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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 Problems volume 2018, Article number: 116 (2018)
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 (\(N\geq1\)).
1 Introduction
We consider the porous medium equation with weighted nonlocal source and inner absorption terms
subject to null Dirichlet boundary and initial conditions
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
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
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
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
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
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
Compute the derivative and use Green’s formula and Hölder’s inequality to derive
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
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
Choosing \(\delta_{1} \) satisfies \(0 < \delta_{1} < \frac{{a ( {p + q} )}}{s}|\Omega|^{\frac{2s-(p+q)}{s}}\), then (6) can be rewritten as
where
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
 □
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:
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
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
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
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
Substituting (11) into (10) and using Young’s inequality, we obtain
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
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
where
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
 □
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
Assume that there exist \(I_{1} \ge I_{2} > 2\) such that
If u is the nonnegative solution to problem (1)–(3) and the initial data \(u_{0} ( x )\) satisfies
then u blows up in \(L^{2m}\)-norm, and an upper bound for \(t^{*} \) is
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
Then (16) is equivalent to
Define
Differentiating \(\xi ( t )\) and using (15), (17) and Green’s formula, we obtain
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
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
we can derive
Applying (19) and (21), we can compute
and
Hence, (22) leads to
 □
Remark 2
In the fast and linear diffusion situation (\({0 < m \le1}\)), we set
Suppose that there exist \(I_{3} \ge I_{4} > 2\) such that
If u is a nonnegative solution to problem (1)–(3) and the initial data \(u_{0} ( x )\) satisfies
then u blows up in \(L^{2}\)-norm, and an upper bound for \(t^{*} \) is
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
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
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
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
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
 □
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
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
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
Proof
By virtue of the Rayleigh principle, we know
Now, choosing \(\omega = a^{\frac{1}{2}} ( x )u^{k}\), by (\({a_{3}}\)) and Hölder’s inequality, we have
Hence, we derive
 □
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
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
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
 □
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
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
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
where \(\varepsilon_{1} \) is a positive constant to be determined later.
To begin with, applying Hölder’s inequality, we get the inequality
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
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
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
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
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,
We can obtain the inequality
where
In order to deal with the gradient term in (34), we will use Sobolev’s inequality
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
Now, using (30), (35), and (36) to estimate the second term on the right-hand side of (34), we obtain
Next, applying the inequality
and (31), we derive an estimate for the summation of the bracket in (37) as follows:
Substituting (38) into (37), we have
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
and
where \(\varepsilon_{3} ,\varepsilon_{4} > 0\) are constants to be determined later. Substituting (40), (41) into (39) leads to
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
where
Then, substituting (42) into (44), we can derive
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
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
 □
Remark 3
If the null Dirichlet boundary condition (2) is replaced by the null Neumann boundary condition
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].
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Acknowledgements
The authors would like to deeply thank all the reviewers for their insightful and constructive comments.
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Yunde Shen was supported by the project of Zhejiang Natural Science Foundation of China under (Grant No. LY16E050006).
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Shen, Y., Fang, Z.B. Bounds for the blow-up time of a porous medium equation with weighted nonlocal source and inner absorption terms. Bound Value Probl 2018, 116 (2018). https://doi.org/10.1186/s13661-018-1038-3
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DOI: https://doi.org/10.1186/s13661-018-1038-3