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Global existence and blow-up for a class of nonlinear reaction diffusion problems
Boundary Value Problems volume 2014, Article number: 168 (2014)
Abstract
This paper deals with the global existence and blow-up of the solution for a class of nonlinear reaction diffusion problems. The purpose of this paper is to establish conditions on the data to guarantee the blow-up of the solution at some finite time, and conditions to ensure that the solution remains global. In addition, an upper bound for the ‘blow-up time’, an upper estimate of the ‘blow-up rate’, and an upper estimate of the global solution are also specified. Finally, as applications of the obtained results, some examples are presented.
MSC: 35K57, 35K55, 35B05.
1 Introduction
The global existence and blow-up for nonlinear reaction diffusion equations have been widely studied in recent years (see, for instance, [1]–[8]). In this paper, we consider the following problem:
where () is a bounded domain with smooth boundary ∂D, is the closure of D, is the outward normal derivative on ∂D, T is the maximal existence time of u. Set . We assume, throughout this paper, that is a positive function, is a positive function, is a positive function, is a function, for any , and is a positive function. Under the above assumptions, it is well known from the classical parabolic equation theory [5] and maximum principle [9] that there exists a unique local positive solution for problem (1.1). Moreover, by the regularity theorem [10], .
Many authors discussed the global existence and blow-up for nonlinear reaction diffusion equations with Neumann boundary conditions and obtained a lot of interesting results [11]–[24]. Some special cases of (1.1) have been studied already. Lair and Oxley [25] investigated the following problem:
where () is a bounded domain with smooth boundary ∂D. The necessary and sufficient conditions characterized by functions a and f were given for the global existence and blow-up solution. Zhang [26] dealt with the following problem:
where () is a bounded domain with smooth boundary ∂D. The sufficient conditions were obtained there for the existence of global and blow-up solutions. Gao et al.[27] considered the following problem:
where () is a bounded domain with smooth boundary ∂D. The sufficient conditions were developed for the existence of global and blow-up solutions. Meanwhile, the upper estimate of the global solution, the upper bound of the ‘blow-up time’, and the upper estimate of the ‘blow-up rate’ were also given.
In this paper, we study reaction diffusion problem (1.1). Note that , and are nonlinear reaction, nonlinear diffusion and nonlinear convection, respectively. Since the diffusion function depends not only on the concentration variable u but also on the space variable x, it seems that the methods of [26], [27] are not applicable for the problem (1.1). In this paper, by constructing completely different auxiliary functions from those in [26], [27] and technically using maximum principles, we obtain the conditions on the data to guarantee the blow-up of the solution at some finite time, and conditions to ensure that the solution remains global. In addition, an upper bound for the ‘blow-up time’, an upper estimate of the ‘blow-up rate’, and an upper estimate of the global solution are also given. Our results extend and supplement those obtained in [26], [27].
We proceed as follows. In Section 2 we study the blow-up solution of (1.1). Section 3 is devoted to the global solution of (1.1). A few examples are given in Section 4 to illustrate the applications of the obtained results.
2 Blow-up solution
In this section we establish sufficient conditions on the data of the problem (1.1) to produce a blow-up of the solution at some finite time T and under these conditions we derive an explicit upper bound for T and an explicit upper estimate of the ‘blow-up rate’. The main result of this section is formulated in the following theorem.
Theorem 2.1
Letbe a solution of the problem (1.1). Assume that the data of the problem (1.1) satisfies the following conditions:
-
(i)
for any ,
(2.1)
-
(ii)
the constant
(2.2)
-
(iii)
the integration
(2.3)
Then must blow up in a finite time T and
as well as
where
andis the inverse function of Φ.
Proof
Consider the auxiliary function
Now we have
and
It follows from (2.9) and (2.10) that
By the first equation of (1.1), we have
Substitute (2.12) into (2.11) to obtain
It follows from (2.8) that
Substituting (2.14) into (2.13), we get
With (2.7), we have
Substitute (2.16) into (2.15) to get
The assumptions (2.1) ensure that the right-hand side of (2.17) is nonpositive; that is,
Now, by (1.1), we have
Furthermore, it follows from (2.2) that
Combining (2.18)-(2.20) and applying the maximum principle [9], we find that the minimum of Q in is zero. Hence,
that is,
At the point , where , integrating (2.21) over , we get
By the assumption (2.3), we know that must blow up in finite time , moreover,
For each fixed x, integrating the inequality (2.21) over (), we obtain
Letting , we have
which implies
The proof is complete. □
3 Global solution
In this section we establish sufficient conditions on the data of the problem (1.1) in order to ensure that the solution has global existence. Under these conditions, we derive an explicit upper estimate of the global solution. The main results of this section are the following theorem.
Theorem 3.1
Letbe a solution of the problem (1.1). Assume that the data of the problem (1.1) satisfies the following conditions:
-
(i)
for any ,
(3.1)
-
(ii)
the constant
(3.2)
-
(iii)
the integration
(3.3)
Then must be a global solution and
where
andis the inverse function of Ψ.
Proof
Consider an auxiliary function
In (2.17), by replacing Q and α by P and β, respectively, we have
It follows from (3.1) that the right-hand side of (3.7) is nonnegative; that is,
With (1.1), we have
It follows from (3.2) that
Combining (3.8)-(3.10) and applying the maximum principle, we know that the maximum of P in is zero; that is,
From (3.11), we get
For each fixed , by integrating (3.12) over , we have
It follows from (3.13) and (3.3) that must be a global solution. Furthermore, by (3.13), we have
Hence,
The proof is complete. □
4 Applications
When , , and or and , the conclusions of Theorems 2.1 and 3.1 are valid. In this sense, our results extend and supplement the results of [26], [27].
In what follows, we present several examples to demonstrate the applications of the obtained results.
Example 4.1
Let be a solution of the following problem:
where is the unit ball of . The above problem may be turned into the following problem:
Now
By setting
we have and
It is easy to check that (2.1)-(2.3) hold. By Theorem 2.1, must blow up in a finite time T and
as well as
Example 4.2
Let be a solution of the following problem:
where is the unit ball of . The above problem can be transformed into the following problem:
Now we have
In order to determine the constant β, we assume
then and
Again, it is easy to check that (3.1)-(3.3) hold. By Theorem 3.1, must be a global solution and
Author’s contributions
All results belong to Juntang Ding.
References
Quittner P, Souplet P: Superlinear Parabolic Problems: Blow-Up, Global Existence and Steady States. Birkhäuser, Basel; 2007.
Galaktionov VA, Vázquez JL: The problem of blow-up in nonlinear parabolic equations. Discrete Contin. Dyn. Syst. 2002, 8: 399-433. 10.3934/dcds.2002.8.399
Deng K, Levine HA: The role of critical exponents in blow-up theorems: the sequel. J. Math. Anal. Appl. 2000, 243: 85-126. 10.1006/jmaa.1999.6663
Bandle C, Brunner H: Blow-up in diffusion equations: a survey. J. Comput. Appl. Math. 1998, 97: 3-22. 10.1016/S0377-0427(98)00100-9
Samarskii AA, Galaktionov VA, Kurdyumov SP, Mikhailov AP: Blow-Up in Problems for Quasilinear Parabolic Equations. Nauka, Moscow; 1987.
Levine HA: The role of critical exponents in blow-up theorems. SIAM Rev. 1990, 32: 262-288. 10.1137/1032046
Ding JT: Blow-up solutions and global existence for quasilinear parabolic problems with Robin boundary conditions. Abstr. Appl. Anal. 2014., 2014:
Zhang LL: Blow-up of solutions for a class of nonlinear parabolic equations. Z. Anal. Anwend. 2006, 25: 479-486.
Protter MH, Weinberger HF: Maximum Principles in Differential Equations. Prentice-Hall, Englewood Cliffs; 1967.
Friedman A: Partial Differential Equation of Parabolic Type. Prentice-Hall, Englewood Cliffs; 1964.
Qu CY, Bai XL, Zheng SN: Blow-up versus extinction in a nonlocal p -Laplace equation with Neumann boundary conditions. J. Math. Anal. Appl. 2014, 412: 326-333. 10.1016/j.jmaa.2013.10.040
Li FS, Li JL: Global existence and blow-up phenomena for nonlinear divergence form parabolic equations with inhomogeneous Neumann boundary conditions. J. Math. Anal. Appl. 2012, 385: 1005-1014. 10.1016/j.jmaa.2011.07.018
Gao WJ, Han YZ: Blow-up of a nonlocal semilinear parabolic equation with positive initial energy. Appl. Math. Lett. 2011, 24: 784-788. 10.1016/j.aml.2010.12.040
Song JC: Lower bounds for the blow-up time in a non-local reaction-diffusion problem. Appl. Math. Lett. 2011, 24: 793-796. 10.1016/j.aml.2010.12.042
Ding JT, Li SJ: Blow-up and global solutions for nonlinear reaction-diffusion equations with Neumann boundary conditions. Nonlinear Anal. TMA 2008, 68: 507-514. 10.1016/j.na.2006.11.016
Jazar M, Kiwan R: Blow-up of a non-local semilinear parabolic equation with Neumann boundary conditions. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 2008, 25: 215-218. 10.1016/j.anihpc.2006.12.002
Soufi AE, Jazar M, Monneau R: A Gamma-convergence argument for the blow-up of a non-local semilinear parabolic equation with Neumann boundary conditions. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 2007, 24: 17-39. 10.1016/j.anihpc.2005.09.005
Payne LE, Schaefer PW: Lower bounds for blow-up time in parabolic problems under Neumann conditions. Appl. Anal. 2006, 85: 1301-1311. 10.1080/00036810600915730
Ishige K, Yagisita H: Blow-up problems for a semilinear heat equations with large diffusion. J. Differ. Equ. 2005, 212: 114-128. 10.1016/j.jde.2004.10.021
Ishige K, Mizoguchi N: Blow-up behavior for semilinear heat equations with boundary conditions. Differ. Integral Equ. 2003, 16: 663-690.
Mizoguchi N: Blow-up rate of solutions for a semilinear heat equation with Neumann boundary condition. J. Differ. Equ. 2003, 193: 212-238. 10.1016/S0022-0396(03)00128-1
Mizoguchi N, Yanagida E: Blow-up of solutions with sign changes for a smilinear diffusion equation. J. Math. Anal. Appl. 1996, 204: 283-290. 10.1006/jmaa.1996.0436
Deng K: Blow-up behavior of the heat equations with Neumann boundary conditions. J. Math. Anal. Appl. 1994, 188: 641-650. 10.1006/jmaa.1994.1450
Chen XY, Matano H: Convergence, asymptotic periodicity, and finite-point blow-up in one-dimensional semilinear heat equations. J. Differ. Equ. 1989, 78: 160-190. 10.1016/0022-0396(89)90081-8
Lair AV, Oxley ME: A necessary and sufficient condition for global existence for degenerate parabolic boundary value problem. J. Math. Anal. Appl. 1998, 221: 338-348. 10.1006/jmaa.1997.5900
Zhang HL: Blow-up solutions and global solutions for nonlinear parabolic problems. Nonlinear Anal. TMA 2008, 69: 4567-4574. 10.1016/j.na.2007.11.013
Gao XY, Ding JT, Guo BZ: Blow-up and global solutions for quasilinear parabolic equations with Neumann boundary conditions. Appl. Anal. 2009, 88: 183-191. 10.1080/00036810802713818
Acknowledgements
This work was supported by the National Natural Science Foundation of China (Nos. 61074048 and 61174082), the Research Project Supported by Shanxi Scholarship Council of China (Nos. 2011-011 and 2012-011), and the Higher School ‘131’ Leading Talent Project of Shanxi Province.
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Ding, J. Global existence and blow-up for a class of nonlinear reaction diffusion problems. Bound Value Probl 2014, 168 (2014). https://doi.org/10.1186/s13661-014-0168-5
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DOI: https://doi.org/10.1186/s13661-014-0168-5