+1)(p-1)},\qquad \beta=\frac{p-q}{q+1+\mu_{2}}\alpha, \end{aligned}$$
(16)
$$\begin{aligned} \bigl(\bigl\vert V'\bigr\vert ^{q-1}V' \bigr)'+\frac{n-1}{r}\bigl\vert V'\bigr\vert ^{q-1}V'+ \beta r^{\mu_{1}+1}V'+\alpha r^{\mu_{1}}V+r^{\mu_{2}}V^{p}=0, \quad r>0. \end{aligned}$$
(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
where
is a supersolution to equation (17), i.e.
$$\begin{aligned} V(r)=\varepsilon \bigl(1+\rho(\varepsilon)r^{\lambda}\bigr)^{-\gamma}, \quad r>0, \end{aligned}$$
$$\begin{aligned} \lambda=1+\frac{\mu_{1}+1}{q},\qquad \gamma=\frac{q}{1-q}, \qquad\rho (\varepsilon)= \frac{1}{\gamma\lambda}\beta^{1/q}\varepsilon^{(1-q)/q}, \end{aligned}$$
$$\begin{aligned} \bigl(\bigl\vert V'\bigr\vert ^{q-1}V' \bigr)'+ \frac{n-1}{r}\bigl\vert V'\bigr\vert ^{q-1}V'+\beta r^{\mu_{1}+1}V'+\alpha r^{\mu_{1}}V+r^{\mu_{2}}V^{p} \le0, \quad r>0. \end{aligned}$$
Proof
It is not hard to show that it suffices to verify namely From the definition of \([\varepsilon\gamma\lambda\rho(\varepsilon )]^{q}=\beta\varepsilon\), we have Due to \(p>p_{c}=q+(q+1+\mu_{2})/(n+\mu_{1})\) and \((n-1)/q-(n+1)<\mu_{1}\le\mu_{2}\), Hence, holds for sufficiently small \(\varepsilon>0\), and (18) is obtained. □
$$\begin{aligned} &q(\gamma+1)\rho(\varepsilon)\lambda \bigl[\varepsilon\gamma \lambda\rho( \varepsilon) \bigr]^{q} \bigl[1+\rho(\varepsilon)r^{\lambda}\bigr]^{-q(\gamma+1)-1}r^{(q+1)(\lambda-1)} \\ &\quad{} -q(\lambda-1) \bigl[\varepsilon\gamma\lambda\rho (\varepsilon) \bigr]^{q} \bigl[1+\rho(\varepsilon)r^{\lambda}\bigr]^{-q(\gamma+1)}r^{q(\lambda-1)-1} \\ &\quad{} -(n-1) \bigl[\varepsilon\gamma\lambda\rho(\varepsilon) \bigr]^{q} \bigl[1+\rho(\varepsilon)r^{\lambda}\bigr]^{-q(\gamma+1)}r^{q(\lambda-1)-1} \\ &\quad{} -\beta\varepsilon\gamma\lambda\rho(\varepsilon) \bigl[1+\rho( \varepsilon)r^{\lambda}\bigr]^{-(\gamma+1)}r^{\mu_{1}+\lambda} +\alpha \varepsilon \bigl[1+\rho(\varepsilon)r^{\lambda}\bigr]^{-\gamma }r^{\mu_{1}} \\ &\quad{} +\varepsilon^{p} \bigl[1+\rho(\varepsilon)r^{\lambda}\bigr]^{-p\gamma }r^{\mu_{2}} \le0,\quad r>0, \end{aligned}$$
$$\begin{aligned} & \bigl\{ \bigl[\varepsilon\gamma\lambda\rho (\varepsilon) \bigr]^{q}-\beta\varepsilon \bigr\} \gamma\lambda\rho(\varepsilon) \bigl[1+\rho(\varepsilon)r^{\lambda}\bigr]^{-(\gamma+1)}r^{\mu _{1}+\lambda} \\ &\quad{} - \bigl\{ \bigl[q(\lambda-1)+n-1 \bigr]\varepsilon ^{-1} \bigl[ \varepsilon\gamma\lambda\rho(\varepsilon) \bigr]^{q} -\alpha- \varepsilon^{p-1} \bigl[1+\rho(\varepsilon)r^{\lambda}\bigr]^{-(p-1)\gamma}r^{\mu_{2}-\mu_{1}} \bigr\} \\ &\quad{} \cdot\varepsilon \bigl[1+\rho(\varepsilon)r^{\lambda}\bigr]^{-\gamma }r^{\mu_{1}} \le0, \quad r>0. \end{aligned}$$
(18)
$$\begin{aligned} & \bigl[q(\lambda-1)+n-1 \bigr]\varepsilon^{-1} \bigl[\varepsilon \gamma \lambda\rho(\varepsilon) \bigr]^{q} -\alpha-\varepsilon^{p-1} \bigl[1+\rho(\varepsilon)r^{\lambda}\bigr]^{-(p-1)\gamma}r^{\mu _{2}-\mu_{1}} \\ &\quad= (n+\mu_{1})\beta-\alpha-\varepsilon^{p-1} \bigl[1+ \rho(\varepsilon)r^{\lambda}\bigr]^{-(p-1)\gamma}r^{\mu_{2}-\mu_{1}} \\ &\quad\ge(n+\mu_{1})\beta-\alpha-\varepsilon^{p-1} \bigl[ \rho(\varepsilon) \bigr]^{-(p-1)\gamma}r^{\mu_{2}-\mu_{1}-\lambda (p-1)\gamma}. \end{aligned}$$
$$\begin{aligned} (n+\mu_{1})\beta>\alpha, \quad\lambda(p-1)\gamma\ge\mu_{2}- \mu_{1}\ge0. \end{aligned}$$
$$\begin{aligned} (n+\mu_{1})\beta-\alpha-\varepsilon^{p-1} \bigl[\rho( \varepsilon) \bigr]^{-(p-1)\gamma}r^{\mu_{2}-\mu_{1}-\lambda (p-1)\gamma}>0,\quad r>0 \end{aligned}$$
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 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 \(u_{0}\).
$$\begin{aligned} V'(r)< 0, \quad r>0, \end{aligned}$$
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 holds for some \(l_{0}>1\), then from (11), we get and for some \(\delta_{0}>0\). Therefore, u is a blow-up solution.
$$\begin{aligned} M_{0}M_{1}l_{0}^{n-q-1-(n+\mu_{2})q/p} < \frac{1}{2}M_{2}^{q-p}l_{0}^{[-p(n+\mu_{1})+(n+\mu_{2})](p-q)/p} \eta_{l_{0}}^{p-q}(0) \end{aligned}$$
$$\begin{aligned} \eta_{l_{0}}(t)\ge\eta_{l_{0}}(0),\quad t>0, \end{aligned}$$
$$\begin{aligned} \frac{d\eta_{l_{0}}}{dt} \ge{}& \bigl(M_{2}^{-p}l_{0}^{-p(n+\mu _{1})+(n+\mu_{2})} \bigr)^{q/p}\eta_{l_{0}}^{q} \\ &{} \cdot \biggl(\frac{1}{2} M_{2}^{q-p}l_{0}^{[-p(n+\mu_{1})+(n+\mu_{2})](p-q)/p} \eta_{l_{0}}^{p-q} \biggr) \\ ={}&\delta_{0}\eta_{l_{0}}^{p} \end{aligned}$$
For 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. □
4 Critical case
Now, let us deal with the critical case \(p=p_{c}=q+(q+1+\mu_{2})/(n+\mu_{1})\). Let \(\psi_{l}\), \(\eta_{l}\), \(M_{1}\), \(M_{2}\) be defined as in the previous section.
Lemma 4.1
Assume that
\(p=p_{c}=q+(q+1+\mu_{2})/(n+\mu_{1})\)
and
u
is a nontrivial, global, radial, and nonincreasing solution to the problem (1), (2). Then
holds for some constant
\(M>0\)
independent of
t.
$$\begin{aligned} \int_{\mathbb {R}^{n}}\bigl(\vert x\vert +1\bigr)^{\mu_{1}}u(x,t)\,dx\le M, \quad t>0, \end{aligned}$$
(19)
Proof
\(p=p_{c}\) yields Then, for the global, radial and nonincreasing solution u, from (11), we have Otherwise, u blows up in a finite time. Therefore (19) holds for some constant \(M>0\) owing to □
$$\begin{aligned} n-q-1-(n+\mu_{2})q/{p_{c}}= \bigl[-p_{c}(n+ \mu_{1})+(n+\mu_{2}) \bigr](p_{c}-q)/{p_{c}}. \end{aligned}$$
$$\begin{aligned} M_{2}^{q-p_{c}}\eta_{l}^{p_{c}-q}(t) \le2M_{0}M_{1},\quad l>1, t>0. \end{aligned}$$
$$\begin{aligned} \lim_{l\to+\infty}\eta_{l}(t) = \int_{\mathbb {R}^{n}}\bigl(\vert x\vert +1\bigr)^{\mu_{1}}u(x,t)\,dx. \end{aligned}$$
Lemma 4.2
Assume that
\(p=p_{c}=q+(q+1+\mu_{2})/(n+\mu_{1})\), u
be a nontrivial, radial, and nonincreasing solution to the problem (1), (2) and
\(0<\theta<1\). Then
holds for any
\(l>1\)
with a constant
\(M_{3}>0\)
independent of
l.
$$\begin{aligned} \frac{d\eta_{l}}{dt}\ge{}& M_{2}^{-q(1-\theta)} l^{(n+\mu_{2})-p_{c}(n+\mu_{1})}\eta_{l}^{q(1-\theta)} \\ &{} \cdot \biggl\{ -M_{3} \biggl( \int_{B_{2l}\setminus B_{l}}\bigl(\vert x\vert +1\bigr)^{\mu_{1}} u \psi_{l} \,dx \biggr)^{q\theta}+ M_{2}^{q(1-\theta)-p_{c}} \eta_{l}^{p_{c}-q(1-\theta)} \biggr\} \end{aligned}$$
(20)
Proof
For any \(l>1\), the Hölder inequality yields with \(M>0\) a constant independent of l, which, together with (10) and (8), implies Then (20) holds due to □
$$\begin{aligned} & \biggl\vert \int_{B_{2l}} u\Delta\psi_{l}\,dx \biggr\vert ^{q} \\ &\quad\le \biggl( \int_{B_{2l}\setminus B_{l}} u\vert \Delta\psi_{l}\vert \,dx \biggr)^{q} \\ &\quad\le \biggl( \int_{B_{2l}\setminus B_{l}} \bigl(\vert x\vert +1\bigr)^{-(\theta p_{c}\mu_{1}+(1-\theta)\mu_{2})/[(p_{c}-1)(1-\theta)]} \\ &\qquad{} \cdot \vert \Delta\psi_{l}\vert ^{p_{c}/[(p_{c}-1)(1-\theta)]} \psi _{l}^{-(1-\theta+p_{c}\theta)/[(p_{c}-1)(1-\theta)]}\,dx \biggr)^{q(p_{c}-1)(1-\theta)/{p_{c}}} \\ &\qquad{}\cdot \biggl( \int_{B_{2l}\setminus B_{l}}\bigl(\vert x\vert +1\bigr)^{\mu_{2}} u^{p_{c}} \psi_{l} \,dx \biggr)^{q(1-\theta)/{p_{c}}} \biggl( \int_{B_{2l}\setminus B_{l}}\bigl(\vert x\vert +1\bigr)^{\mu_{1}} u \psi_{l} \,dx \biggr)^{q\theta} \\ &\quad\le Ml^{q(n-2)-q(n+\mu_{2})/{p_{c}} +q\theta(\mu_{2}-p_{c}\mu_{1}-(p_{c}-1)n)/{p_{c}}} \\ &\qquad{} \cdot \biggl( \int_{\mathbb {R}^{n}}\bigl(\vert x\vert +1\bigr)^{\mu_{2}} u^{p_{c}} \psi_{l} \,dx \biggr)^{q(1-\theta)/{p_{c}}} \biggl( \int_{B_{2l}\setminus B_{l}}\bigl(\vert x\vert +1\bigr)^{\mu_{1}} u \psi_{l} \,dx \biggr)^{q\theta}, \end{aligned}$$
$$\begin{aligned} \frac{d\eta_{l}}{dt} \ge{}& \biggl( \int_{\mathbb {R}^{n}}\bigl(\vert x\vert +1\bigr)^{\mu_{2}}u^{p_{c}} \psi_{l} \,dx \biggr)^{q(1-\theta)/{p_{c}}} \\ &{} \cdot \biggl\{ -M_{0}Ml^{(n-1)(1-q)+q(n-2)-q(n+\mu_{2})/{p_{c}} +q\theta[\mu_{2}-p_{c}\mu_{1}-(p_{c}-1)n]/{p_{c}}} \\ &{} \cdot \biggl( \int_{B_{2l}\setminus B_{l}}\bigl(\vert x\vert +1\bigr)^{\mu_{1}} u \psi_{l} \,dx \biggr)^{q\theta} + \biggl( \int_{\mathbb {R}^{n}}\bigl(\vert x\vert +1\bigr)^{\mu_{2}}u^{p_{c}} \psi_{l} \,dx \biggr)^{[p_{c}-q(1-\theta)]/{p_{c}}} \biggr\} \\ \ge{}& \bigl(M_{2}^{-p_{c}}l^{-p_{c}(n+\mu_{1})+(n+\mu_{2})} \bigr)^{q(1-\theta)/{p_{c}}} \eta_{l}^{q(1-\theta)} \\ &{} \cdot \biggl\{ -M_{0}Ml^{(n-1)(1-q)+q(n-2)-q(n+\mu_{2})/{p_{c}} +q\theta[\mu_{2}-p_{c}\mu_{1}-(p_{c}-1)n]/{p_{c}}} \\ &{} \cdot \biggl( \int_{B_{2l}\setminus B_{l}}\bigl(\vert x\vert +1\bigr)^{\mu_{1}} u \psi_{l} \,dx \biggr)^{q\theta} \\ &{} + M_{2}^{q(1-\theta)-p_{c}}\eta_{l}^{p_{c}-q(1-\theta)}l^{[-p_{c}(n+\mu _{1})+(n+\mu_{2})] [p_{c}-q(1-\theta)]/{p_{c}}} \biggr\} . \end{aligned}$$
$$\begin{aligned} &(n-1) (1-q)+q(n-2)-q(n+\mu_{2})/{p_{c}} +q\theta \bigl[ \mu_{2}-p_{c}\mu_{1}-(p_{c}-1)n \bigr]/{p_{c}} \\ &\quad= \bigl[-p_{c}(n+\mu_{1})+(n+\mu_{2}) \bigr] \bigl[p_{c}-q(1-\theta) \bigr]/{p_{c}}. \end{aligned}$$
Lemma 4.3
Assume that
\(p=p_{c}=q+(q+1+\mu_{2})/(n+\mu_{1})\), u
be a nontrivial, radial and nonincreasing solution to the problem (1), (2). Then
holds for some constant
\(M_{4}>0\)
independent of
l.
$$\begin{aligned} \frac{d\eta_{l}}{dt}\ge-{M_{4}}l^{[p_{c}(n-q-1)-q(n+\mu _{2})]/(p_{c}-q)},\quad l>1 \end{aligned}$$
(21)
Proof
From (9), we have Then the Young inequality gives □
$$\begin{aligned} \frac{d\eta_{l}}{dt} \ge{}& \int_{\mathbb {R}^{n}}\bigl(\vert x\vert +1\bigr)^{\mu_{2}}u^{p_{c}} \psi_{l} \,dx \\ &{} -Ml^{(n-1)(1-q)+q[n-2-(n+\mu_{2})/p_{c}]} \biggl( \int_{\mathbb {R}^{n}}\bigl(\vert x\vert +1\bigr)^{\mu_{2}}u^{p_{c}} \psi_{l}\,dx \biggr)^{q/{p_{c}}}. \end{aligned}$$
$$\begin{aligned} \frac{d\eta_{l}}{dt} \ge{}& \int_{\mathbb {R}^{n}}\bigl(\vert x\vert +1\bigr)^{\mu_{2}}u^{p_{c}} \psi_{l} \,dx-\frac {q}{p_{c}} \int_{\mathbb {R}^{n}}\bigl(\vert x\vert +1\bigr)^{\mu_{2}}u^{p_{c}} \psi_{l}\,dx \\ &{} -\frac{p_{c}-q}{p_{c}}M^{{p_{c}}/(p_{c}-q)} l^{[p_{c}(n-q-1)-q(n+\mu_{2})]/(p_{c}-q)} \\ \ge{}&{-}{M_{4}}l^{[p_{c}(n-q-1)-q(n+\mu_{2})]/(p_{c}-q)}. \end{aligned}$$
We are ready to prove the blow-up theorem of Fujita type for the critical case \(p=p_{c}=q+(q+1+\mu_{2})/(n+\mu_{1})\).
Theorem 4.1
Assume that \(p=p_{c}=q+(q+1+\mu_{2})/(n+\mu_{1})\) and u is a solution to the problem (1), (2) with \(0\le u_{0}\in C_{0}(\mathbb {R}^{n})\) nontrivial. Then the problem (1), (2) admits a blow-up solution.
Proof
Similarly to the proof of Theorem 3.1, at first assume \(u_{0}\) is radial and nonincreasing. Then u is radial, given by (6). Denote From (19) and the nontriviality of u, \(0<\varLambda<+\infty\). For any \(0<\sigma<\varLambda\), due to (22) and \(\eta_{l}\) being nondecreasing with respect to \(l\in(1,+\infty)\), there exist \(\omega_{0}\ge0\) and \(l_{0}>2\) such that Then it follows from (21) that Thus Choosing \(l=l_{0}\) in (20), we have Fix \(\sigma_{0}\in(0,\varLambda)\) and \(M_{5}>0\), independent of \(l_{0}\), such that Then where Hence Notice one gets with a positive constant independent of \(l_{0}\). Similarly, we reason The same argument yields with Repeating the procedure, one can show that with Therefore which contradicts (19).
$$\begin{aligned} \varLambda=\sup_{l>1,t>0}\eta_{l}(t)=\sup _{t>0} \int_{\mathbb {R}^{n}}\bigl(\vert x\vert +1\bigr)^{\mu_{1}}u(x,t)\,dx. \end{aligned}$$
(22)
$$\begin{aligned} \eta_{{l_{0}}/2}(\omega_{0})\ge\varLambda-\sigma. \end{aligned}$$
$$\begin{aligned} & \int_{\mathbb {R}^{n}}\bigl(\vert x\vert +1\bigr)^{\mu_{1}}u(x,s) \psi_{{l_{0}}/2}(x)\,dx \\ &\quad\ge \int_{\mathbb {R}^{n}}\vert x\vert ^{\mu_{1}}u(x, \omega_{0}) \psi_{{l_{0}}/2}(x)\,dx -{M_{4}}({l_{0}}/2)^{[p_{c}(n-q-1)-q(n+\mu_{2})]/(p_{c}-q)}(s- \omega_{0}) \\ &\quad\ge\varLambda-\sigma-{M_{4}}({l_{0}}/2)^{[p_{c}(n-q-1)-q(n+\mu _{2})]/(p_{c}-q)}(s- \omega_{0}),\quad s\ge\omega_{0}. \end{aligned}$$
$$\begin{aligned} & \int_{B_{2l_{0}}\setminus B_{l_{0}}}\bigl(\vert x\vert +1\bigr)^{\mu_{1}}u(x,s) \psi_{l_{0}}(x)\,dx \\ &\quad\le \int_{\mathbb {R}^{n}}\bigl(\vert x\vert +1\bigr)^{\mu_{1}}u(x,s)\,dx- \int_{\mathbb {R}^{n}}\bigl(\vert x\vert +1\bigr)^{\mu_{1}}u(x,s) \psi_{{l_{0}}/2}(x)\,dx \\ &\quad\le\sigma+{M_{4}}({l_{0}}/2)^{[p_{c}(n-q-1)-q(n+\mu _{2})]/(p_{c}-q)}(s- \omega_{0}),\quad s\ge\omega_{0}. \end{aligned}$$
$$\begin{aligned} \frac{d\eta_{l_{0}}}{dt} \ge{}& M_{2}^{-q(1-\theta)}l_{0}^{(n+\mu _{2})-p_{c}(n+\mu_{1})} \eta_{l_{0}}^{q(1-\theta)} \\ &{} \cdot \biggl\{ -M_{3} \biggl( \int_{B_{2l_{0}}\setminus B_{l_{0}}}\bigl(\vert x\vert +1\bigr)^{\mu_{1}}u \psi_{l_{0}}\,dx \biggr)^{q\theta}+ M_{2}^{q(1-\theta)-p_{c}} \eta_{l_{0}}^{p_{c}-q(1-\theta)} \biggr\} \\ \ge{}& M_{2}^{-q(1-\theta)}l_{0}^{(n+\mu_{2})-p_{c}(n+\mu_{1})}\eta _{l_{0}}^{q(1-\theta)} \\ &{} \cdot \bigl\{ -M_{3} \bigl(\sigma +{M_{4}}({l_{0}}/2)^{[p_{c}(n-q-1)-q(n+\mu_{2})]/(p_{c}-q)}(s- \omega_{0}) \bigr\} \bigr)^{q\theta} \\ &{} +M_{2}^{q(1-\theta)-p_{c}}\eta_{l_{0}}^{p_{c}-q(1-\theta )} \},\quad t>\omega_{0}. \end{aligned}$$
$$\begin{aligned} M_{3} (\sigma_{0}+M_{5} )^{q\theta}\le \frac{1}{2} M_{2}^{q(1-\theta)-p_{c}} (\varLambda- \sigma_{0})^{p_{c}-q(1-\theta)}. \end{aligned}$$
$$\begin{aligned} \frac{d\eta_{l_{0}}}{dt} \ge\frac{1}{2} M_{2}^{-p_{c}}l_{0}^{(n+\mu _{2})-p_{c}(n+\mu_{1})} \eta_{l_{0}}^{p_{c}},\quad \omega_{0}< t< \omega_{1}, \end{aligned}$$
$$\begin{aligned} \omega_{1}=\omega_{0}+\frac{M_{5}}{M_{4}}({l_{0}}/2)^{-[p_{c}(n-q-1)-q(n+\mu _{2})]/(p_{c}-q)}. \end{aligned}$$
$$\begin{aligned} \eta_{l_{0}}(\omega_{1}) \ge{}&\eta_{l_{0}}( \omega_{0})+\frac{1}{2} M_{2}^{-p_{c}}l_{0}^{(n+\mu_{2})-p_{c}(n+\mu_{1})} (\varLambda-\sigma_{0})^{p_{c}}(\omega_{1}- \omega_{0}) \\ \ge{}&\eta_{l_{0}}(\omega_{0})+\frac{1}{2} M_{2}^{-p_{c}}l_{0}^{(n+\mu _{2})-p_{c}(n+\mu_{1})} (\varLambda- \sigma_{0})^{p_{c}} \\ &{} \cdot\frac{M_{5}}{M_{4}}({l_{0}}/2)^{-[p_{c}(n-q-1)-q(n+\mu_{2})]/(p_{c}-q)}. \end{aligned}$$
$$\begin{aligned} (n+\mu_{2})-p_{c}(n+\mu_{1})- \bigl[p_{c}(n-q-1)-q(n+\mu_{2}) \bigr]/(p_{c}-q)=0, \end{aligned}$$
$$\begin{aligned} \int_{\mathbb {R}^{n}}\bigl(\vert x\vert +1\bigr)^{\mu_{1}} u(x, \omega_{1})\,dx \ge\eta_{l_{0}}(\omega_{1}) \ge \eta_{l_{0}}( \omega_{0})+\delta_{0} \ge \varLambda-\sigma_{0}+ \delta_{0}, \end{aligned}$$
$$\begin{aligned} \delta_{0}=\frac{M_{2}^{-p_{c}}M_{5}}{2M_{4}}(\varLambda-\sigma _{0})^{p_{c}}2^{[p_{c}(n-q-1)-q(n+\mu_{2})]/(p_{c}-q)} \end{aligned}$$
$$\begin{aligned} \eta_{(2l_{0})/2}(\omega_{1})=\eta_{l_{0}}( \omega_{1})\ge\varLambda-\sigma_{0}+\delta_{0} \ge\varLambda-\sigma_{0}. \end{aligned}$$
$$\begin{aligned} \int_{\mathbb {R}^{n}}\bigl(\vert x\vert +1\bigr)^{\mu_{1}}u(x,t_{2})\,dx \ge \eta_{2l_{0}}(\omega_{2})\ge\eta_{2l_{0}}( \omega_{1})+\delta_{0} \ge\varLambda-\sigma_{0}+2 \delta_{0} \end{aligned}$$
$$\begin{aligned} \omega_{2}=\omega_{1}+\frac{M_{5}}{M_{4}}{l_{0}}^{-[p_{c}(n-q-1)-q(n+\mu_{2})]/(p_{c}-q)}. \end{aligned}$$
$$\int_{\mathbb {R}^{n}}\bigl(\vert x\vert +1\bigr)^{\mu_{1}}u(x, \omega_{k})\,dx \ge\eta_{2^{k-1}l_{0}}(\omega_{k}) \ge \eta_{2^{k-1}l_{0}}(\omega _{k-1})+\delta_{0} \ge\varLambda- \sigma_{0}+k \delta_{0} $$
$$\begin{aligned} \omega_{k}=\omega_{k-1}+\frac {M_{5}}{M_{4}} \bigl(2^{k-2}l_{0} \bigr)^{-[p_{c}(n-q-1)-q(n+\mu_{2})]/(p_{c}-q)},\quad k=1, 2, \ldots. \end{aligned}$$
$$\begin{aligned} \sup_{t>0} \int_{\mathbb {R}^{n}}\bigl(\vert x\vert +1\bigr)^{\mu_{1}} u(x,t)\,dx=+\infty, \end{aligned}$$
Now, for the general case without the assumption that \(u(x,t)\) is nonincreasing, consider \(\underline{u}(x,t)\) defined by (12), which is nonincreasing and satisfies (13) and (14). Therefore, the conclusions of Lemmas 4.1-4.3 are all valid for \(\underline{u}\). Similar to the above argument, one can show that \(\underline{u}\) blows up in some \(0< T_{*}<+\infty\), and thus u is a blow-up solution. □
Declarations
Acknowledgements
This work is supported by the National Natural Science Foundation of China (Grant Nos. 11222106 and 11571137).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
(1)
School of Mathematics, Jilin University, Changchun, China
(2)
School of Mathematics and Statistics, Beihua University, Jilin, China
(3)
College of Computer Science and Technology, Jilin University, Changchun, China
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