A Quasilinear Parabolic System with Nonlocal Boundary Condition
© Botao Chen et al. 2011
Received: 8 May 2010
Accepted: 11 August 2010
Published: 18 August 2010
We investigate the blow-up properties of the positive solutions to a quasilinear parabolic system with nonlocal boundary condition. We first give the criteria for finite time blowup or global existence, which shows the important influence of nonlocal boundary. And then we establish the precise blow-up rate estimate. These extend the resent results of Wang et al. (2009), which considered the special case , and Wang et al. (2007), which studied the single equation.
where , and is a bounded connected domain with smooth boundary. and for the sake of the meaning of nonlocal boundary are nonnegative continuous functions defined for and , while the initial data , are positive continuous functions and satisfy the compatibility conditions and for , respectively.
with , , , and . They obtained that solutions of (1.6) are global if , and may blow up in finite time if . For the critical case of , there should be some additional assumptions on the geometry of .
Recently, the genuine degenerate situation with zero boundary values for (1.7) has been discussed by Lei and Zheng . Clearly, problem (1.6) is just the special case by taking in (1.7) with zero boundary condition.
with nonlocal boundary condition (1.2). The typical characterization of systems (1.10) and (1.11) is the complete couple of the nonlocal sources, which leads to the analysis of simultaneous blowup.
where and are positive parameters. They gave the criteria for finite time blowup or global existence, and established blow-up rate estimate.
To our knowledge, there is no work dealing with the parabolic system (1.1) with nonlocal boundary condition (1.2) except for the single equation case, although this is a very classical model. Therefore, the main purpose of this paper is to understand how the reaction terms, the weight functions and the nonlinear diffusion affect the blow-up properties for the problem (1.1)–(1.3). We will show that the weight functions play substantial roles in determining blowup or not of solutions. Firstly, we establish the global existence and finite time blow-up of the solution. Secondly, we establish the precise blowup rate estimates for all solutions which blow up.
Our main results could be stated as follows.
Suppose that for any . If and hold, then any solution to (1.1)–(1.3) with positive initial data blows up in finite time.
Theorem 1.2 ..
Suppose that for any .
(1)If , and , then every nonnegative solution of (1.1)–(1.3) is global.
(2)If , or , then the nonnegative solution of (1.1)–(1.3) exists globally for sufficiently small initial values and blows up in finite time for sufficiently large initial values.
To establish blow-up rate of the blow-up solution, we need the following assumptions on the initial data
(1) for some ;
where , , and will be given in Section 4.
Theorem 1.3 ..
This paper is organized as follows. In the next section, we give the comparison principle of the solution of problem (1.1)–(1.3) and some important lemmas. In Section 3, we concern the global existence and nonexistence of solution of problem (1.1)–(1.3) and show the proofs of Theorems 1.1 and 1.2. In Section 4, we will give the estimate of the blow-up rate.
In this section, we give some basic preliminaries. For convenience, we denote that for . As it is now well known that degenerate equations need not posses classical solutions, we begin by giving a precise definition of a weak solution for problem (1.1)–(1.3).
Definition 2.1 ..
A vector function defined on , for some , is called a sub (or super) solution of ( 1.1 )–( 1.3 ), if all the following hold:
A weak solution of (1.1) is a vector function which is both a subsolution and a supersolution of (1.1)-(1.3).
Lemma 2.2 (Comparison principle).
Let and be a subsolution and supersolution of (1.1)–(1.3) in , respectively. Then in , if
By Gronwall's inequality, we know that , can be obtained in similar way, then .
Local in time existence of positive classical solutions of the problem (1.1)–(1.3) can be obtained using fixed point theorem (see ), the representation formula and the contraction mapping principle as in . By the above comparison principle, we get the uniqueness of the solution to the problem. The proof is more or less standard, so is omitted here.
From Lemma 2.2, it is easy to see that the solution of (1.1)–(1.3) is unique if .
where are bounded functions and , and and is not identically zero. Then for imply that in . Moreover, if or if , then for imply that in
We give some lemmas that will be used in the following section. Please see  for their proofs.
If , and , then there exist two positive constants , such that . Moreover, for any .
Lemma 2.6 ..
If , or , then there exist two positive constants , such that . Moreover, for any .
3. Global Existence and Blowup in Finite Time
Compared with usual homogeneous Dirichlet boundary data, the weight functions and play an important role in the global existence or global nonexistence results for problem (1.1)–(1.3).
Proof of Theorem 1.1..
where , and we use the assumption
It is easy to check that is the unique solution of the ODE problem (3.1), then and imply that blows up in finite time. Under the assumption that for any , is a subsolution of problem (1.1)–(1.3). Therefore, by Lemma 2.2, we see that the solution of problem (1.1)–(1.3) satisfies and then blows up in finite time.
- (1)Let be the positive solution of the linear elliptic problem(3.4)
where are positive constant such that . We remark that and ensure the existence of such .
here, we used , and .
Now, it follows from (3.8)–(3.15) that defined by (3.7) is a positive supersolution of (1.1)–(1.3).
- (2)If , or , by Lemma 2.6, there exist positive constants such that(3.16)
So we can choose . Furthermore, assume that are small enough to satisfy (3.15). It follows that defined by (3.7) is a positive supersolution of (1.1)–(1.3). Hence, exists globally.
Due to the requirement of the comparison principle we will construct blow-up subsolutions in some subdomain of in which . We use an idea from Souplet  and apply it to degenerate equations. Let be a nontrivial nonnegative continuous function and vanished on . Without loss of generality, we may assume that and . We will construct a blow-up positive subsolution to complete the proof.
notice that is sufficiently small.
Since and is continuous, there exist two positive constants and such that , for all . Choose small enough to insure , hence on . Under the assumption that and for any , we have and Furthermore, choose so large that . By comparison principle, we have . It shows that solution to (1.1)–(1.3) blows up in finite time.
4. Blow-Up Rate Estimates
where , ; , ; , , , ; , ; , . By the conditions (4.1), we have and satisfy that . Under this transformation, assumptions - become
() , for some ;
where will be given later.
By the standard method [16, 42], we can show that system (4.2) has a smooth nonnegative solution , provided that satisfy the hypotheses - . We thus assume that the solution of problem (4.2) blows up in the finite time . Denote . We can obtain the blow-up rate from the following lemmas.
by virtue of Young's inequality. Integrating (4.6) from to , we can obtain (4.4).
where are positive constants independent of . It follows from Lemma 4.1 and (4.19), we have the following lemma.
According the transform and Lemma 4.3, we can obtain Theorem 1.3.
The authors would like to thank the anonymous referees for their suggestions and comments on the original manuscript. This work was partially supported by NSF of China (10771226) and partially supported by the Educational Science Foundation of Chongqing (KJ101303), China.
- Diaz JI, Kersner R: On a nonlinear degenerate parabolic equation in infiltration or evaporation through a porous medium. Journal of Differential Equations 1987, 69(3):368-403. 10.1016/0022-0396(87)90125-2View ArticleMathSciNetGoogle Scholar
- Deng K, Levine HA: The role of critical exponents in blow-up theorems: the sequel. Journal of Mathematical Analysis and Applications 2000, 243(1):85-126. 10.1006/jmaa.1999.6663View ArticleMathSciNetGoogle Scholar
- Levine HA: The role of critical exponents in blowup theorems. SIAM Review 1990, 32(2):262-288. 10.1137/1032046View ArticleMathSciNetGoogle Scholar
- Chen H: Global existence and blow-up for a nonlinear reaction-diffusion system. Journal of Mathematical Analysis and Applications 1997, 212(2):481-492. 10.1006/jmaa.1997.5522View ArticleMathSciNetGoogle Scholar
- Escobedo M, Herrero MA: Boundedness and blow up for a semilinear reaction-diffusion system. Journal of Differential Equations 1991, 89(1):176-202. 10.1016/0022-0396(91)90118-SView ArticleMathSciNetGoogle Scholar
- Escobedo M, Levine HA: Critical blowup and global existence numbers for a weakly coupled system of reaction-diffusion equations. Archive for Rational Mechanics and Analysis 1995, 129(1):47-100. 10.1007/BF00375126View ArticleMathSciNetGoogle Scholar
- Levine HA: A Fujita type global existence–global nonexistence theorem for a weakly coupled system of reaction-diffusion equations. Zeitschrift für Angewandte Mathematik und Physik 1991, 42(3):408-430. 10.1007/BF00945712View ArticleGoogle Scholar
- Quirós F, Rossi JD: Non-simultaneous blow-up in a semilinear parabolic system. Zeitschrift für Angewandte Mathematik und Physik 2001, 52(2):342-346. 10.1007/PL00001549View ArticleGoogle Scholar
- Wang M: Global existence and finite time blow up for a reaction-diffusion system. Zeitschrift für Angewandte Mathematik und Physik 2000, 51(1):160-167. 10.1007/PL00001504View ArticleGoogle Scholar
- Zheng S: Global boundedness of solutions to a reaction-diffusion system. Mathematical Methods in the Applied Sciences 1999, 22(1):43-54. 10.1002/(SICI)1099-1476(19990110)22:1<43::AID-MMA19>3.0.CO;2-8View ArticleMathSciNetGoogle Scholar
- Zheng S: Global existence and global non-existence of solutions to a reaction-diffusion system. Nonlinear Analysis: Theory, Methods & Applications 2000, 39(3):327-340. 10.1016/S0362-546X(98)00171-0View ArticleMathSciNetGoogle Scholar
- Galaktionov VA, Kurdyumov SP, Samarskiĭ AA: A parabolic system of quasilinear equations. I. Differentsial'nye Uravneniya 1983, 19(12):2123-2140.Google Scholar
- Galaktionov VA, Kurdyumov SP, Samarskiĭ AA: A parabolic system of quasilinear equations. II. Differentsial'nye Uravneniya 1985, 21(9):1049-1062.Google Scholar
- Song X, Zheng S, Jiang Z: Blow-up analysis for a nonlinear diffusion system. Zeitschrift für Angewandte Mathematik und Physik 2005, 56(1):1-10. 10.1007/s00033-004-1152-1View ArticleMathSciNetGoogle Scholar
- Lei P, Zheng S: Global and nonglobal weak solutions to a degenerate parabolic system. Journal of Mathematical Analysis and Applications 2006, 324(1):177-198. 10.1016/j.jmaa.2005.12.012View ArticleMathSciNetGoogle Scholar
- Duan Z, Deng W, Xie C: Uniform blow-up profile for a degenerate parabolic system with nonlocal source. Computers & Mathematics with Applications 2004, 47(6-7):977-995. 10.1016/S0898-1221(04)90081-8View ArticleMathSciNetGoogle Scholar
- Li Z, Mu C, Cui Z: Critical curves for a fast diffusive polytropic filtration system coupled via nonlinear boundary flux. Zeitschrift fur Angewandte Mathematik und Physik 2009, 60(2):284-296. 10.1007/s00033-008-7095-1View ArticleMathSciNetGoogle Scholar
- Li Z, Cui Z, Mu C: Critical curves for fast diffusive polytropic filtration equations coupled through boundary. Applicable Analysis 2008, 87(9):1041-1052. 10.1080/00036810802428912View ArticleMathSciNetGoogle Scholar
- Zhou J, Mu C: On the critical Fujita exponent for a degenerate parabolic system coupled via nonlinear boundary flux. Proceedings of the Edinburgh Mathematical Society. Series II 2008, 51(3):785-805. 10.1017/S0013091505001537View ArticleMathSciNetGoogle Scholar
- Zhou J, Mu C: The critical curve for a non-Newtonian polytropic filtration system coupled via nonlinear boundary flux. Nonlinear Analysis: Theory, Methods & Applications 2008, 68(1):1-11. 10.1016/j.na.2006.10.022View ArticleMathSciNetGoogle Scholar
- Day WA: A decreasing property of solutions of parabolic equations with applications to thermoelasticity. Quarterly of Applied Mathematics 1983, 40(4):468-475.Google Scholar
- Day WA: Heat Conduction within Linear Thermoelasticity, Springer Tracts in Natural Philosophy. Volume 30. Springer, New York, NY, USA; 1985:viii+83.View ArticleGoogle Scholar
- Friedman A: Monotonic decay of solutions of parabolic equations with nonlocal boundary conditions. Quarterly of Applied Mathematics 1986, 44(3):401-407.MathSciNetGoogle Scholar
- Deng K: Comparison principle for some nonlocal problems. Quarterly of Applied Mathematics 1992, 50(3):517-522.MathSciNetGoogle Scholar
- Pao CV: Asymptotic behavior of solutions of reaction-diffusion equations with nonlocal boundary conditions. Journal of Computational and Applied Mathematics 1998, 88(1):225-238. 10.1016/S0377-0427(97)00215-XView ArticleMathSciNetGoogle Scholar
- Seo S: Blowup of solutions to heat equations with nonlocal boundary conditions. Kobe Journal of Mathematics 1996, 13(2):123-132.MathSciNetGoogle Scholar
- Seo S: Global existence and decreasing property of boundary values of solutions to parabolic equations with nonlocal boundary conditions. Pacific Journal of Mathematics 2000, 193(1):219-226. 10.2140/pjm.2000.193.219View ArticleMathSciNetGoogle Scholar
- Wang Y, Mu C, Xiang Z: Blowup of solutions to a porous medium equation with nonlocal boundary condition. Applied Mathematics and Computation 2007, 192(2):579-585. 10.1016/j.amc.2007.03.036View ArticleMathSciNetGoogle Scholar
- Kong L, Wang M: Global existence and blow-up of solutions to a parabolic system with nonlocal sources and boundaries. Science in China. Series A 2007, 50(9):1251-1266. 10.1007/s11425-007-0105-5View ArticleMathSciNetGoogle Scholar
- Wang Y, Xiang Z: Blowup analysis for a semilinear parabolic system with nonlocal boundary condition. Boundary Value Problems 2009, 2009:-14.Google Scholar
- Wang Y, Mu C, Xiang Z: Properties of positive solution for nonlocal reaction-diffusion equation with nonlocal boundary. Boundary Value Problems 2007, 2007:-12.Google Scholar
- Yin H-M: On a class of parabolic equations with nonlocal boundary conditions. Journal of Mathematical Analysis and Applications 2004, 294(2):712-728. 10.1016/j.jmaa.2004.03.021View ArticleMathSciNetGoogle Scholar
- Yin Y: On nonlinear parabolic equations with nonlocal boundary condition. Journal of Mathematical Analysis and Applications 1994, 185(1):161-174. 10.1006/jmaa.1994.1239View ArticleMathSciNetGoogle Scholar
- Zheng S, Kong L: Roles of weight functions in a nonlinear nonlocal parabolic system. Nonlinear Analysis: Theory, Methods & Applications 2008, 68(8):2406-2416. 10.1016/j.na.2007.01.067View ArticleMathSciNetGoogle Scholar
- Souplet P: Uniform blow-up profiles and boundary behavior for diffusion equations with nonlocal nonlinear source. Journal of Differential Equations 1999, 153(2):374-406. 10.1006/jdeq.1998.3535View ArticleMathSciNetGoogle Scholar
- Anderson JR: Local existence and uniqueness of solutions of degenerate parabolic equations. Communications in Partial Differential Equations 1991, 16(1):105-143. 10.1080/03605309108820753View ArticleMathSciNetGoogle Scholar
- Day WA: Extensions of a property of the heat equation to linear thermoelasticity and other theories. Quarterly of Applied Mathematics 1982, 40(3):319-330.MathSciNetGoogle Scholar
- Lin Z, Liu Y: Uniform blowup profiles for diffusion equations with nonlocal source and nonlocal boundary. Acta Mathematica Scientia. Series B 2004, 24(3):443-450.MathSciNetGoogle Scholar
- Pao CV: Nonlinear Parabolic and Elliptic Equations. Plenum Press, New York, NY, USA; 1992:xvi+777.Google Scholar
- Pao CV: Blowing-up of solution for a nonlocal reaction-diffusion problem in combustion theory. Journal of Mathematical Analysis and Applications 1992, 166(2):591-600. 10.1016/0022-247X(92)90318-8View ArticleMathSciNetGoogle Scholar
- Deng W: Global existence and finite time blow up for a degenerate reaction-diffusion system. Nonlinear Analysis: Theory, Methods & Applications 2005, 60(5):977-991. 10.1016/j.na.2004.10.016View ArticleMathSciNetGoogle Scholar
- Souplet P: Blow-up in nonlocal reaction-diffusion equations. SIAM Journal on Mathematical Analysis 1998, 29(6):1301-1334. 10.1137/S0036141097318900View ArticleMathSciNetGoogle Scholar
- Friedman A, McLeod B: Blow-up of positive solutions of semilinear heat equations. Indiana University Mathematics Journal 1985, 34(2):425-447. 10.1512/iumj.1985.34.34025View ArticleMathSciNetGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.