Research Article | Open Access
Jun Zhou, "Global Existence and Blowup for a Reaction-Diffusion System with Nonlocal Boundary Condition", International Scholarly Research Notices, vol. 2013, Article ID 852902, 9 pages, 2013. https://doi.org/10.1155/2013/852902
Global Existence and Blowup for a Reaction-Diffusion System with Nonlocal Boundary Condition
This paper considers the singularity properties of positive solutions for a reaction-diffusion system with nonlocal boundary condition. The conditions on the existence and nonexistence of global positive solutions are given. Moreover, we establish the blow-up rate estimate for the blow-up solution.
This paper studies the singularity properties of the following reaction-diffusion system with nonlocal boundary condition:
where is a bounded domain of , , with smooth boundary and is the closure of . , and are positive numbers which ensure that the equations in (1) are completely coupled with the nonlinear reaction terms. The functions defined for , are nonnegative and continuous. The initial values and are nonnegative, which are mathematically convenient and currently followed throughout the paper. We also assume that satisfies the compatibility condition on , and that and for any for the sake of the meaning of nonlocal boundary.
Denote that , , and , where . A pair of functions is called a classical solution of problem (1) if for some , , and satisfies (1). The local existence of classical solution of (1) is standard (see [1, 2]). If , it is easy to see , and we say that the solution of problem (1) blows up at finite time . If , is called a global solution of problem (1).
Over the past few years, many physical phenomena were formulated as nonlocal mathematical models (see [3, 4]). It has also being suggested that nonlocal growth terms present a more realistic model in physics for compressible reactive gases. Problem (1) arises in the study of the heat transfer with local source (see [5, 6]) and in the study of population dynamics (see [7, 8]).
In recent years many authors have investigated the following initial boundary value problem of reaction-diffusion system:
with Dirichlet, Neumanns or Robin boundary condition, which can be used to describe heat propagation on the boundary of container (see [2, 4, 9–23] and the literatures cited therein). Specially, when have the form
However, there are some important phenomena formulated as parabolic equations which are coupled with nonlocal boundary conditions in mathematical modeling such as thermoelasticity theory (see [24–26]). In this case, the solution could be used to describe the entropy per volume of the material. The problem of nonlocal boundary conditions for linear parabolic equation of the type
with uniformly elliptic operator
As for more general discussions on the dynamic of parabolic problem with nonlocal boundary conditions, one can see Pao , where the following problem:
was considered, and recently Pao  gave the numerical solutions for diffusion equation with nonlocal conditions.
In particular, the following single equation:
under the assumption the for any was consider, by Seo  and the following blow-up rate estimate is established:
where is the blow-up time, and is any constant satisfying .
Furthermore, Zheng and Kong  gave the condition for global existence or nonexistence of solution to the following system:
Motivated by the above cited works, in this paper, we deal with singularity analysis of the parabolic system (1) with nonlocal boundary condition and it is seems that there is no work dealing with this type of systems except the single equations case, although this is a very classical model. Our main results read as follows.
Theorem 1. If , , and , then every nonnegative solution of (1) is global.
Theorem 2. Suppose or or .(i)For any nonnegative functions and , the nonnegative solution of (1) blows up in finite time provided the initial values are large enough.(ii)If and for any , the nonnegative solution of (1) blows up in finite time with any positive initial values.(iii)If and for any , the nonnegative solution of (1) is global with small initial values.
To estimate the blow-up rate of the blow-up solution of (1), we need to add some assumptions for initial data as follows.(H1) for some .(H2) If , then there exists a a sufficient small constant (which will be given in Section 4) such that and on , where (H3) If , then there exists a sufficient small constant (which will be given in Section 4) such that and on , where
Theorem 3. Suppose , and , for all and assumptions (H1)-(H2) hold. If is the smooth solution of (1) and blows up in finite time , then there exist positive constants such that
Theorem 4. Suppose , and , for all and assumptions (H1) and (H3) hold. If is the smooth solution of (1) and blows up in finite time , then there exist positive constants such that
The rest of this paper is organized as follows. In the next section, we give some preliminaries, which include the comparison principle related to system (1). In Section 3, we will study the conditions for the solution to blowup and exist globally and prove Theorems 1 and 2. In Section 4, we will establish the precise blowup rate estimate for small weighted nonlocal boundary and prove Theorems 3 and 4.
In this section, we give some basic preliminaries. We begin with the definition of upper and lower solutions of (1).
Definition 5. A pair of nonnegative functions is called an upper solution of problem (1) if and satisfies
Similarly, is called a lower solution of (1) if it satisfied all the reversed inequalities in (16).
Lemma 6. Let , and be continuous and nonnegative functions, and let satisfy
and then on .
Proof. Set . Since , by continuous, there exists such that for all . Thus, .
We claim that will lead to a contradiction. Indeed, suggests that or for some . Without loss of generality, we suppose that .
If , we first notice that
In addition, it is clear that on the boundary and at initial state . Then it follows from the strong maximum principle that in , which contradicts .
If , we will have a contradiction
In the last inequality, we have used the facts that for any and for any , which is a direct result of the previous case.
Therefore, the claim is true and thus , which implies that on . The proof is complete.
Remark 7. If and for any in Lemma 6, we can obtain on under the assumption that for any . Indeed, for any , we can conclude that on as the proof of Lemma 6. Then the desired result follows from the limit procedure .
Lemma 8. Let and be a upper and lower solution of (1) in , respectively. If for , then on .
Proof. Let and , and then
So, the functions and satisfy
Lemma 6 ensures that on , that is, on . The proof is complete.
3. Global Existence and Blowup
In this section, we will use the upper and lower solutions and their corresponding comparison principle developed in Section 2 to get the global existence or finite blowup of the solution to (1). Let us first give the proof of Theorem 1.
Proof of Theorem 1. Using the condition , and , we have . Thus, we can choose two positive constant and such that
Then, let be a continuous function such that and set
We consider the following auxiliary problem:
where is the measure of and . It follows from [32, Theorem 4.2] that exists globally and indeed on [32, Theorem 2.1].
Our aim is to show that is a global upper solution of (1). Indeed, a direct computation yields
Here, we have used that conclusion and the inequality (23). We still have to consider the boundary and initial conditions. When , we have
Similarly, we have
It is clear that and . Therefore, we get that is a global upper solution of (1) and hence the solution of (1) exists globally by Lemma 8. The proof is complete.
Proof of Theorem 2. (i) Let be the solution of (2) and (3) with homogeneous Dirichlet boundary. Then it is well known for sufficiently large initial data that the solution blows up in finite time when or or (Theorem A). On the other hand, it is obvious that is a lower solution of (1). Hence, the solution of (1) with large initial data blows up in finite time.
(ii) We consider the following ODE system:
If or , it is clear that the solution of (29) blows up in finite time. For the case and , it follows that
Thus, we get
Then implies that blows up in finite time, and so does . From the above analysis, we see that or or implies that blows up in finite time. Under the assumption and for any , is a lower solution of problem (1). Therefore, by Lemma 8, we see that the solution of (1) satisfies and then blows up in finite time.
(iii) Let be the positive solution of the linear elliptic problem
and be the positive solution of the linear elliptic problem
Since and for any , we can choose such that , .
Let and , where are positive constants which satisfy . We remark that under the assumption or or , we can choose such easily. We now show that is an upper solution of (1) for small initial data . Indeed, for any , we have
Here, we have used , . The above inequalities show that is an upper solution of (1) whenever and . The proof is complete.
4. Blowup Rate
In this section, we will estimate the blow-up rate of (1). By the standard methods (see [1, 2, 6]), we can show that system (1) has a smooth solution provided that satisfy the hypotheses (H1). We thus assume that the smooth solution of (1) blows up at finite time and set , . We can obtain the blow-up rate from the following lemmas.
Lemma 9. Suppose that satisfy (H1), and then there exists a positive constant such that
Lemma 10. If , , and for any , then there exists a positive constant such that the solution of (1) with positive initial value satisfies
Proof. Let , where is a positive constant to be chosen. For , a series of calculations shows that
If we choose , then . So, we have
where is a function of and , which lines between and .
When , we have
Denote that for any . Since for any , . It follows from Jensen's inequality and that
Combining the above inequality with (42), we obtain
For the initial condition, we have
on provided that .
Summarily, if we take small enough such that
it follows from (41), (44), (45), and [32, Theorem 2.1] that , which implies (39). The proof is complete.
if , where ,, , and . It is easy to see that and if .
Let be the solution of the following system:
It is easy to see that by Remark 7 if for any .
Proof. Let and . For , a series of calculations shows that
For , using the boundary conditions we have
It follows from for any and Jensen's inequality that
Similarly, we have
For the initial condition, under assumption (H2), we have
if is small enough such that and . Then (49) follows from (50)–(55). The proof is complete.
Proof of Theorem 3. Integrating the inequality for in (49) on yields
where . Since , we obtain
Combining (36) and (57), we get
where and are two positive constants.
Since , it follows from (57) that
Integrating this inequality from to , we get
where is a positive constant. Combining (36) and (60), we get
where and are two positive constants. We completed the proof of Theorem 3.
This study is partially supported by the NSFC Grant 11201380, the Fundamental Research Funds for the Central Universities Grant XDJK2012B007, Doctor Fund of Southwest University Grant SWU111021, and Educational Fund of Southwest University Grant 2010JY053.
- O. A. Ladyženskaja, V. A. Solonik, and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, vol. 23 of Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, USA, 1967.
- C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, NY, USA, 1992.
- K. Bimpong-Bota, P. Ortoleva, and J. Ross, “Far-from-equilibrium phenomena at local sites of reaction,” The Journal of Chemical Physics, vol. 60, no. 8, pp. 3124–3133, 1974.
- P. Souplet, “Uniform blow-up profiles and boundary behavior for diffusion equations with nonlocal nonlinear source,” Journal of Differential Equations, vol. 153, no. 2, pp. 374–406, 1999.
- V. A. Galaktionov, “On asymptotic self-similar behaviour for a quasilinear heat equation: single point blow-up,” SIAM Journal on Mathematical Analysis, vol. 26, no. 3, pp. 675–693, 1995.
- A. A. Samarskii, V. A. Galaktionov, S. P. Kurdyumov, and A. P. Mikhailoi, Blow-Up in Quasilinear Parabolic Equations, Nauka, Moscow, Russia, 1987.
- R. S. Cantrell and C. Cosner, “Diffusive logistic equations with indefinite weights: population models in disrupted environments. II,” SIAM Journal on Mathematical Analysis, vol. 22, no. 4, pp. 1043–1064, 1989.
- J. Furter and M. Grinfeld, “Local vs. nonlocal interactions in population dynamics,” Journal of Mathematical Biology, vol. 27, no. 1, pp. 65–80, 1989.
- J. M. Chadam, A. Peirce, and H.-M. Yin, “The blowup property of solutions to some diffusion equations with localized nonlinear reactions,” Journal of Mathematical Analysis and Applications, vol. 169, no. 2, pp. 313–328, 1992.
- H. Chen, “Global existence and blow-up for a nonlinear reaction-diffusion system,” Journal of Mathematical Analysis and Applications, vol. 212, no. 2, pp. 481–492, 1997.
- F. Dickstein and M. Escobedo, “A maximum principle for semilinear parabolic systems and applications,” Nonlinear Analysis: Theory, Methods & Applications, vol. 45, no. 7, pp. 825–837, 2001.
- M. Escobedo and M. A. Herrero, “A semilinear parabolic system in a bounded domain,” Annali di Matematica Pura ed Applicata, vol. 165, pp. 315–336, 1993.
- V. A. Galaktionov and J. L. Vázquez, “The problem of blow-up in nonlinear parabolic equations,” Discrete and Continuous Dynamical Systems A, vol. 8, no. 2, pp. 399–433, 2002.
- F.-C. Li, S.-X. Huang, and C.-H. Xie, “Global existence and blow-up of solutions to a nonlocal reaction-diffusion system,” Discrete and Continuous Dynamical Systems A, vol. 9, no. 6, pp. 1519–1532, 2003.
- L. Gang and B. D. Sleeman, “Non-existence of global solutions to systems of semi-linear parabolic equations,” Journal of Differential Equations, vol. 104, no. 1, pp. 147–168, 1993.
- F. Merle and H. Zaag, “O.D.E. type behavior of blow-up solutions of nonlinear heat equations,” Discrete and Continuous Dynamical Systems A, vol. 8, no. 2, pp. 435–450, 2002.
- J. D. Rossi and N. Wolanski, “Blow-up vs. global existence for a semilinear reaction-diffusion system in a bounded domain,” Communications in Partial Differential Equations, vol. 20, no. 11-12, pp. 1991–2004, 1995.
- P. Souplet, “Blow-up in nonlocal reaction-diffusion equations,” SIAM Journal on Mathematical Analysis, vol. 29, no. 6, pp. 1301–1334, 1998.
- M. Wang and Y. Wang, “Properties of positive solutions for non-local reaction-diffusion problems,” Mathematical Methods in the Applied Sciences, vol. 19, no. 14, pp. 1141–1156, 1996.
- M. Wang, “Global existence and finite time blow up for a reaction-diffusion system,” Zeitschrift für Angewandte Mathematik und Physik, vol. 51, no. 1, pp. 160–167, 2000.
- J. Zhang, “Boundedness and blow-up behavior for reaction-diffusion systems in a bounded domain,” Nonlinear Analysis: Theory, Methods & Applications, vol. 35, no. 7, pp. 833–844, 1999.
- S. Zheng, “Global boundedness of solutions to a reaction-diffusion system,” Mathematical Methods in the Applied Sciences, vol. 22, no. 1, pp. 43–54, 1999.
- S. Zheng, “Global existence and global non-existence of solutions to a reaction-diffusion system,” Nonlinear Analysis: Theory, Methods & Applications, vol. 39, no. 3, pp. 327–340, 2000.
- W. A. Day, “A decreasing property of solutions of parabolic equations with applications to thermoelasticity,” Quarterly of Applied Mathematics, vol. 40, no. 4, pp. 468–475, 1983.
- W. A. Day, Heat Conduction within Linear Thermoelasticity, vol. 30 of Springer Tracts in Natural Philosophy, Springer, New York, NY, USA, 1985.
- A. Friedman, “Monotonic decay of solutions of parabolic equations with nonlocal boundary conditions,” Quarterly of Applied Mathematics, vol. 44, no. 3, pp. 401–407, 1986.
- C. V. Pao, “Asymptotic behavior of solutions of reaction-diffusion equations with nonlocal boundary conditions,” Journal of Computational and Applied Mathematics, vol. 88, no. 1, pp. 225–238, 1998.
- C. V. Pao, “Numerical solutions of reaction-diffusion equations with nonlocal boundary conditions,” Journal of Computational and Applied Mathematics, vol. 136, no. 1-2, pp. 227–243, 2001.
- S. Seo, “Blowup of solutions to heat equations with nonlocal boundary conditions,” Kobe Journal of Mathematics, vol. 13, no. 2, pp. 123–132, 1996.
- L.-H. Kong and M.-X. Wang, “Global existence and blow-up of solutions to a parabolic system with nonlocal sources and boundaries,” Science in China A, vol. 50, no. 9, pp. 1251–1266, 2007.
- S. Zheng and L. Kong, “Roles of weight functions in a nonlinear nonlocal parabolic system,” Nonlinear Analysis: Theory, Methods & Applications, vol. 68, no. 8, pp. 2406–2416, 2008.
- K. Deng, “Comparison principle for some nonlocal problems,” Quarterly of Applied Mathematics, vol. 50, no. 3, pp. 517–522, 1992.
- A. Friedman and B. McLeod, “Blow-up of positive solutions of semilinear heat equations,” Indiana University Mathematics Journal, vol. 34, no. 2, pp. 425–447, 1985.
Copyright © 2013 Jun Zhou. 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.