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Boundary Value Problems
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Bounds for the blow-up time of a porous medium equation with weighted nonlocal source and inner absorption terms

Boundary Value Problems20182018:116

https://doi.org/10.1186/s13661-018-1038-3

©  The Author(s) 2018

  • Received: 15 March 2018
  • Accepted: 12 July 2018
  • Published:

Abstract

We investigate the blow-up phenomena for a porous medium equation with weighted nonlocal source and inner absorption terms subject to null Dirichlet boundary condition. Based on a modified differential inequality technique, we establish some sufficient conditions to guarantee the existence of non-global solutions to the model and also derive the upper bounds for the blow-up time. Moreover, the lower bounds for the blow-up time are obtained under some appropriate measure in the whole-dimensional space (\(N\geq1\)).

Keywords

  • Porous medium equation
  • Weighted nonlocal source
  • Inner absorption
  • Bounds for the blow-up time

MSC

  • 35K59
  • 35B30
  • 35B40

1 Introduction

We consider the porous medium equation with weighted nonlocal source and inner absorption terms
$$ u_{t} = \Delta u^{m} + a ( x )u^{p} \int_{\Omega}{u^{q} \,dx} - u^{s} , \quad ( {x,t} ) \in\Omega\times \bigl(0,t^{\ast} \bigr), $$
(1)
subject to null Dirichlet boundary and initial conditions
$$\begin{aligned}& u (x,t ) = 0,\quad ( {x,t} ) \in\partial \Omega \times \bigl( {0,t^{*} } \bigr), \end{aligned}$$
(2)
$$\begin{aligned}& u ( {x,0} ) = u_{0} ( x ) \ge0,\quad x \in \Omega, \end{aligned}$$
(3)
where \(\Omega\subset\mathbb{R}^{N}\) (\(N\geq1\)) is a bounded region with smooth boundary \(\partial\Omega, m, p, q>0\), \(s\geq1\). \(t^{\ast}\) is a possible blow-up time when blow-up occurs, otherwise \(t^{\ast}=+\infty\). The weight function \(a ( x ) \in C^{1} ( \Omega ) \cap C^{0} ( {\bar{\Omega}} )\) satisfies
(\(a_{1}\)): 

\(a ( x ) \ge a > 0\) for all \(x \in\bar{\Omega}\), where a is a positive constant,

or
(\(a_{2}\)): 

\(a ( x ) > 0 \), \(x \in\Omega\), and \(a ( x ) = 0\), \(x \in\partial\Omega\).

Moreover, the initial data \(u_{0}(x)\) is a positive \(C^{1}\)-function which satisfies a compatibility condition. Therefore, by the parabolic theory, it follows that the local weak solution to problem (1)–(3) exists uniquely, and is nonnegative. For convenience, we may assume that the appropriate weak solution is smooth, and no longer consider approximation problem.

Equation (1) describes the diffusion of concentration of some Newtonian fluids through porous medium or the density of some biological species in many physical phenomena and biological species theories (cf. [1–6]). It has been known that the nonlocal source term presents a more realistic model for population dynamics, see [1–3].

During the past decades, there have been many works to deal with the existence and nonexistence of global solutions, blow-up of solutions, bounds for the blow-up time, blow-up rates, blow-up sets, and asymptotic behavior of the solutions to the parabolic equations. We refer the readers to the monographs [7–9] as well as the survey paper [10] and the references therein. Roughly, the existence and nonexistence of global solutions and behavior of the solutions to parabolic equations depend on space dimension, nonlinearity, initial data, and nonlinear boundary flux. Specially, Quittner and Souplet [9, Chap. 5] introduced the qualitative properties of the solution to nonlocal semilinear parabolic equation with homogeneous Dirichlet boundary condition in detail. In a sense, the nonlocal models are closer to the practical problems than the local ones, and now many local theories are no longer holding, hence, nonlocal problems are more challenging and difficult. In this paper, we would like to investigate the blow-up phenomena for the solution to a class of nonlocal problems, and our main aim is to derive the bounds for blow-up time if blow-up occurs in finite time. As far as we know, a variety of methods have been used to investigate the upper bounds for the blow-up time to the above problems (cf. [11]), while the lower bounds for the blow-up time may be harder to be determined and fewer methods can be chosen to deal with them. Recently, the study on the lower bounds for blow-up time has made some progress, while more attention was paid to the local model with constant coefficients. We provide the readers with the literature [12–14] for a three-dimensional case and [15] for a high-dimensional case.

For some research on the nonlocal reaction–diffusion equations with constant coefficients, Song [16] considered the semilinear parabolic equation with nonlocal source and local absorption
$$u_{t} = \Delta u + \int_{\Omega}{u^{q}\, {{d}}x} - ku^{s} ,\quad (x,t) \in\Omega \times \bigl( {0,t^{*} } \bigr), $$
under homogeneous Dirichlet or Neumann boundary condition, and obtained the bounds for blow-up time of the solution to the initial boundary value problem in a three-dimensional space. Later, Liu [17] considered the problem with nonlinear Neumann boundary condition and derived the lower bounds for blow-up time of the blow-up solution in a three-dimensional space. Recently, Tang et al. [18] generalized the results in [16] to the case of a high-dimensional space. Liu et al. [19] studied the porous medium equation with nonlocal source term
$$u_{t} = \Delta u^{m} + \int_{\Omega}{u^{q} \, {{d}}x} ,\quad ( {x,t} ) \in \Omega \times \bigl( {0,t^{*} } \bigr), $$
under homogeneous Dirichlet or Neumann boundary condition. They obtained the lower bounds for blow-up time of the blow-up solution to the initial boundary value problem in a three-dimensional space. Specially, Fang et al. [20] studied
$$u_{t} = \Delta u^{m} + u^{p} \int_{\Omega}{u^{q} \,dx} - u^{s} ,\quad ( {x,t} ) \in\Omega \times \bigl( {0,t^{*} } \bigr), $$
under homogeneous Dirichlet or Neumann boundary condition and obtained the lower bounds for blow-up time of the blow-up solution in a three-dimensional space. Bao and Song [21] considered the initial boundary value problem of quasilinear parabolic equation under homogeneous Dirichlet or Neumann boundary condition, and the slow diffusion case with nonlocal source term was also included in their results. Besides, one can refer to [22–27] for the results about scalar equation with time-dependent coefficients, nonlocal reaction systems, and models of quasilinear equations.
For some research on the nonlocal reaction–diffusion equations with weight functions, Song and Lv [28, Sect. 5] studied the semilinear parabolic equation with weighted inner source and absorption
$$u_{t} = \Delta u + a ( x ) \int_{\Omega}{u^{q} \,dx} - u^{s} ,\quad ( {x,t} ) \in\Omega \times \bigl( {0,t^{*} } \bigr), $$
where the weight function \(a ( x ) \in C^{1} ( \Omega ) \cap C^{0} ( {\bar{\Omega}} )\) satisfies (\(a_{1}\)) or (\(a_{2}\)). They considered the problem under homogeneous Dirichlet or Neumann boundary condition and obtained the estimate for the blow-up rate and bounds for the blow-up time of solution to the initial boundary value problem in a high-dimensional space (\(N \ge3\)). However, their results did not include the influence of weight functions on the blow-up phenomenon. Besides, one can refer to [29–31] for more results about reaction–diffusion models with space-dependent coefficients.

In view of the works mentioned above, there are few results about bounds for the blow-up time of blow-up solution to the initial boundary value problem (1)–(3). The main difficulties are to seek the competitive relationship among nonlinear diffusion term, source term, and absorption, as well as to investigate the influence of space dimension and weight functions on the blow-up solution. Motivated by these observations, using a modified differential inequality technique, we can establish sufficient conditions for the blow-up of solution to problem (1)–(3) under appropriate measure and obtain the upper bounds for the blow-up time. Meanwhile, we can also derive the lower bounds for the blow-up time of blow-up solution in the whole-dimensional space (\({N \ge1}\)). Indeed, for the case \(p + q \le\max \{ {m,s} \}\), we can easily construct the global supersolution for the solution to problem (1)–(3), so we only consider the case \(p + q > \max \{ {m,s} \}\) in our paper.

Our paper is organized as follows. In Sect. 2, we assume some conditions on the weight function \(a(x)\) to guarantee that the solution to problem (1)–(3) blows up in finite time under appropriate measure, and derive the upper bounds for blow-up time. In Sect. 3, we obtain the lower bounds for blow-up time of the solution to problem (1)–(3) in the whole-dimensional space.

2 Upper bounds for the blow-up time

In this section, we establish some sufficient conditions for the solution to problem (1)–(3) to blow up in finite time under different measures, and then derive the upper bounds for the blow-up time.

We firstly give a sufficient condition for the solution to problem (1)–(3) to blow up in \(L^{1}\)-norm and establish an upper bound for the blow-up time.

Theorem 1

Suppose that \(m>1\), \(\min\{p,q\}>s\), and the weight function \(a(x)\) satisfies (\(a_{1}\)). If u is the nonnegative solution to problem (1)–(3), then u blows up in \(L^{1}\)-norm, and an upper bound for \(t^{*} \) is
$$\int_{J_{1} ( 0 )}^{\infty}{\frac{{d\eta}}{{M_{1} \eta^{p + q} - M_{2} }}} , $$
where \(J_{1} ( 0 ) = \int_{\Omega}{u_{0} ( x )\,dx} \), the initial data \(u_{0} ( x )\) is large enough and positive constants \(M_{1}\), \(M_{2}\) will be given in the proof.

Proof

Define
$$J_{1} ( t ) = \int_{\Omega}{u\,dx} . $$
Compute the derivative and use Green’s formula and Hölder’s inequality to derive
$$\begin{aligned} J_{1} ^{\prime}( t ) &= \int_{\Omega}{u_{t} \,dx} \\ &= \int_{\Omega}{\Delta u^{m} \,dx} + \int_{\Omega}{a ( x )u^{p} \,dx} \int_{\Omega}{u^{q} \,dx} - \int_{\Omega}{u^{s} \,dx} \\ & \ge a |\Omega|^{\frac{2s-(p+q)}{s}} \biggl( { \int_{\Omega}{u^{s} \,dx} } \biggr)^{\frac{p+q}{s}} - \int_{\Omega}u^{s} \,dx \\ & ={ \int_{\Omega}{u^{s} \,dx} } \biggl[ a| \Omega|^{\frac {2s-(p+q)}{s}} \biggl( { \int_{\Omega}{u^{s} \,dx} } \biggr)^{\frac{p+q-s}{s}} - 1 \biggr]. \end{aligned}$$
(4)
Obviously, since \(p+q>s\), we can get that the function \(f ( \eta ) = \eta^{\frac{p + q - s }{{s}}} \) is monotone increasing and if \(u_{0} ( x )\) satisfies
$$ a|\Omega|^{\frac{2s-(p+q)}{s}} \biggl( { \int_{\Omega}{u_{0}^{s} \,dx} } \biggr)^{\frac{p+q-s}{s}}>1, $$
(5)
then we can know that the solution to problem (1)–(3) blows up in finite time.
On the other hand, by (4) and Young’s and Hölder’s inequalities, we can derive
$$\begin{aligned} J_{1} ^{\prime}( t ) & \geq a |\Omega|^{\frac{2s-(p+q)}{s}} \biggl( { \int_{\Omega}{u^{s} \,dx} } \biggr)^{\frac{p+q}{s}} - \int_{\Omega}u^{s} \,dx \\ &= a |\Omega|^{\frac{2s-(p+q)}{s}} \biggl( { \int_{\Omega}{u^{s} \,dx} } \biggr)^{\frac{p+q}{s}} - \biggl( \delta_{1} \biggl( \int_{\Omega}u^{s} \,dx \biggr)^{\frac{p+q}{s}} \biggr)^{\frac{s}{p+q}} \bigl(\delta_{1}^{-\frac{s}{p+q-s}} \bigr)^{\frac {p+q-s}{p+q}} \\ &\ge \biggl[a|\Omega|^{\frac{2s-(p+q)}{s}} -{\frac{s\delta_{1}}{p+q}} \biggr] \biggl( \int_{\Omega}u^{s} \,dx \biggr)^{\frac{p+q}{s}} -{ \frac {p+q-s}{p+q}}\delta_{1} ^{-\frac{s}{p+q-s}} \\ &\ge \biggl[a|\Omega|^{\frac{2s-(p+q)}{s}} -{\frac{s\delta_{1}}{p+q}} \biggr] | \Omega|^{\frac{(1-s)(p+q)}{s}} \biggl( \int_{\Omega}u \,dx \biggr)^{p+q} -{\frac{p+q-s}{p+q}} \delta_{1} ^{-\frac{s}{p+q-s}}. \end{aligned}$$
(6)
Choosing \(\delta_{1} \) satisfies \(0 < \delta_{1} < \frac{{a ( {p + q} )}}{s}|\Omega|^{\frac{2s-(p+q)}{s}}\), then (6) can be rewritten as
$$ J_{1} ^{\prime}( t ) \ge M_{1} J_{1}^{p + q} - M_{2} , $$
(7)
where
$$M_{1} = \biggl(a|\Omega|^{\frac{2s-(p+q)}{s}} -{\frac{s\delta_{1}}{p+q}} \biggr) | \Omega|^{\frac{(1-s)(p+q)}{s}} > 0,\qquad M_{2} = \frac{{p + q - s}}{{p + q}} \delta_{1}^{ - \frac{s}{{p + q - s}}} > 0. $$
Hence, if \(u_{0} ( x )\) is large enough satisfying \(J_{1} ( 0 ) = \int_{\Omega}{u_{0} ( x )\,dx} > ( {\frac{{M_{2} }}{{M_{1} }}} )^{\frac{1}{{p + q}}} \) and (5), by virtue of (7), we can derive that the blow-up time \(t^{*}\) satisfies
$$t^{*} \le \int_{J_{1} ( 0 )}^{\infty}\frac{{d\eta}}{{M_{1} \eta^{p + q} - M_{2} }}. $$
 □

Next we will use Kaplan’s method to investigate the upper bound for blow-up time of blow-up solution to problem (1)–(3).

Denote by \(\lambda_{1} \) and \(\phi_{1} \) the first eigenvalue and the corresponding eigenfunction of the following fixed membrane problem:
$$\begin{aligned}& \Delta\phi_{1} + \lambda_{1} \phi_{1} = 0, \quad x \in\Omega, \end{aligned}$$
(8)
$$\begin{aligned}& \phi_{1} ( x ) = 0, \quad x \in\partial\Omega. \end{aligned}$$
(9)
It is well known that \(\phi_{1} \) may be normalized as \(\sup_{\Omega} \vert {\phi _{1} } \vert = 1\).

Theorem 2

Suppose that \(1\leq m< s<\min\{p,q\}\), and the weight function \(a(x)\) satisfies \(( {a_{1} } )\). Define an auxiliary function
$$J_{2} ( t ) = \int_{\Omega}{u\phi_{1} \,dx} . $$
If u is the nonnegative solution to problem (1)–(3), then u blows up in the measure of \(J_{2}\), and an upper bound for \(t^{*} \) is
$$\int_{J_{2} ( 0 )}^{\infty}{\frac{{d\eta}}{{M_{3} \eta^{p+q} - M_{4} }}} , $$
where \(J_{2} ( 0 ) = \int_{\Omega}u_{0} ( x )\phi_{1} ( x )\,dx\), the initial data \(u_{0} ( x )\) is large enough and positive constants \(M_{3}\), \(M_{4}\) will be given in the proof.

Proof

We compute the derivative and use Green’s formula to obtain
$$\begin{aligned} J_{2} ^{\prime}( t ) &= \int_{\Omega}{u_{t} \phi_{1} \,dx} \\ &= \int_{\Omega}{\Delta u^{m} \phi_{1} \,dx} + \int_{\Omega}{a ( x )u^{p} \phi_{1} \,dx} \int_{\Omega}{u^{q} \,dx} - \int_{\Omega}{u^{s} \phi_{1} \,dx} \\ &\ge - \lambda_{1} \int_{\Omega}{u^{m} \phi_{1} \,dx} + a { \int_{\Omega}{u^{ p} \phi_{1} \,dx} } \int_{\Omega}u^{q} \phi_{1} \,dx - \int_{\Omega}{u^{s} \phi_{1} \,dx} . \end{aligned}$$
(10)
Notice that \(m< s<\min\{p,q \}\), by Hölder’s inequality to estimate the first and second terms on the right-hand side of (10), we have
$$\begin{aligned} & \int_{\Omega}u^{m} \phi_{1} \,dx\leq \biggl( \int_{\Omega}u^{s} \phi_{1} \,dx \biggr)^{\frac{m}{s}} \biggl( \int_{\Omega}\phi_{1} \,dx \biggr)^{\frac{s-m}{s}} \leq \biggl( \int_{\Omega}u^{s} \phi_{1} \,dx \biggr)^{\frac{m}{s}} |\Omega |^{\frac{s-m}{s}}, \\ & \int_{\Omega}u^{p} \phi_{1} \,dx \int_{\Omega}u^{q} \phi_{1} \,dx \geq \biggl( \int _{\Omega}u^{s} \phi_{1} \,dx \biggr)^{\frac{p+q}{s}} \biggl( \int_{\Omega}\phi_{1} \,dx \biggr)^{\frac{2s-(p+q)}{s}} \\ & \hphantom{\int_{\Omega}u^{p} \phi_{1} \,dx \int_{\Omega}u^{q} \phi_{1} \,dx}\geq \biggl( \int_{\Omega}u^{s} \phi_{1} \,dx \biggr)^{\frac{p+q}{s}} |\Omega|^{\frac{2s-(p+q)}{s}}. \end{aligned}$$
(11)
Substituting (11) into (10) and using Young’s inequality, we obtain
$$\begin{aligned} J_{2} ^{\prime}( t ) \geq& a|\Omega|^{\frac{2s-(p+q)}{s}} \biggl( \int_{\Omega}u^{s} \phi_{1} \,dx \biggr)^{\frac{p+q}{s}} -\lambda_{1} |\Omega|^{\frac{s-m}{s}} \biggl( \int_{\Omega}u^{s} \phi_{1} \,dx \biggr)^{\frac{m}{s}} \\ &{} - \int_{\Omega}u^{s} \phi_{1} \,dx \\ \geq& a|\Omega|^{\frac{2s-(p+q)}{s}} \biggl( \int_{\Omega}u^{s} \phi_{1} \,dx \biggr)^{\frac{p+q}{s}} - \biggl(\frac{m}{s} \lambda_{1}^{\frac{s}{m}} +1 \biggr) \int_{\Omega}u^{s} \phi_{1} \,dx - \frac {s-m}{s}|\Omega|. \end{aligned}$$
(12)
By the property of quadratic function and \(a>0 \), if \(u_{0} ( x )\) is large enough, we can derive that the solution to problem (1)–(3) blows up in finite time.
Now, applying the same argument in the proof of Theorem 2 to (12), we obtain
$$ J_{2} ^{\prime}( t ) \ge \biggl( {a| \Omega|^{\frac {2s-(p+q)}{s}} - \frac{{m\delta_{2}+s\delta_{3} }}{{p + q}} } \biggr) \biggl( { \int_{\Omega}{u^{s} \phi_{1} \,dx} } \biggr)^{\frac {p+q}{s}} -M_{4}. $$
(13)
Choosing \(\delta_{2} \), \(\delta_{3} \) satisfies \(0 < \delta _{2} < \frac{{ ( {p + q} )a|\Omega|^{\frac {2s-(p+q)}{s}}}}{2m}\) and \(0 < \delta_{3} < \frac{{ ( {p + q} )a|\Omega|^{\frac{2s-(p+q)}{s}}}}{2s}\), using Hölder’s inequality, (13) can be rewritten as
$$ J_{2} ^{\prime}( t ) \ge M_{3} J_{2}^{p + q} - M_{4} , $$
(14)
where
$$\begin{aligned} &M_{3} = \biggl( {a|\Omega|^{\frac{2s-(p+q)}{s}} - \frac{{m\delta _{2}+s\delta_{3} }}{ {p + q}} } \biggr)|\Omega|^{\frac{(1-s)(p+q)}{s}}> 0, \\ &M_{4} = {\frac{p+q-m}{p+q}} \delta_{2}^{- \frac{m}{p+q-m}} \lambda_{1} ^{\frac{p+q}{p+q-m}} |\Omega|^{\frac{(s-m)(p+q)}{s(p+q-m)}} +{\frac{p+q-s}{p+q}} \delta_{3}^{-\frac{s}{p+q-s}}> 0. \end{aligned}$$
Hence, if \(u_{0} ( x )\) is large enough and satisfies \(J_{2} ( 0 ) = \int_{\Omega}{u_{0} ( x )\phi_{1} ( x )\,dx} > ( {\frac{{M_{4} }}{{M_{3} }}} )^{\frac{1}{{p + q}}}\) and the above requirement, by (14), we obtain
$$t^{*} \le \int_{J_{2} ( 0 )}^{\infty}\frac{{d\eta}}{{M_{3} \eta^{p + q} - M_{4} }}. $$
 □

Afterwards, we will seek a sufficient condition for the solution to problem (1)–(3) to blow up in \(L^{2m}\)-norm, and then obtain an upper bound for the blow-up time.

Theorem 3

Suppose that \(m > 1\), \(p \ge s > 1\), and the weight function \(a ( x )\) satisfies (\({a_{1}}\)) or (\({a_{2}}\)). Define auxiliary functions
$$\begin{aligned}& F \bigl( {u^{m} } \bigr) = \int_{0}^{u^{m} } {\xi^{1 + \frac{{p - 1}}{m}} \biggl( { \int_{\Omega}\xi ^{\frac{q}{m}} \,dx} \biggr)\,d\xi} , \\& G \bigl( {u^{m} } \bigr) = \int_{0}^{u^{m} } {\xi^{1 + \frac{{s - 1}}{m}} \,d\xi} = \frac{m}{{2m + s - 1}}u^{2m + s - 1} . \end{aligned}$$
Assume that there exist \(I_{1} \ge I_{2} > 2\) such that
$$ u^{2m + p - 1} \int_{\Omega}{u^{q} \,dx} \ge I_{1} F \bigl( {u^{m} } \bigr), \qquad u^{2m + s - 1} \le I_{2} G \bigl( {u^{m} } \bigr). $$
(15)
If u is the nonnegative solution to problem (1)–(3) and the initial data \(u_{0} ( x )\) satisfies
$$- \frac{1}{2} \int_{\Omega}{\nabla u_{0}^{2m - 1} \nabla u_{0}^{m} } \,dx + \int _{\Omega}{a ( x )F \bigl( {u_{0}^{m} } \bigr)\,dx - \int_{\Omega}{G \bigl( {u_{0}^{m} } \bigr) \,dx} } > 0, $$
then u blows up in \(L^{2m}\)-norm, and an upper bound for \(t^{*} \) is
$$\frac{{2\xi ( 0 )}}{{I_{2} ( {I_{2} - 2} )\eta ( 0 )}}, $$
where functions \(\xi,\eta\) will be given in the proof.

Remark 1

Since \(p \ge s > 1\), we can choose \(I_{1} = \frac{{2m + p - 1}}{m}\), \(I_{2} = \frac{{2m + s - 1}}{m}\), which satisfy condition (15).

Proof

Applying the transformation \(v = u^{m} \) in (1), we have
$$ \bigl( {v^{\frac{1}{m}} } \bigr)_{t} = \Delta v + a ( x )v^{\frac{p}{m}} \int_{\Omega}{v^{\frac{q}{m}} \,dx} - v^{\frac{s}{m}} . $$
(16)
Then (16) is equivalent to
$$ v_{t} = mv^{1 - \frac{1}{m}} \biggl( {\Delta v + a ( x )v^{\frac{p}{m}} \int_{\Omega}{v^{\frac{q}{m}} \,dx} - v^{\frac{s}{m}} } \biggr). $$
(17)
Define
$$ \xi ( t ) = \int_{\Omega}{u^{2m} \,dx} = \int_{\Omega}{v^{2} \,dx} . $$
(18)
Differentiating \(\xi ( t )\) and using (15), (17) and Green’s formula, we obtain
$$\begin{aligned} \xi' ( t ) =& 2 \int_{\Omega}{vv_{t} } \,dx \\ =& - 2m \int_{\Omega}{\nabla v^{2 - \frac{1}{m}} \cdot\nabla v} \,dx + 2m \int_{\Omega}{a ( x )v^{2 + \frac{{p - 1}}{m}} \,dx \int_{\Omega}{v^{\frac {q}{m}} \,dx} } \\ &{} - 2m \int_{\Omega}{v^{2 + \frac{{s - 1}}{m}} \,dx} \\ \ge& - I_{2} m \int_{\Omega}{\nabla v^{2 - \frac{1}{m}} \cdot\nabla v} \,dx + 2I_{1} m \int_{\Omega}{a ( x )F ( v )\,dx - 2I_{2} m \int_{\Omega}{G ( v )\,dx} } \\ \ge&2I_{2} \eta ( t ), \end{aligned}$$
(19)
where \(\eta ( t ): = - \frac{m}{2}\int_{\Omega}{\nabla v^{2 - \frac{1}{m}} \cdot\nabla v} \,dx + m\int_{\Omega}{a ( x )F ( v )\,dx - m\int_{\Omega}{G ( v )\,dx} } \).
Next, differentiating \(\eta ( t )\) and using Green’s formula and \(m > 1\), we have
$$\begin{aligned} \eta' ( t ) =& - \frac{m}{2} \int_{\Omega}{\nabla \bigl( {v^{2 - \frac{1}{m}} } \bigr)_{t} \cdot\nabla v} \,dx - \frac{m}{2} \int_{\Omega}{\nabla v^{2 - \frac{1}{m}} \cdot\nabla v_{t} } \,dx \\ &{} + m \int_{\Omega}{a ( x )v^{1 + \frac{{p - 1}}{m}} \biggl( { \int_{\Omega}{v^{\frac{q}{m}} \,dx} } \biggr)v_{t} \,dx}- m \int_{\Omega}{v^{1 + \frac{{s - 1}}{m}} v_{t} \,dx} \\ =& m \biggl( {2 - \frac{1}{m}} \biggr) \int_{\Omega}{v^{1 - \frac{1}{m}} v_{t} \Delta v \,dx} \\ &{} + \frac{m}{2} \biggl( {2 - \frac{1}{m}} \biggr) \biggl( {1 - \frac{1}{m}} \biggr) \int_{\Omega}{v^{ - \frac{1}{m}} \vert {\nabla v} \vert ^{2} v_{t} \,dx} \\ &{} + m \int_{\Omega}{a ( x )v^{1 + \frac{{p - 1}}{m}} \biggl( { \int_{\Omega}{v^{\frac{q}{m}} \,dx} } \biggr)v_{t} \,dx} - m \int_{\Omega}{v^{1 + \frac{{s - 1}}{m}} v_{t} \,dx} \\ \ge& m \int_{\Omega}{v^{1 - \frac{1}{m}} \Delta v \cdot v_{t} \,dx} + m \int_{\Omega}{a ( x )v^{1 + \frac{{p - 1}}{m}} \biggl( { \int_{\Omega}{v^{\frac{q}{m}} \,dx} } \biggr)v_{t} \,dx} \\ &{} - m \int_{\Omega}{v^{1 + \frac{{s - 1}}{m}} v_{t} \,dx} \\ =& \int_{\Omega}{ ( {v_{t} } )^{2} } \,dx \ge0. \end{aligned}$$
(20)
By the hypotheses in Theorem 3, we can easily see that \(\eta ( 0 ) > 0\), and then \(\eta ( t ) > 0\), \(\forall t \ge0\). Therefore, (19) implies that ξ is monotone increasing, and then v blows up in finite time \(t^{*} \) in \(L^{2}\)-norm.
Combining (18)–(20) and using Schwarz’s inequality
$$\biggl( { \int_{\Omega}{vv_{t} \,dx} } \biggr)^{2} \le \int_{\Omega}{v^{2} \,dx} \int_{\Omega}{ ( {v_{t} } )^{2} \,dx} , $$
we can derive
$$ 4\xi ( t )\eta' ( t ) \ge \bigl( {\xi' ( t )} \bigr)^{2} \ge2I_{2} \xi' ( t )\eta ( t ). $$
(21)
Applying (19) and (21), we can compute
$$\eta ( t ) \ge\frac{{\eta ( 0 )}}{{ ( {\xi ( 0 )} )^{\frac{{I_{2} }}{2}} }} \bigl( {\xi ( t )} \bigr)^{\frac{{I_{2} }}{2}} $$
and
$$ \xi' ( t ) \ge\frac{{2I_{2} \eta ( 0 )}}{{ ( {\xi ( 0 )} )^{\frac{{I_{2} }}{2}} }} \bigl( {\xi ( t )} \bigr)^{\frac{{I_{2} }}{2}} . $$
(22)
Hence, (22) leads to
$$t^{*} \le\frac{{\xi ( 0 )}}{{I_{2} ( {I_{2} - 2} )\eta ( 0 )}}. $$
 □

Remark 2

In the fast and linear diffusion situation (\({0 < m \le1}\)), we set
$$F ( u ) = \int_{0}^{u} {\xi^{p} \biggl( { \int_{\Omega}{\xi^{q} \,dx} } \biggr)\,d\xi} ,\qquad G ( u ) = \int_{0}^{u} {\xi^{s} \,ds} = \frac{1}{{s + 1}}u^{s + 1} . $$
Suppose that there exist \(I_{3} \ge I_{4} > 2\) such that
$$u^{p + 1} \int_{\Omega}{u^{q} \,dx} \ge I_{3} F ( u ),\qquad u^{s + 1} \le I_{4} G ( u ). $$
If u is a nonnegative solution to problem (1)–(3) and the initial data \(u_{0} ( x )\) satisfies
$$- \frac{1}{2} \int_{\Omega}{\nabla u_{0} \nabla u_{0}^{m} } \,dx + \int_{\Omega}{a ( x )F} ( {u_{0} } )\,dx - \int_{\Omega}{G ( {u_{0} } )\,dx} > 0, $$
then u blows up in \(L^{2}\)-norm, and an upper bound for \(t^{*} \) is
$$t^{*} \le\frac{{\chi ( 0 )}}{{I_{4} ( {I_{4} - 2} )\zeta ( 0 )}}, $$
where \(\chi ( t ) = \int_{\Omega}{u^{2} \,dx}\), \(\zeta ( t ) = - \frac{1}{2}\int_{\Omega}\nabla u\nabla u^{m} \,dx + \int_{\Omega}{a ( x )F ( u )\,dx} - \int_{\Omega}{G ( u )\,dx} \).

The proof is the same as Theorem 3, so we omit it.

3 Lower bounds for the blow-up time

In this section, we seek the lower bounds for the blow-up time of the solution to problem (1)–(3) in an N-dimensional space \(\Omega \subset{\mathbf{R}}^{N}\) (\({N \ge1}\)).

3.1 \(N=1\) case

Suppose \(\Omega = ( {0,l} )\), \(N = 1\), then problem (1)–(3) can be rewritten as
$$\begin{aligned}& u_{t} = \bigl( {u^{m} } \bigr)_{xx} + a ( x )u^{p} \int_{0}^{l} {u^{q} \,dx - u^{s} } ,\quad x \in ( {0,l} ),t \in \bigl( {0,t^{*} } \bigr), \end{aligned}$$
(23)
$$\begin{aligned}& u ( {0,t} ) = u ( {l,t} ) = 0,\quad t \in \bigl( {0,t^{*} } \bigr), \end{aligned}$$
(24)
$$\begin{aligned}& u ( {x,0} ) = u_{0} ( x ) \ge0,\quad x \in ( {0,l} ). \end{aligned}$$
(25)

Theorem 4

Suppose that \(m > 0\), \(0 \le p < 1\), \(q > 1\), \(\Omega = ( {0,l} )\) and the weight function \(a ( x )\) satisfies (\({a_{1} }\)) or (\({a_{2}}\)). Define an auxiliary function
$$\psi_{1} ( t ) = \int_{0}^{l} {u^{k + 1} \,dx} , $$
where \(k > \max \{ {2 - m,q - 1} \}\). If the solution u to problem (23)–(25) blows up in \(L^{k+1}\)-norm at \(t^{*} \), then \(t^{*} \) is bounded below by
$$\int_{\psi_{1} ( 0 )}^{\infty}{\frac{{d\eta}}{{H_{1} \eta ^{\frac{{k + p + q}}{{k + 1}}} }}} , $$
where \(\psi_{1} ( 0 ) = \int_{0}^{l} u_{0}^{k + 1} ( x )\,dx \), positive constant \(H_{1}\) will be given in the proof.

Proof

Differentiating \(\psi_{1} ( t )\) and using Green’s formula and Hölder’s inequality, we have
$$\begin{aligned} \psi_{1} ^{\prime}( t ) =& ( {k + 1} ) \int_{0}^{l} {u^{k} u_{t} \,dx} \\ =& - mk ( {k + 1} ) \int_{0}^{l} {u^{k + m - 2} u_{x}^{2} \,dx} + ( {k + 1} ) \int_{0}^{l} {a ( x )} u^{k + p} \,dx \int_{0}^{l} {u^{q} \,dx} \\ &{} - ( {k + 1} ) \int_{0}^{l} {u^{k + s} \,dx} \\ \le& H_{1} \biggl( { \int_{0}^{l} {u^{k + 1} \,dx} } \biggr)^{\frac{{k + p + q}}{{k + 1}}} , \end{aligned}$$
(26)
where \(H_{1} = ( {k + 1} )l^{\frac{{k + 1 - q}}{{k + 1}}} ( {\int_{0}^{l} { ( {a ( x )} )} ^{\frac{{k + 1}}{{1 - p}}} \,dx} )^{\frac{{1 - p}}{{k + 1}}} \).
Hence, applying (26), we can derive that the lower bound for \(t^{*}\) satisfies
$$t^{*} \ge \int_{\psi_{1} ( 0 )}^{\infty}{\frac{{d\eta}}{{H_{1} \eta^{\frac{{k + p + q}}{{k + 1}}} }}} . $$
 □

3.2 \(N=2\) case

Theorem 5

Suppose that \(m > 0\), \(0\leq p <1\), \(q > 1\), \(\Omega \subset{\mathbf{R}}^{2} \) and the weight function satisfies (\({a_{2}}\)) and
(\({a_{3}}\)): 

there exists \(A = ( {A_{1} ,A_{2} } )\) such that \(- a ( x )A \le\nabla a ( x ) \le a ( x )A\),

where \(x \in\Omega\), \(A_{i} > 0\), \(i = 1,2\), while A satisfies \(\vert A \vert ^{2} < \lambda_{1} \). Define a weight function
$$\psi_{2} ( t ) = \int_{\Omega}{a ( x )} u^{k + 1} \,dx, $$
where \(\lambda_{1} \) is the first eigenvalue of the fixed membrane problem (8)–(9) for a two-dimensional space, \(k > \max \{ {q - 1,\frac{{ \vert A \vert m}}{{2 ( {\sqrt{\lambda _{1} } - \vert A \vert } )}}} \}\). If the solution u to problem (1)–(3) blows up in the measure \(\psi_{2} \) at \(t^{*} \), then \(t^{*} \) is bounded below by
$$\int_{\psi_{2} ( 0 )}^{\infty}{\frac{{d\eta}}{{H_{2} \eta ^{\frac{{k + p + q}}{{k + 1}}} }}} , $$
where \(\psi_{2} ( 0 ) = \int_{\Omega}{a ( x )u_{0}^{k + 1} ( x )} \,dx\), positive constant \(H_{2}\) will be given in the proof.

In order to prove Theorem 5, we firstly need to give a lemma.

Lemma 1

Suppose that \(a ( x )\) satisfies (\({a_{2}}\)), (\({a_{3}}\)), and \(\vert A \vert ^{2} < \lambda_{1} \). If \(u \in C^{1} ( \Omega )\) is nonnegative, then we have the differential inequality
$$\biggl( {\sqrt{\lambda_{1} } - \frac{{ \vert A \vert }}{2}} \biggr) \biggl( { \int_{\Omega}{a ( x )u^{2k} \,dx} } \biggr)^{\frac{1}{2}} \le \biggl( { \int_{\Omega}{a ( x ) \bigl\vert {\nabla u^{k} } \bigr\vert ^{2} \,dx} } \biggr)^{\frac{1}{2}} ,\quad k > 0,x \in \Omega. $$

Proof

By virtue of the Rayleigh principle, we know
$$\lambda_{1} \int_{\Omega}{\omega^{2} \,dx} \le \int_{\Omega}{ \vert {\nabla \omega} \vert ^{2} \,dx},\quad \mbox{where } \omega|_{\partial\Omega} = 0. $$
Now, choosing \(\omega = a^{\frac{1}{2}} ( x )u^{k}\), by (\({a_{3}}\)) and Hölder’s inequality, we have
$$\begin{aligned} &\lambda_{1} \int_{\Omega}{a ( x )u^{2k} \,dx} \\ &\quad \le \int_{\Omega}{ \bigl\vert {\nabla \bigl( {a^{\frac{1}{2}} ( x )u^{k} } \bigr)} \bigr\vert } ^{2} \,dx \\ &\quad = \frac{1}{4} \int_{\Omega}{\frac{{ \vert {\nabla a ( x )} \vert ^{2} }}{{a ( x )}}u^{2k} \,dx} + \int_{\Omega}{a ( x ) \bigl\vert {\nabla u^{k} } \bigr\vert ^{2} \,dx} + \int_{\Omega}{u^{k} \nabla a ( x )\nabla u^{k} \,dx} \\ &\quad \le\frac{{ \vert A \vert ^{2} }}{4} \int_{\Omega}{a ( x )u^{2k} \,dx} + \int_{\Omega}{a ( x ) \bigl\vert {\nabla u^{k} } \bigr\vert ^{2} \,dx} \\ &\qquad {} + \vert A \vert \biggl( { \int_{\Omega}{a ( x )u^{2k} \,dx} } \biggr)^{\frac{1}{2}} \biggl( { \int_{\Omega}{a ( x ) \bigl\vert {\nabla u^{k} } \bigr\vert ^{2} \,dx} } \biggr)^{\frac{1}{2}} \\ &\quad \le \biggl[ {\frac{{ \vert A \vert }}{2} \biggl( { \int_{\Omega}{a ( x )u^{2k} \,dx} } \biggr)^{\frac{1}{2}} + \biggl( { \int_{\Omega}{a ( x ) \bigl\vert {\nabla u^{k} } \bigr\vert ^{2} \,dx} } \biggr)^{\frac{1}{2}} } \biggr]^{2} . \end{aligned}$$
Hence, we derive
$$\biggl( {\sqrt{\lambda_{1} } - \frac{{ \vert A \vert }}{2}} \biggr) \biggl( { \int_{\Omega}{a ( x )u^{2k} \,dx} } \biggr)^{\frac{1}{2}} \le \biggl( { \int_{\Omega}{a ( x ) \bigl\vert {\nabla u^{k} } \bigr\vert ^{2} \,dx} } \biggr)^{\frac{1}{2}} . $$
 □

The proof of Theorem 5 can be given as follows.

Proof

Differentiating \(\psi_{2} ( t )\) and using Green’s formula, Hölder’s inequality, and Lemma 1, we have
$$\begin{aligned} \psi_{2} ^{\prime}( t ) =& ( {k + 1} ) \int_{\Omega}{a ( x )u^{k} u_{t} \,dx} \\ =& - ( {k + 1} ) \int_{\Omega}{ \bigl( {\nabla a ( x )u^{k} + ka ( x )u^{k - 1} \nabla u} \bigr)mu^{m - 1} \nabla u\,dx} \\ &{} + ( {k + 1} ) \int_{\Omega}{a^{2} ( x )u^{k + p} \,dx} \int_{\Omega}{u^{q} \,dx} - ( {k + 1} ) \int_{\Omega}{a ( x )u^{k + s} \,dx} \\ \le&\frac{{2m ( {k + 1} ) \vert A \vert }}{{k + m}} \int _{\Omega}{a ( x )u^{\frac{{k + m}}{2}} \bigl\vert {\nabla u^{\frac {{k + m}}{2}} } \bigr\vert \,dx} \\ &{} -\frac{{4mk ( {k + 1} )}}{{ ( {k + m} )^{2} }} \int _{\Omega}{a ( x ) \bigl\vert {\nabla u^{\frac{{k + m}}{2}} } \bigr\vert ^{2} \,dx} \\ &{} + ( {k + 1} ) \int_{\Omega}{a^{2} ( x )u^{k + p} \,dx} \int_{\Omega}{u^{q} \,dx} - ( {k + 1} ) \int_{\Omega}{a ( x )u^{k + s} \,dx} \\ \le&\frac{{2m ( {k + 1} ) \vert A \vert }}{{k + m}} \biggl( { \int_{\Omega}{a ( x )u^{k + m} \,dx} } \biggr)^{\frac{1}{2}} \biggl( { \int_{\Omega}{a ( x ) \bigl\vert {\nabla u^{\frac{{k + m}}{2}} } \bigr\vert ^{2} \,dx} } \biggr)^{\frac{1}{2}} \\ &{} - \frac{{4mk ( {k + 1} )}}{{ ( {k + m} )^{2} }} \int _{\Omega}{a ( x ) \bigl\vert {\nabla u^{\frac{{k + m}}{2}} } \bigr\vert ^{2} \,dx} \\ &{} + ( {k + 1} ) \int_{\Omega}{a^{2} ( x )u^{k + p} \,dx} \int_{\Omega}{u^{q} \,dx} \\ \le& \biggl[ {\frac{{4m ( {k + 1} ) \vert A \vert }}{{ ( {2\sqrt{\lambda_{1} } - \vert A \vert } ) ( {k + m} )}} - \frac{{4mk ( {k + 1} )}}{{ ( {k + m} )^{2} }}} \biggr] \int_{\Omega}{a ( x ) \bigl\vert {\nabla u^{\frac{{k + m}}{2}} } \bigr\vert ^{2} \,dx} \\ &{} + ( {k + 1} ) \int_{\Omega}{a^{2} ( x )u^{k + p} \,dx} \int_{\Omega}{u^{q} \,dx} . \end{aligned}$$
(27)
By the value of k, we can easily get that the coefficient of \(\int_{\Omega}a ( x ) \vert {\nabla u^{\frac{{k + m}}{2}} } \vert ^{2} \,dx\) in (28) is negative. Then, applying Hölder’s inequality, we have
$$ \psi_{2} ^{\prime}( t ) \le H_{2} \psi_{2}^{\frac{{k + p + q}}{{k + 1}}} , $$
(28)
where \(H_{2} = ( {k + 1} ) ( {\int_{\Omega}( {a ( x )} )^{\frac{{k + 2 - p}}{{1-p}}} \,dx } )^{\frac{{1 - p}}{{k + 1}}} (\int_{\Omega}{ ( {a ( x )} )^{ - \frac{q}{{k + 1 - q}}} \,dx} )^{\frac{{k + 1 - q}}{{k + 1}}} \).
Hence, using (27), we can obtain that the lower bound for \(t^{*}\) satisfies
$$t^{*} \ge \int_{\psi_{2} ( 0 )}^{\infty}\frac{{d\eta }}{{H_{2} \eta^{\frac{{k + p + q}}{{k + 1}}} }} . $$
 □

3.3 \(N \geq3\) case

Theorem 6

Suppose that \(m > 0\), \(p + q > \max \{ {m,s} \}\), \(\Omega \subset{\mathbf{R}}^{N}\) (\({N \ge3}\)) is a bounded convex region with smooth boundary, the weight function \(a ( x )\) satisfies (\(a_{1}\)) or (\(a_{2}\)) and
\((a_{3})^{\prime}\) : 

there exists \(A = ( {A_{1} , \ldots,A_{N} } )\) such that \(- a ( x )A \le\nabla a ( x ) \le a ( x )A\),

where \(x \in\Omega\), \(A_{i} > 0\), \(i = 1, \ldots,N\), \(N \geq3\).
Define a weight function
$$\psi_{3} ( t ) = \int_{\Omega}{a ( x )} u^{Nk} \,dx, $$
where \(\lambda_{1} \) is the first eigenvalue of the fixed membrane problem (8)–(9) for the N-dimensional space, \(k > \max \{ {\frac{1}{N},\frac{{2 ( {N - 2} ) ( {p + q - 1} )}}{N},\frac{{2 - 2m}}{3}} \}\). If the solution u to problem (1)–(3) blows up in the measure \(\psi_{3} \) at \(t^{*} \), then \(t^{*} \) is bounded below by
$$\int_{\psi_{3} ( 0 )}^{\infty}{\frac{{d\eta}}{{H_{3} \eta ^{\frac{{ ( {N - 2} )\theta}}{{N - 2 - N ( {1 - \theta} )\gamma}}} + H_{4} \eta^{\frac{{ ( {N - 2} )\theta}}{{N - 2 - N ( {1 - \theta} )}}} + H_{5} }}} , $$
where \(\psi_{3} ( 0 ) = \int_{\Omega}{a ( x )u_{0}^{Nk} ( x )} \,dx\) and positive constants \(H_{3} \), \(H_{4} \), \(H_{5} \), γ, θ will be given in the proof.

Proof

Differentiating \(\psi_{3} ( t )\) and using Green’s formula, Hölder’s, and Young’s inequalities, and condition \((a_{3})^{\prime}\), we have
$$\begin{aligned} \psi_{3} ^{\prime}( t ) =& Nk \int_{\Omega}{a ( x )u^{Nk - 1} u_{t} \,dx} \\ =& Nk \int_{\Omega}{a ( x )u^{Nk - 1} \Delta u^{m} } \,dx + Nk \int _{\Omega}{a^{2} ( x )} u^{Nk + p - 1} \,dx \int_{\Omega}{u^{q} \,dx} \\ &{} - Nk \int_{\Omega}{a ( x )} u^{Nk + s - 1} \,dx \\ =& - mNk ( {Nk - 1} ) \int_{\Omega}{a ( x )u^{Nk + m - 3} \vert {\nabla u} \vert ^{2} \,dx} \\ &{} - mNk \int_{\Omega}{\nabla a ( x )u^{Nk + m - 2} \nabla u\,dx} \\ &{} + Nk \int_{\Omega}{a^{2} ( x )} u^{Nk + p - 1} \,dx \int_{\Omega}{u^{q} \,dx - Nk} \int_{\Omega}{a ( x )} u^{Nk + s - 1} \,dx \\ \le& - \frac{{4mNk ( {Nk - 1} )}}{{ ( {Nk + m - 1} )^{2} }} \int_{\Omega}{a ( x ) \bigl\vert {\nabla u^{\frac{{Nk + m - 1}}{2}} } \bigr\vert ^{2} \,dx} \\ &{} + mNk \vert A \vert \biggl( { \int_{\Omega}{a ( x )u^{Nk + m - 1} \,dx} } \biggr)^{\frac{1}{2}} \\ &{} \times \biggl( {\frac{4}{{ ( {Nk + m - 1} )^{2} }} \int_{\Omega}{a ( x ) \bigl\vert {\nabla u^{\frac{{Nk + m - 1}}{2}} } \bigr\vert ^{2} \,dx} } \biggr)^{\frac{1}{2}} \\ &{} + Nk \int_{\Omega}{a^{2} ( x )} u^{Nk + p - 1} \,dx \int_{\Omega}{u^{q} \,dx - Nk} \int_{\Omega}{a ( x )} u^{Nk + s - 1} \,dx \\ \le& \biggl[ { - \frac{{4mNk ( {Nk - 1} )}}{{ ( {Nk + m - 1} )^{2} }} + \frac{{2mNk \vert A \vert \varepsilon_{1} }}{{ ( {Nk + m - 1} )^{2} }}} \biggr] \int_{\Omega}{a ( x ) \bigl\vert {\nabla u^{\frac{{Nk + m - 1}}{2}} } \bigr\vert ^{2} \,dx} \\ &{} + \frac{{mNk \vert A \vert }}{{2\varepsilon_{1} }} \int_{\Omega}{a ( x )u^{Nk + m - 1} \,dx} + Nk \int_{\Omega}{a^{2} ( x )} u^{Nk + p - 1} \,dx \int_{\Omega}{u^{q} \,dx} \\ &{} - Nk \int_{\Omega}{a ( x )} u^{Nk + s - 1} \,dx, \end{aligned}$$
(29)
where \(\varepsilon_{1} \) is a positive constant to be determined later.
To begin with, applying Hölder’s inequality, we get the inequality
$$\begin{aligned} & \int_{\Omega}{a^{\frac{{N - 2\theta}}{{N - 2}}} ( x )u^{ \frac{{Nk ( {2N - 3} )}}{{2 ( {N - 2} )}}} \,dx} \\ &\quad \le \biggl( { \int_{\Omega}{a ( x )u^{Nk} \,dx} } \biggr)^{\theta}\biggl( { \int_{\Omega}{ \bigl( {a^{\frac{1}{2}} ( x )u^{\frac{{Nk + m - 1}}{2}} } \bigr)} ^{\frac{{2N}}{{N - 2}}} \,dx} \biggr)^{1 - \theta} , \end{aligned}$$
(30)
where \(\theta = \frac{{3k + 2m - 2}}{{4k + 2m - 2}}\). Using Hölder’s and Young’s inequalities to estimate the second term on the right-hand side of (29), we have
$$\begin{aligned} \int_{\Omega}{a ( x )u^{Nk + m - 1} \,dx} &\le \biggl( { \int _{\Omega}{a^{\frac{{N - 2\theta}}{{N - 2}}} ( x )u^{\frac {{Nk ( {2N - 3} )}}{{2 ( {N - 2} )}}} \,dx} } \biggr)^{\gamma}\biggl( { \int_{\Omega}{ \bigl( {a ( x )} \bigr)^{\sigma_{1} } \,dx} } \biggr)^{1 - \gamma} \\ & \le\gamma \int_{\Omega}{a^{\frac{{N - 2\theta}}{{N - 2}}} ( x )u^{\frac{{Nk ( {2N - 3} )}}{{2 ( {N - 2} )}}} \,dx} + ( {1 - \gamma} ) \int_{\Omega}\bigl( {a ( x )} \bigr)^{\sigma_{1} } \,dx , \end{aligned}$$
(31)
where \(\gamma = \frac{{2 ( {N - 2} ) ( {Nk + m - 1} )}}{{Nk ( {2N - 3} )}}\), \(\sigma_{1}\) such that \(1 = \frac{{N - 2\theta}}{{N - 2}}\gamma + \sigma_{1} ( {1 - \gamma} )\).
Next, applying Hölder’s inequality to the third term on the right-hand side of (29), we obtain the inequality
$$\begin{aligned} & \int_{\Omega}{a^{2} ( x )} u^{Nk + p - 1} \,dx \int_{\Omega}{u^{q} \,dx} \\ &\quad \le \biggl( { \int_{\Omega}{ \bigl( {a ( x )} \bigr)^{\sigma _{2} } \,dx} } \biggr)^{\frac{q}{{Nk + p + q - 1}}} \biggl( { \int_{\Omega}{ \bigl( {a ( x )} \bigr)^{\sigma_{3} } \,dx} } \biggr)^{\frac {{Nk + p - 1}}{{Nk + p + q - 1}}} \\ &\qquad {}\times \int_{\Omega}{a^{m_{1} + \frac{{N - 2\theta}}{{N - 2}}m_{2} } ( x )} u^{Nk + p + q - 1} \,dx, \end{aligned}$$
(32)
where \(m_{1} = \frac{{Nk - 2N ( {p + q} ) + 2N + 4 ( {p + q} ) - 4}}{{Nk - 2Ns + 2N + 4s - 4}}\), \(m_{2} = \frac{{2 ( {p + q - s} ) ( {N - 2} )}}{{Nk - 2Ns + 2N + 4s - 4}}\), \(\sigma_{2} ,\sigma_{3} \) such that
$$\begin{aligned}& 2 = \frac{{Nk + p - 1}}{{Nk + p + q - 1}} \biggl( {m_{1} + \frac{{N - 2\theta}}{{N - 2}}m_{2} } \biggr) + \frac{q}{{Nk + p + q - 1}}\sigma_{2}, \\& 0 = \frac{q}{{Nk + p + q - 1}} \biggl( {m_{1} + \frac{{N - 2\theta }}{{N - 2}}m_{2} } \biggr) + \frac{{Nk + p - 1}}{{Nk + p + q - 1}}\sigma_{3} . \end{aligned}$$
Thus, applying Hölder’s and Young’s inequalities to the third integral term on the right-hand side of (32), we can derive the inequality
$$\begin{aligned} & \int_{\Omega}{a^{m_{1} + \frac{{N - 2\theta}}{{N - 2}}m_{2} } ( x )} u^{Nk + p + q - 1} \,dx \\ &\quad \le \biggl( { \int_{\Omega}{a ( x )} u^{Nk + s - 1} \,dx} \biggr)^{m_{1} } \biggl( { \int_{\Omega}{a^{\frac{{N - 2\theta}}{{N - 2}}} ( x )u^{\frac{{Nk ( {2N - 3} )}}{{2 ( {N - 2} )}}} \,dx} } \biggr)^{m_{2} } \\ &\quad \le m_{1} \varepsilon_{2} \int_{\Omega}{a ( x )} u^{Nk + s - 1} \,dx + m_{2} \varepsilon_{2} ^{ - \frac{{m_{1} }}{{m_{2} }}} \int_{\Omega}a^{\frac{{N - 2\theta}}{{N - 2}}} ( x )u^{\frac{{Nk ( {2N - 3} )}}{{2 ( {N - 2} )}}} \,dx , \end{aligned}$$
(33)
where \(\varepsilon_{2} > 0\) is a constant to be determined later.
Now, we substitute (31)–(33) into (29) and choose suitable \(\varepsilon_{2} > 0\) to make the coefficient of \(\int_{\Omega}{a ( x )} u^{Nk + s - 1} \,dx\) in (29) vanish, that is,
$$m_{1} \biggl( { \int_{\Omega}{ \bigl( {a ( x )} \bigr)^{\sigma _{2} } } \,dx} \biggr)^{\frac{p}{{Nk + p + q - 1}}} \biggl( { \int_{\Omega}{ \bigl( {a ( x )} \bigr)^{\sigma_{3} } } \,dx} \biggr)^{\frac{{Nk + q - 1}}{{Nk + p + q - 1}}} \varepsilon_{2} = 1. $$
We can obtain the inequality
$$ \psi_{3} ^{\prime}( t ) \le C_{1} \int_{\Omega}{a ( x ) \bigl\vert {\nabla u^{\frac{{Nk + m - 1}}{2}} } \bigr\vert ^{2} \,dx} + C_{2} \int_{\Omega}{a^{\frac{{N - 2\theta}}{{N - 2}}} ( x )u^{\frac{{Nk ( {2N - 3} )}}{{2 ( {N - 2} )}}} \,dx} + C_{3} , $$
(34)
where
$$\begin{aligned} &C_{1} = - \frac{{4mNk ( {Nk - 1} )}}{{ ( {Nk + m - 1} )^{2} }} + \frac{{2mNk \vert A \vert \varepsilon_{1} }}{{ ( {Nk + m - 1} )^{2} }}, \\ &C_{2} = Nkm_{2} \varepsilon_{2} ^{ - \frac{{m_{1} }}{{m_{2} }}} \biggl( { \int _{\Omega}{ \bigl( {a ( x )} \bigr)^{\sigma_{2} } \,dx} } \biggr)^{\frac{p}{{Nk + p + q - 1}}} \biggl( { \int_{\Omega}{ \bigl( {a ( x )} \bigr)^{\sigma_{3} } \,dx} } \biggr)^{\frac{{Nk + q - 1}}{{Nk + p + q - 1}}} + \frac{{mNk \vert A \vert \gamma}}{{2\varepsilon_{1} }}, \\ &C_{3} = \frac{{mNk \vert A \vert ( {1 - \gamma} )}}{{2\varepsilon_{1} }} \int_{\Omega}\bigl( {a ( x )} \bigr)^{\sigma_{1} } \,dx. \end{aligned}$$
In order to deal with the gradient term in (34), we will use Sobolev’s inequality
$$ \biggl( { \int_{\Omega}{ \bigl( {a^{\frac{1}{2}} ( x )u^{\frac{{Nk + m - 1}}{2}} } \bigr)} ^{\frac{{2N}}{{N - 2}}} \,dx} \biggr)^{\frac{{N - 2}}{{2N}}} \le C_{s} \biggl( { \int_{\Omega}{ \bigl\vert {\nabla \bigl( {a^{\frac{1}{2}} ( x )u^{\frac{{Nk + m - 1}}{2}} } \bigr)} \bigr\vert ^{2} \,dx} } \biggr)^{\frac{1}{2}} , $$
(35)
where \(C_{s} \) is the optimal Sobolev constant. Applying condition \((a_{3})^{\prime}\) and Hölder’s inequality to the right-hand side of (35), we have
$$\begin{aligned} & \int_{\Omega}{ \bigl\vert {\nabla \bigl( {a^{\frac{1}{2}} ( x )u^{\frac{{Nk + m - 1}}{2}} } \bigr)} \bigr\vert ^{2} \,dx} \\ &\quad \le\frac{{ \vert A \vert ^{2} }}{4} \int_{\Omega}{a ( x )u^{Nk + m - 1} \,dx} + \int_{\Omega}{a ( x ) \bigl\vert {\nabla u^{\frac{{Nk + m - 1}}{2}} } \bigr\vert ^{2} \,dx} \\ &\qquad {} + \vert A \vert \int_{\Omega}{a ( x )u^{\frac{{Nk + m - 1}}{2}} \bigl\vert {\nabla u^{\frac{{Nk + m - 1}}{2}} } \bigr\vert \,dx} \\ &\quad \le \biggl[ {\frac{{ \vert A \vert }}{2} \biggl( { \int_{\Omega}{a ( x )u^{Nk + m - 1} \,dx} } \biggr)^{\frac{1}{2}} + \biggl( { \int_{\Omega}{a ( x ) \bigl\vert {\nabla u^{\frac{{Nk + m - 1}}{2}} } \bigr\vert ^{2} \,dx} } \biggr)^{\frac{1}{2}} } \biggr]^{2} . \end{aligned}$$
(36)
Now, using (30), (35), and (36) to estimate the second term on the right-hand side of (34), we obtain
$$\begin{aligned} & \int_{\Omega}{a^{\frac{{N - 2\theta}}{{N - 2}}} ( x )u^{\frac{{Nk ( {2N - 3} )}}{{2 ( {N - 2} )}}} \,dx} \\ &\quad \le C_{s}^{\frac{{2N ( {1 - \theta} )}}{{N - 2}}} \biggl( { \int _{\Omega}{a ( x )u^{Nk} \,dx} } \biggr)^{\theta}\biggl( { \int _{\Omega}{ \bigl\vert {\nabla a^{\frac{1}{2}} ( x )u^{\frac{{Nk + m - 1}}{2}} } \bigr\vert ^{2} \,dx} } \biggr)^{\frac{{N ( {1 - \theta} )}}{{N - 2}}} \\ &\quad \le C_{s} ^{\frac{{2N ( {1 - \theta} )}}{{N - 2}}} \biggl( { \int_{\Omega}{a ( x )u^{Nk} \,dx} } \biggr)^{\theta} \\ & \qquad {}\times \biggl[ {\frac{{ \vert A \vert }}{2} \biggl( { \int_{\Omega}{a ( x )u^{Nk + m - 1} \,dx} } \biggr)^{\frac{1}{2}} + \biggl( { \int_{\Omega}{a ( x ) \bigl\vert {\nabla u^{\frac{{Nk + m - 1}}{2}} } \bigr\vert ^{2} \,dx} } \biggr)^{\frac{1}{2}} } \biggr]^{\frac{{2N ( {1 - \theta} )}}{{N - 2}}} . \end{aligned}$$
(37)
Next, applying the inequality
$$( {x + y} )^{k} \le \bigl( {2\max \{ {x,y} \}} \bigr)^{k} \le2^{k} \max \bigl\{ {x^{k} ,y^{k} } \bigr\} \le2^{k} \bigl( {x^{k} + y^{k} } \bigr), \quad \forall x,y,k \ge0, $$
and (31), we derive an estimate for the summation of the bracket in (37) as follows:
$$\begin{aligned} & \biggl[ {\frac{{ \vert A \vert }}{2} \biggl( { \int_{\Omega}{a ( x )u^{Nk + m - 1} \,dx} } \biggr)^{\frac{1}{2}} + \biggl( { \int _{\Omega}{a ( x ) \bigl\vert {\nabla u^{\frac{{Nk + m - 1}}{2}} } \bigr\vert ^{2} \,dx} } \biggr)^{\frac{1}{2}} } \biggr]^{\frac{{2N ( {1 - \theta} )}}{{N - 2}}} \\ &\quad \le \vert A \vert ^{\frac{{2N ( {1 - \theta} )}}{{N - 2}}} \biggl( { \int_{\Omega}{ \bigl( {a ( x )} \bigr)^{\sigma _{1} } \,dx} } \biggr)^{\frac{{N ( {1 - \theta} ) ( {1 - \gamma} )}}{{N - 2}}} \biggl( { \int_{\Omega}{a^{\frac{{N - 2\theta}}{{N - 2}}} ( x )u^{\frac{{Nk ( {2N - 3} )}}{{2 ( {N - 2} )}}} \,dx} } \biggr)^{\frac{{N ( {1 - \theta} )\gamma}}{{N - 2}}} \\ &\qquad {} + 2^{\frac{{2N ( {1 - \theta} )}}{{N - 2}}} \biggl( { \int_{\Omega}{a ( x ) \bigl\vert {\nabla u^{\frac{{Nk + m - 1}}{2}} } \bigr\vert ^{2} \,dx} } \biggr)^{\frac{{N ( {1 - \theta} )}}{{N - 2}}} . \end{aligned}$$
(38)
Substituting (38) into (37), we have
$$\begin{aligned} & \int_{\Omega}{a^{\frac{{N - 2\theta}}{{N - 2}}} ( x )u^{\frac{{Nk ( {2N - 3} )}}{{2 ( {N - 2} )}}} \,dx} \\ &\quad \le C_{4} \biggl( { \int_{\Omega}{a ( x )u^{Nk} \,dx} } \biggr)^{\theta}\biggl( { \int_{\Omega}{a^{\frac{{N - 2\theta}}{{N - 2}}} ( x )u^{\frac{{Nk ( {2N - 3} )}}{{2 ( {N - 2} )}}} \,dx} } \biggr)^{\frac{{N ( {1 - \theta} )\gamma}}{{N - 2}}} \\ &\qquad {} + C_{5} \biggl( { \int_{\Omega}{a ( x )u^{Nk} \,dx} } \biggr)^{\theta}\biggl( { \int_{\Omega}{a ( x ) \bigl\vert {\nabla u^{\frac{{Nk + m - 1}}{2}} } \bigr\vert ^{2} \,dx} } \biggr)^{\frac{{N ( {1 - \theta} )}}{{N - 2}}} , \end{aligned}$$
(39)
where \(C_{4} = ( {C_{s} \vert A \vert } )^{\frac{{2N ( {1 - \theta} )}}{{N - 2}}} ( {\int_{\Omega}{ ( {a ( x )} )^{\sigma_{1} } \,dx} } )^{\frac{{N ( {1 - \theta} ) ( {1 - \gamma} )}}{{N - 2}}} \), \(C_{5} = ( {2C_{s} } )^{\frac{{2N ( {1 - \theta} )}}{{N - 2}}} \).
Now, applying Young’s inequality to the two terms on the right-hand side of (39), we can get
$$\begin{aligned} & \biggl( { \int_{\Omega}{a ( x )u^{Nk} \,dx} } \biggr)^{\theta}\biggl( { \int_{\Omega}{a^{\frac{{N - 2\theta}}{{N - 2}}} ( x )u^{\frac{{Nk ( {2N - 3} )}}{{2 ( {N - 2} )}}} \,dx} } \biggr)^{\frac{{N ( {1 - \theta} )\gamma}}{{N - 2}}} \\ &\quad \le\frac{{N - 2 - N ( {1 - \theta} )\gamma}}{{N - 2}}\varepsilon_{3} ^{ - \frac{{N ( {1 - \theta} )\gamma }}{{N - 2 - N ( {1 - \theta} )\gamma}}} \biggl( { \int_{\Omega}{a ( x )u^{Nk} \,dx} } \biggr)^{\frac {{ ( {N - 2} )\theta}}{{N - 2 - N ( {1 - \theta} )\gamma}}} \\ &\qquad {} + \frac{{N ( {1 - \theta} )\gamma\varepsilon_{3} }}{{N - 2}} \int_{\Omega}{a^{\frac{{N - 2\theta}}{{N - 2}}} ( x )u^{\frac{{Nk ( {2N - 3} )}}{{2 ( {N - 2} )}}} \,dx} , \end{aligned}$$
(40)
and
$$\begin{aligned} & \biggl( { \int_{\Omega}{a ( x )u^{Nk} \,dx} } \biggr)^{\theta}\biggl( { \int_{\Omega}{a ( x ) \bigl\vert {\nabla u^{\frac{{Nk + m - 1}}{2}} } \bigr\vert ^{2} \,dx} } \biggr)^{\frac{{N ( {1 - \theta} )}}{{N - 2}}} \\ &\quad \le\frac{{N - 2 - N ( {1 - \theta} )}}{{N - 2}}\varepsilon_{4} ^{ - \frac{{N ( {1 - \theta} )}}{{N - 2 - N ( {1 - \theta} )}}} \biggl( { \int_{\Omega}{a ( x )u^{Nk} \,dx} } \biggr)^{\frac {{ ( {N - 2} )\theta}}{{N - 2 - N ( {1 - \theta} )}}} \\ &\qquad {} + \frac{{N ( {1 - \theta} )\varepsilon_{4} }}{{N - 2}} \int_{\Omega}{a ( x ) \bigl\vert {\nabla u^{\frac{{Nk + m - 1}}{2}} } \bigr\vert ^{2} \,dx} , \end{aligned}$$
(41)
where \(\varepsilon_{3} ,\varepsilon_{4} > 0\) are constants to be determined later. Substituting (40), (41) into (39) leads to
$$\begin{aligned} & \biggl( {1 - \frac{{C_{4} N ( {1 - \theta} )\gamma \varepsilon_{3} }}{{N - 2}}} \biggr) \int_{\Omega}{a^{\frac{{N - 2\theta}}{{N - 2}}} ( x )u^{\frac{{Nk ( {2N - 3} )}}{{2 ( {N - 2} )}}} \,dx} \\ &\quad \le\frac{{N - 2 - N ( {1 - \theta} )\gamma}}{{N - 2}}C_{4} \varepsilon_{3}^{ - \frac{{N ( {1 - \theta} )\gamma }}{{N - 2 - N ( {1 - \theta} )\gamma}}} \biggl( { \int_{\Omega}{a ( x )u^{Nk} \,dx} } \biggr)^{\frac {{ ( {N - 2} )\theta}}{{N - 2 - N ( {1 - \theta} )\gamma}}} \\ &\qquad {} + \frac{{N - 2 - N ( {1 - \theta} )}}{{N - 2}}C_{5} \varepsilon_{4} ^{ - \frac{{N ( {1 - \theta} )}}{{N - 2 - N ( {1 - \theta} )}}} \biggl( { \int_{\Omega}{a ( x )u^{Nk} \,dx} } \biggr)^{\frac{{ ( {N - 2} )\theta}}{{N - 2 - N ( {1 - \theta } )}}} \\ &\qquad {} + \frac{{C_{5} N ( {1 - \theta} )\varepsilon_{4} }}{{N - 2}} \int_{\Omega}{a ( x ) \bigl\vert {\nabla u^{\frac{{Nk + m - 1}}{2}} } \bigr\vert ^{2} \,dx} . \end{aligned}$$
Choose \(\varepsilon_{3} > 0\) small enough such that \(\rho: = 1 - \frac{{C_{4} N ( {1 - \theta} )\gamma \varepsilon_{3} }}{{N - 2}} > 0\).
It follows that the second term on the right-hand side of (34) satisfies
$$\begin{aligned} & \int_{\Omega}{a^{\frac{{N - 2\theta}}{{N - 2}}} ( x )u^{\frac{{Nk ( {2N - 3} )}}{{2 ( {N - 2} )}}} \,dx} \\ &\quad \le C_{6} \biggl( { \int_{\Omega}{a ( x )u^{Nk} \,dx} } \biggr)^{\frac{{ ( {N - 2} )\theta}}{{N - 2 - N ( {1 - \theta} )\gamma}}} + C_{7} \biggl( { \int_{\Omega}{a ( x )u^{Nk} \,dx} } \biggr)^{\frac{{ ( {N - 2} )\theta}}{{N - 2 - N ( {1 - \theta} )}}} \\ &\qquad {} + C_{8} \int_{\Omega}{a ( x ) \bigl\vert {\nabla u^{\frac{{Nk + m - 1}}{2}} } \bigr\vert ^{2} \,dx} , \end{aligned}$$
(42)
where
$$\begin{aligned} &C_{6} = \frac{{N - 2 - N ( {1 - \theta} )\gamma }}{{ ( {N - 2} )\rho}}C_{4} \varepsilon_{3} ^{ - \frac{{\gamma N ( {1 - \theta} )}}{{N - 2 - \gamma N ( {1 - \theta} )}}} , \\ &C_{7} = C_{5} \frac{{N - 2 - N ( {1 - \theta} )}}{{ ( {N - 2} )\rho}} \varepsilon_{4} ^{\frac{{N ( {\theta - 1} )}}{{N - 2 - N ( {1 - \theta} )}}} , \\ &C_{8} = \frac{{C_{5} N ( {1 - \theta} )\varepsilon_{4} }}{{ ( {N - 2} )\rho}}. \end{aligned}$$
Then, substituting (42) into (44), we can derive
$$\begin{aligned} \psi_{3} ^{\prime}( t ) &\le ( {C_{1} + C_{2} C_{8} } ) \int_{\Omega}{a ( x ) \bigl\vert {\nabla u^{\frac{{Nk + m - 1}}{2}} } \bigr\vert ^{2} \,dx} \\ & \quad {}+ C_{2} C_{6} \psi_{3} ^{\frac{{ ( {N - 2} )\theta}}{{N - 2 - N ( {1 - \theta} )\gamma}}} + C_{2} C_{7} \psi_{3} ^{\frac{{ ( {N - 2} )\theta}}{{N - 2 - N ( {1 - \theta} )}}} + C_{3} . \end{aligned}$$
(43)
Choose \(\varepsilon_{1} \) small enough such that \(C_{1} < 0\) and \(\varepsilon_{4} \) such that \(C_{1} + C_{2} C_{8} = 0\). Therefore, (43) can be rewritten as
$$ \psi_{3} ^{\prime}( t ) \le H_{3} \psi_{3} ^{\frac{{ ( {N - 2} )\theta}}{{N - 2 - N ( {1 - \theta} )\gamma }}} + H_{4} \psi_{3} ^{\frac{{ ( {N - 2} )\theta}}{{N - 2 - N ( {1 - \theta} )}}} + H_{5} , $$
(44)
where \(H_{3} = C_{2} C_{6} \), \(H_{4} = C_{2} C_{7} \), \(H_{5} = C_{3} \).
Note that \(\frac{{ ( {N - 2} )\theta}}{{N - 2 - N ( {1 - \theta} )}} > 1\), then integrating (44) from 0 to \(t^{*} \), we derive
$$t^{*} \ge \int_{\psi_{3} ( 0 )}^{\infty}{\frac{{d\eta }}{{H_{3} \eta^{\frac{{ ( {N - 2} )\theta}}{{N - 2 - N ( {1 - \theta} )\gamma}}} + H_{4} \eta^{\frac{{ ( {N - 2} )\theta}}{{N - 2 - N ( {1 - \theta} )}}} + H_{5} }}} . $$
 □

Remark 3

If the null Dirichlet boundary condition (2) is replaced by the null Neumann boundary condition
$$ \frac{{\partial u}}{{\partial\nu}} ( {x,t} ) = 0,\quad (x,t)\in \partial\Omega\times \bigl(0,t^{\ast}\bigr), $$
(2′)
where ν is the unit outward normal vector on ∂Ω, then Theorem 1 is valid for the case \(m \ge1\), and Theorems 4–6 are also valid.

4 Conclusion

Bounds for the blow-up time to a porous medium equation with weighted nonlocal source and inner absorption terms under some appropriate measure in the whole-dimensional space (\(N\geq1\)) are derived in this paper. Note that the methods for a semilinear parabolic equation in [16, 28] are not necessarily applicable to our quasilinear parabolic model, and our results extend the results for the model in [20, 28].

Declarations

Acknowledgements

The authors would like to deeply thank all the reviewers for their insightful and constructive comments.

Availability of data and materials

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

Funding

Yunde Shen was supported by the project of Zhejiang Natural Science Foundation of China under (Grant No. LY16E050006).

Authors’ contributions

All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

Competing interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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 College of Mechanical Electrical Engineering, Wenzhou University, Wenzhou, P.R. China
(2)
School of Mathematical Sciences, Ocean University of China, Qingdao, P.R. China

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