A PARABOLIC SYSTEM WITH NONLOCAL BOUNDARY CONDITIONS AND NONLOCAL SOURCES

- Journal title : Communications of the Korean Mathematical Society
- Volume 27, Issue 3, 2012, pp.629-644
- Publisher : The Korean Mathematical Society
- DOI : 10.4134/CKMS.2012.27.3.629

Title & Authors

A PARABOLIC SYSTEM WITH NONLOCAL BOUNDARY CONDITIONS AND NONLOCAL SOURCES

Gao, Wenjie; Han, Yuzhu;

Gao, Wenjie; Han, Yuzhu;

Abstract

In this work, the authors study the blow-up properties of solutions to a parabolic system with nonlocal boundary conditions and nonlocal sources. Conditions for the existence of global or blow-up solutions are given. Global blow-up property and precise blow-up rate estimates are also obtained.

Keywords

global existence;blow-up;nonlocal sources;nonlocal boundary conditions;global blow-up;blow-up rate;

Language

English

References

1.

J. R. Anderson, Local existence and uniqueness of solutions of degenerate parabolic equations, Comm. Partial Differential Equations 16 (1991), no. 1, 105-143.

2.

J. Bebernes, A. Bressan, and A. Lacey, Total blow-up versus single point blow-up, J. Differential Equations 73 (1988), no. 1, 30-44.

3.

S. Carl and V. Lakshmikantham, Generalized quasilinearization method for reaction- diffusion equations under nonlinear and nonlocal flux conditions, J. Math. Anal. Appl. 271 (2002), no. 1, 182-205.

4.

D. E. Carlson, Linear thermoelasticity, in Encyclopedia of Physics, Springer, Berlin, 1972.

5.

Y. J. Chen and M. X. Wang, A class of nonlocal and degenerate quasilinear parabolic system not in divergence form, Nonlinear Anal. 71 (2009), no. 7-8, 3530-3537.

6.

Z. J. Cui and Z. D. Yang, Roles of weight functions to a nonlinear porous medium equation with nonlocal source and nonlocal boundary condition, J. Math. Anal. Appl. 342 (2008), no. 1, 559-570.

7.

W. A. Day, Extensions of a property of the heat equation to linear thermoelasticity and other theories, Quart. Appl. Math. 40 (1982/83), no. 3, 319-330.

8.

W. A. Day, A decreasing property of solutions of parabolic equations with applications to thermoelasticity, Quart. Appl. Math. 40 (1982/83), no. 4, 468-475.

9.

K. Deng, Comparison principle for some nonlocal problems, Quart. Appl. Math. 50 (1992), no. 3, 517-522.

10.

W. B. Deng, Y. X. Li, and C. H. Xie, Blow-up and global existence for a nonlocal degenerate parabolic system, J. Math. Anal. Appl. 277 (2003), no. 1, 199-217.

11.

L. L. Du, Blow-up for a degenerate reaction-diffusion system with nonlinear nonlocal sources, J. Comput. Appl. Math. 202 (2007), no. 2, 237-247.

12.

A. Friedman, Monotonic decay of solutions of parabolic equations with nonlocal boundary conditions, Quart. Appl. Math. 44 (1986), no. 3, 401-407.

13.

A. Friedman and J. B. Mcleod, Blow-up of positive solutions of semilinear heat equations, Indiana Univ. Math. J. 34 (1985), no. 2, 425-447.

14.

L. J. Jiang and H. L. Li, Uniform blow-up profiles and boundary layer for a parabolic system with nonlocal sources, Math. Comput. Modelling 45 (2007), no. 7-8, 814-824.

15.

Z. G. Lin and Y. R. Lin, Uniform blowup profile for diffusion equations with nonlocal source and nonlocal boundary, Acta Math. Sci. Ser. B Engl. Ed. 24 (2004), no. 3, 443-450.

16.

C. V. Pao, Dynamics of reaction-diffusion equations with nonlocal boundary conditions, Quart. Appl. Math. 50 (1995), no. 1, 173-186.

17.

C. V. Pao, Asymptotic behavior of solutions of reaction-diffusion equations with nonlocal boundary conditions, J. Comput. Appl. Math. 88 (1998), no. 1, 225-238.

18.

C. V. Pao, Numerical solutions of reaction-diffusion equations with nonlocal boundary conditions, J. Comput. Appl. Math. 136 (2001), no. 1-2, 227-243.

19.

S. Seo, Blowup of solutions to heat equations with nonlocal boundary conditions, Kobe J. Math. 13 (1996), no. 2, 123-132.

20.

S. Seo, Global existence and decreasing property of boundary values of solutions to parabolic equations with nonlocal boundary conditions, Pacific J. Math. 193 (2000), no. 1, 219-226.

21.

P. Souplet, Uniform blow-up profiles and boundary behavior for diffusion equations with nonlocal nonlinear source, J. Differential Equations 153 (1999), no. 2, 374-406.

22.

Y. L. Wang, C. L. Mu, and Z. Y. Xiang, Blowup of solutions to a porous medium equation with nonlocal boundary condition, Appl. Math. Comput. 192 (2007), no. 2, 579-585.

23.

Z. Y. Xiang, X. G. Hu, and C. L. Mu, Neumann problem for reaction-diffusion systems with nonlocal nonlinear sources, Nonlinear Anal. 61 (2005), no. 7, 1209-1224.

24.

H. M. Yin, On a class of parabolic equations with nonlocal boundary conditions, J. Math. Anal. Appl. 294 (2004), no. 2, 712-728.