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Lower bounds for the blow-up time of the nonlinear non-local reaction diffusion problems in R N ( N 3 )

Boundary Value Problems20142014:265

https://doi.org/10.1186/s13661-014-0265-5

Received: 14 August 2014

Accepted: 8 December 2014

Published: 20 December 2014

Abstract

This paper deals with the blow-up of the solution to a non-local reaction diffusion problem in R N for N 3 under nonlinear boundary conditions. Utilizing the technique of a differential inequality, lower bounds for the blow-up time are derived when the blow-up does occur under some suitable assumptions.

MSC: 35K20, 35K55, 35K65.

Keywords

blow-uplower boundsnon-local reaction diffusion problem

1 Introduction

There is a vast literature on the question of blow-up of solutions to nonlinear parabolic equations and systems. Readers can refer to the books of Straughan [1] and Quittner and Souple [2], as well as to the survey paper of Bandle and Brunner [3]. For more recent work, one can refer to [4]–[12].

In practical situations, one would like to know among other things whether the solution blows up. In this paper, we consider the blow-up for the solution of the following nonlinear non-local reaction diffusion problems, which have been studied by Song in [4]:
u t = u + Ω u p d x k u q in  Ω × ( 0 , t ) ,
(1.1)
u = 0 in  Ω × ( 0 , t ) ,
(1.2)
u ( x , 0 ) = f ( x ) 0 in  Ω ,
(1.3)

where is the Laplace operator, Ω the boundary of Ω and t the possible blow-up time, p , q > 1 . In [4], [13]–[16], the authors have studied the question of blow-up for the solution of parabolic problems by imposing two different nonlinear boundary conditions: homogeneous Dirichlet boundary conditions or homogeneous Neumann boundary conditions. They determine, for solutions that blow up, a lower bound for the blow-up time t in a bounded domain Ω R N for N = 3 . Besides, some authors have also started to consider the blow-up phenomena of those problems under Robin boundary conditions (see [17]–[19]). However, for the case N 3 , the Sobolev inequality, which is important for the result obtained in [4], is no longer applicable. Recently, some papers begin to pay attentions to the study of the blow-up phenomena of solution to an equation in Ω R N , for N 3 (see [20]–[22]).

In the present paper, for convenience, we set p = s + 1 , s > 0 and rewrite (1.1) as follows:
u t = u + Ω u s + 1 d x k u q in  Ω × ( 0 , t ) .
(1.4)

As indicated in [23], it is well known that if p q the solution will not blow up in finite time. Also it is well known that if the initial data are small enough the solution will actually decay exponentially as t (see e.g.[1], [24]). Since we are interested in a lower bound for t, in the case of blow-up, we are only concerned with the case q < p .

We see by the parabolic maximum principles [25], [26] that u is nonnegative in x for t [ 0 , t ) .

In Section 2, we derive the lower bound for the blow-up time of the system (1.1)-(1.3) in R N . The obtained results extend the corresponding conclusions in the literature to R N for any N 3 .

2 A lower bound for the blow-up time

In this section we seek the lower bound for the blow-up time t and establish the following theorem.

Theorem 1

Let u ( x , t ) be the classical nonnegative solution of problem (1.1)-(1.3) in a bounded star-shaped domain Ω R N ( N 3 ) and assume that q < p . Then the quantity
φ ( t ) = Ω u n s d x
(2.1)
satisfies the differential inequality
d φ d t k 1 φ 1 + β ,
(2.2)
from which follows that the blow-up time t is bounded from below; i.e., we have
t φ ( 0 ) 1 k 1 η 1 + β d η = 1 k 1 β φ β ( 0 ) ,
(2.3)

where k 1 , β are positive constants which will be defined later.

Proof

Firstly we compute
φ ( t ) = n s Ω u n s 1 [ u + Ω u s + 1 d x k u q ] d x 4 ( n s 1 ) n s Ω | u n s 2 | 2 d x + n s | Ω | Ω u s ( n + 1 ) d x k n s Ω u n s + q 1 d x .
(2.4)
For convenience, we now set
v = u s , α = q 1 s .
(2.5)
Since q < s + 1 , α < 1 . Thus, we obtain
φ ( t ) 4 ( n s 1 ) n s Ω | v n 2 | 2 d x + n s | Ω | Ω v ( n + 1 ) d x k n s Ω v n + α d x .
(2.6)
By the Hölder inequality, we have
Ω v n + 1 d x ( Ω v n + α d x ) γ 1 γ α ( Ω v n + γ d x ) 1 α γ α
(2.7)
for positive constant γ > 1 . By the inequality
a r + b 1 r r a + ( 1 r ) b , a , b > 0 , 0 < r < 1 ,
(2.8)
we have
Ω v n + 1 d x γ 1 γ α ε 1 Ω v n + α d x + 1 α γ α ε 1 γ 1 1 α Ω v n + γ d x ,
(2.9)
where ε 1 is a positive constant. If we insert (2.9) into (2.6) and choose
ε 1 = k ( γ α ) | Ω | ( γ 1 ) ,
then (2.6) yields
φ ( t ) 4 ( n s 1 ) n s Ω | v n 2 | 2 d x + n s | Ω | 1 α γ α ε 1 γ 1 1 α Ω v n + γ d x .
(2.10)
By the Hölder inequality again, we have
Ω v n + γ d x ( Ω v n d x ) 2 n γ ( N 2 ) 2 n ( Ω v n 2 2 N N 2 d x ) γ ( N 2 ) 2 n ,
(2.11)
where we have chosen 2 n > γ N . Now let c 1 be the best imbedding constant defined in [27]. Using the Sobolev inequality for W 0 1 , 2 L 2 N N 2 for N 3 , we have
Ω v n 2 2 N N 2 d x c 1 2 N N 2 ( Ω | v n 2 | 2 d x ) N N 2 .
Therefore, (2.11) may be rewritten as
Ω v n + γ d x c 1 γ N 2 n ( Ω v n d x ) 2 n γ ( N 2 ) 2 n ( Ω | v n 2 | 2 d x ) N γ 2 n .
(2.12)
Using (2.8) again, we have
Ω v n + γ d x 2 n N γ 2 n c 1 γ N 2 n ε 2 N 2 n γ ( N 2 ) ( Ω v n d x ) 2 n γ ( N 2 ) 2 n + N γ 2 n c 1 γ N 2 n N γ ε 2 Ω | v n 2 | 2 d x ,
(2.13)
where ε 2 is a positive constant to be chosen as follows:
ε 2 = 8 ( n s 1 ) ( γ α ) N γ | Ω | n s 2 ( 1 α ) ε 1 γ 1 1 α c 1 γ N 2 n ,
(2.14)
and inserting (2.13) back into (2.10), we have
φ ( t ) k 1 φ 2 n γ ( N 2 ) 2 n N γ ,
(2.15)
where
k 1 = n s | Ω | 1 α γ α ε 1 γ 1 1 α 2 n N γ 2 n c 1 γ N 2 n ε 2 N 2 n γ ( N 2 ) .
(2.16)
If we set
β = 2 γ 2 n N γ > 0 ,
(2.17)
then (2.15) can be written as
φ ( t ) k 1 φ 1 + β ,
(2.18)
or
d φ k 1 φ 1 + β 1 .
(2.19)
Upon integration we have for t < t ,
t φ ( 0 ) 1 k 1 η 1 + β d η = 1 k 1 β φ β ( 0 ) ,
(2.20)

where φ ( 0 ) = φ ( t ) = Ω u 0 n s d x . □

Declarations

Acknowledgements

The authors would like to express their gratitude to the anonymous referees for helpful and very careful reading on this paper. This research was supported by the Natural Science Foundation of Hunan Province (No. 14JJ4044; No. 15JJ2063; No. 13JJ3085) and the Foundation for Distinguished Young Talents in Higher Education of Guangdong, China (Grant 2013LYM0112).

Authors’ Affiliations

(1)
Department of Mathematics, Hunan University of Science and Technology, Xiangtan, P.R. China
(2)
Huashang College, Guangdong University of Finance and Economics, Guangzhou, P.R. Chinax

References

  1. Straughan B: Explosive Instabilities in Mechanics. Springer, Berlin; 1998.View ArticleMATHGoogle Scholar
  2. Quittner R, Souplet P: Super Linear Parabolic Problems: Blow-up, Global Existence and Steady States. Birkhäuser, Basel; 2007.MATHGoogle Scholar
  3. Bandle C, Brunner H: Blow-up in diffusion equations: a survey. J. Comput. Appl. Math. 1998, 97: 3-22. 10.1016/S0377-0427(98)00100-9MathSciNetView ArticleMATHGoogle Scholar
  4. Song JC: Lower bounds for blow-up time in a non-local reaction-diffusion problem. Appl. Math. Lett. 2011, 5: 793-796. 10.1016/j.aml.2010.12.042View ArticleMathSciNetMATHGoogle Scholar
  5. Payne LE, Song JC: Lower bounds for blow-up time in a nonlinear parabolic problem. J. Math. Anal. Appl. 2009, 354: 394-396. 10.1016/j.jmaa.2009.01.010MathSciNetView ArticleMATHGoogle Scholar
  6. Li YF, Liu Y, Lin CH: Blow-up phenomena for some nonlinear parabolic problems under mixed boundary conditions. Nonlinear Anal., Real World Appl. 2010, 113: 815-3823.MathSciNetGoogle Scholar
  7. Payne LE, Philippin GA, Piro SV: Blow-up phenomena for a semilinear heat equation with nonlinear boundary condition, I. Z. Angew. Math. Phys. 2010, 61: 999-1007. 10.1007/s00033-010-0071-6MathSciNetView ArticleMATHGoogle Scholar
  8. Liu Y: Lower bounds for the blow-up time in a non-local reaction diffusion problem under nonlinear boundary conditions. Math. Comput. Model. 2013, 57: 926-931. 10.1016/j.mcm.2012.10.002View ArticleMathSciNetMATHGoogle Scholar
  9. Liu Y: Blow up phenomena for the nonlinear nonlocal porous medium equation under Robin boundary condition. Comput. Math. Appl. 2013, 66: 2092-2095. 10.1016/j.camwa.2013.08.024MathSciNetView ArticleGoogle Scholar
  10. Liu Y, Luo SG, Ye YH: Blow-up phenomena for a parabolic problem with a gradient nonlinearity under nonlinear boundary condition. Comput. Math. Appl. 2013, 65: 1194-1199. 10.1016/j.camwa.2013.02.014MathSciNetView ArticleMATHGoogle Scholar
  11. Schaefer PW: Blow up phenomena in some porous medium problems. Dyn. Syst. Appl. 2009, 18: 103-110.MathSciNetMATHGoogle Scholar
  12. Payne LE, Schaefer PW: Bounds for the blow-up time for the heat equation under nonlinear boundary conditions. Proc. R. Soc. Edinb. A 2009, 139: 1289-1296. 10.1017/S0308210508000802MathSciNetView ArticleMATHGoogle Scholar
  13. Liu DM, Mu CL, Qiao X: Lower bounds estimate for the blow up time of a nonlinear nonlocal porous medium equation. Acta Math. Sci. 2012, 32: 1206-1212. 10.1016/S0252-9602(12)60092-7View ArticleMathSciNetMATHGoogle Scholar
  14. Payne LE, Schaefer PW: Lower bounds for blow-up time in parabolic problems under Neumann conditions. Appl. Anal. 2006, 85: 1301-1311. 10.1080/00036810600915730MathSciNetView ArticleMATHGoogle Scholar
  15. Payne LE, Philippin GA, Schaefer PW: Bounds for blow-up time in nonlinear parabolic problems. J. Math. Anal. Appl. 2008, 338: 438-447. 10.1016/j.jmaa.2007.05.022MathSciNetView ArticleMATHGoogle Scholar
  16. Payne LE, Song JC: Lower bounds for the blow-up time in a temperature dependent Navier-Stokes flow. J. Math. Anal. Appl. 2007, 335: 371-376. 10.1016/j.jmaa.2007.01.083MathSciNetView ArticleMATHGoogle Scholar
  17. Payne LE, Schaefer PW: Blow-up in parabolic problems under Robin boundary conditions. Appl. Anal. 2008, 87: 699-707. 10.1080/00036810802189662MathSciNetView ArticleMATHGoogle Scholar
  18. Ding J: Global and blow-up solutions for nonlinear parabolic equations with Robin boundary conditions. Comput. Math. Appl. 2013, 65(11):1808-1822. 10.1016/j.camwa.2013.03.013MathSciNetView ArticleGoogle Scholar
  19. Enache C: Blow-up phenomena for a class of quasilinear parabolic problems under Robin boundary condition. Appl. Math. Lett. 2011, 24(3):288-292. 10.1016/j.aml.2010.10.006MathSciNetView ArticleMATHGoogle Scholar
  20. Payne LE, Philippin GA, Schaefer PW: Blow-up phenomena for some nonlinear parabolic problems. Nonlinear Anal. 2008, 69: 3495-3502. 10.1016/j.na.2007.09.035MathSciNetView ArticleMATHGoogle Scholar
  21. Bao AG, Song XF: Bounds for the blow-up time of the solutions to quasi-linear parabolic problems. Z. Angew. Math. Phys. 2014, 65: 115-123. 10.1007/s00033-013-0325-1MathSciNetView ArticleMATHGoogle Scholar
  22. Li HX, Gao WJ, Han YZ: Lower bounds for the blow up time of solutions to a nonlinear parabolic problems. Electron. J. Differ. Equ. 2014., 2014: 10.1186/1687-1847-2014-20Google Scholar
  23. Souplet P: Uniform blow-up profile and boundary behaviour for a non-local reaction-diffusion problem with critical damping. Math. Methods Appl. Sci. 2004, 27: 1819-1829. 10.1002/mma.567MathSciNetView ArticleMATHGoogle Scholar
  24. Payne LE, Schaefer PW: Lower bounds for blow-up time in parabolic problems under Dirichlet conditions. J. Math. Anal. Appl. 2007, 328: 1196-1205. 10.1016/j.jmaa.2006.06.015MathSciNetView ArticleMATHGoogle Scholar
  25. Friedman A: Remarks on the maximum principle for parabolic equations and its applications. Pac. J. Math. 1958, 8: 201-211. 10.2140/pjm.1958.8.201View ArticleMathSciNetMATHGoogle Scholar
  26. Nirenberg L: A strong maximum principle for parabolic equations. Commun. Pure Appl. Math. 1953, 6: 167-177. 10.1002/cpa.3160060202MathSciNetView ArticleMATHGoogle Scholar
  27. Gilbarg D, Trudinger NS: Elliptic Partial Differential Equations of Second Order. Springer, Berlin; 2001.MATHGoogle Scholar

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© Tang et al.; licensee Springer 2014

This article is published under license to BioMed Central Ltd.Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.

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