Blow-up and global existence for the non-local reaction diffusion problem with time dependent coefficient
© Ahmed et al.; licensee Springer. 2013
Received: 4 July 2013
Accepted: 18 September 2013
Published: 9 November 2013
Blow-up and global existence for the non-local reaction diffusion problem with time dependent coefficient under the Dirichlet boundary condition are investigated. We derive the conditions on the data of problem (1.1) sufficient to guarantee that blow-up will occur, and obtain an upper bound for . Also we give the condition for global existence of the solution.
Keywordsblow-up global existence non-local reaction diffusion problem Dirichlet boundary condition
where is a bounded domain with a smooth boundary ∂ Ω, Δ is the Laplace operator, and is the possible blow-up time. By the maximum principle, it follows that in the time interval of existence. The coefficient is assumed to be nonnegative. The particular case of of problem (1.1) has already been investigated by many authors, in [1, 2], they studied the question of blow-up for the solution, and in [3–5], they derived lower bounds for blow-up time under different boundary conditions. To deal with problem (1.1) with time dependent coefficient, we make the assumption on the parameters p and q, that is, .
where Ω is a bounded sufficiently smooth domain in , , and the coefficient is assumed nonnegative or strictly positive depending on the situation.
In next, we employ a method used by Kaplan in  to obtain a condition, which leads to blow-up at some finite time and also leads to an upper bound for the blow-up time. In Section 3, we derive the condition on the data of problem (1.1) sufficient to guarantee the global existence of .
2 Conditions for blow-up in finite time
defined in , where is the solution of (1.1) and .
If for some , then blows up at time . This result is summarized in the following theorem.
3 Condition for global existence
valid in for a nonnegative function v that vanishes on ∂ Ω. In this section, our results restricted to for proof of (3.1), see .
This result is summarized in the next theorem.
Theorem 2 Let Ω be a bounded domain in , and assume that the data of problem (1.1) satisfy conditions (3.4), (3.16), (3.17). Then the auxiliary function defined in (3.2) satisfies (3.20), and exists for all time .
The authors would like to express sincere gratitude to the referees for their valuable suggestions and comments on the original manuscript. This work is supported in part by the NSF of PR China (11371384) and in part by the Natural Science Foundation Project of CQ CSTC (2010BB9218).
- 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
- Wang M, Wang Y: Properties of positive solutions for non-local reaction-diffusion problem. Math. Methods Appl. Sci. 1996, 19: 1141-1156. 10.1002/(SICI)1099-1476(19960925)19:14<1141::AID-MMA811>3.0.CO;2-9MathSciNetView ArticleMATHGoogle Scholar
- Song JC: Lower bounds for blow-up time in a non-local reaction-diffusion problem. Appl. Math. Lett. 2011, 5: 793-796.View ArticleGoogle Scholar
- 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 ArticleMATHGoogle Scholar
- Liu D, Mu C, Ahmed I: Blow-up for a semilinear parabolic equation with nonlinear memory and nonlocal nonlinear boundary. Taiwan. J. Math. 2013, 17: 1353-1370.MathSciNetMATHGoogle Scholar
- Payne LE, Philippin GA: Blow-up phenomena in parabolic problems with time dependent coefficients under Dirichlet boundary conditions. Proc. Am. Math. Soc. 2013, 141(7):2309-2318. 10.1090/S0002-9939-2013-11493-0MathSciNetView ArticleMATHGoogle Scholar
- Liu D, Mu C, Xin Q: Lower bounds estimate for the blow-up time of a nonlinear nonlocal porous medium equation. Acta Math. Sci. 2012, 32(3):1206-1212. 10.1016/S0252-9602(12)60092-7MathSciNetView ArticleMATHGoogle Scholar
- Kaplan S: On the growth of solutions of quasilinear parabolic equations. Commun. Pure Appl. Math. 1963, 16: 305-330. 10.1002/cpa.3160160307View ArticleMATHGoogle Scholar
- Mu C, Liu D, Zhou S: Properties of positive solutions for a nonlocal reaction-diffusion equation with nonlocal nonlinear boundary condition. J. Korean Math. Soc. 2010, 47(6):1317-1328. 10.4134/JKMS.2010.47.6.1317MathSciNetView ArticleMATHGoogle Scholar
- Talenti G: Best constant in Sobolev inequality. Ann. Mat. Pura Appl. 1976, 110(1):353-372. 10.1007/BF02418013MathSciNetView ArticleMATHGoogle Scholar
- Payne LE, Philippin GA: Blow-up phenomena for a class of parabolic systems with time dependent coefficients. Appl. Math. 2012, 3: 325-330. 10.4236/am.2012.34049MathSciNetView ArticleGoogle Scholar
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