Reaction diffusion equations with non-linear boundary conditions, blowup and steady states

Mathematical Methods in the Applied Sciences

Volumn 21, Issue 17, 1998, Pages 1593-1617

  Leung, Anthony W ^a   Zhang, Qin ^a  

^a UNIVERSITY OF CINCINNATI   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

BOUNDARY CONDITIONS; FUNCTIONS; MATHEMATICAL MODELS;

ELLIPTIC PROBLEMS; REACTION DIFFUSION EQUATIONS;

PROBLEM SOLVING;

EID: 0032204105     PISSN: 01704214     EISSN: None     Source Type: Journal    
DOI: 10.1002/(SICI)1099-1476(19981125)21:17<1593::AID-MMA12>3.0.CO;2-1     Document Type: Article

References (30)

1. Bebernes, J. and Eberly, D., Mathematical problems from Combustion Theory, Springer, Berlin, 1989.
2. Caffarelli, L. A. and Friedman, A., 'Blowup of solutions of nonlinear heat equations', J. Math. Anal. Appl., 129, 409-419 (1988).
3. Chipot, M., Fila, M. and Quitter, P., 'Stationary solutions, blowup and convergence to stationary solutions for semilinear parabolic equations with nonlinear boundary conditions', Acta Math. Univ. Comenianae, LX, 35-103 (1991).
4. Chipot, M., Fila, M. and Quittner, P. 'Stationary solutions, blowup and convergence to stationary solutions for semilinear parabolic equations with nonlinear boundary conditions', Acta Math. Univ. Comenianae, (N.S) 60, 35-103 (1991).
5. Escher, J., 'Global existence, and nonexistence for semilinear parabolic systems parabolic systems with nonlinear boundary condition', Math. Ann. 284(2), 285-305 (1989).
6. Fila, M. and Quitter, P., 'The blow up rate for the heat equation with a nonlinear boundary condition', Comment Math. Univ. Carolinae, 30, 479-484 (1989).
7. Fila, M. and Quitter, P., 'The blow up rate for the heat equation with a nonlinear boundary condition', Math. Meth. Appl. Sci. 14, 197-205 (1991).
8. Filo, J., 'Diffusivity versus absorption through the boundary', J. Diff. Eqns. 99(2), 281-305 (1992).
9. Friedman, A., Partial Differential Equations of parabolic Type. Prentice-Hall, Englewood Clifts, NJ, 1964.
10. Friedman, A. and Mcleod, B., 'Blow up of positive solutions of semilinear heat equations', Indian Univ. Math. J. 34, 425-477 (1985).
11. Gidas, B. and Spruck, J., 'Global and local behavior of positive solutions of nonlinear elliptic equations', Comm. Pure Appl Math., 34, 525-595 (1981).
12. Gomez, J. L., Marquez, V. and Wolanasi, N., 'Blowup results and localization of blowup points for the heat equations with a nonlinear boundary conditions', J. Diff. Eqns. 92(2), 384-401 (1991).
13. Hu, B., 'Nonexistence of a positive solution of the Laplace equation with a nonlinear boundary condition', Diff. Intergral Eqns. 7, 301-313 (1994).
14. Hu, B. and Yin, H. M., 'The profile near blowup time for solutions of the heat equation with a nonlinear boundary conditions', IMA preprint series # 1116, March 1993.
15. Leung, A., 'Systems of Nonlinear Partial Differential Equations', Kluwer, Dordrecht, Boston, 1989.
16. Levine, H., 'Stability and instability for solutions of Burguers' equation with a semilinear boundary condition', SIAM J. Math. Anal. 19, 312-336 (1988).
17. Levine, H. and Payne, L., 'Nonexistence theorems for the heat equations with nonlinear boundary conditions and for the Porous medium equation backward in time', J. Diff. Eqns. 16, 319-334 (1974).
18. Liu, W., 'The blowup rate of solutions of semilinear heat equations', J. Diff. Eqns. 77, 104-122 (1989).
19. Pao, C. V., Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992.
20. Protter, M. H. and Weinberger, H. F., 'Maximum principles in Differential Equations', Prentice-Hall, Englewood Cliff, NJ, 1967.
21. Quittner, P., 'On global existence and stationary solutions for two classes of semilinear parabolic problems, Comment Math. Univ Carolin 34, 105-124 (1993).
22. Sperb, R., 'Maximum Principles and their Applications', Academic Press, New York, 1981.
23. W. Walter, On existence and nonexistence in the large of solutions of parabolic differential equations with a nonlinear boundary condition', SIAM J. Math. Anal. 61, 85-90 (1975).
24. Wang, M. X. and Wu, Y. H., 'Global exist and blow up problems for quasi-linear parabolic equations with nonlinear boundary conditions, SIAM J. Math. Anal. 24 (1993).
25. Wang, M. X. and Ye, Q., 'Heat conduction problem with nonlinear absorption versus release through the boundary', Nonlinear Anal. T M A submitted (1994).
26. Weissler, F. B., 'Existence and nonexistence of global solutions for a semi-linear heat eqaution', Israel J. Maths., 38, 29-40 (1981).
27. α blow-up estimate for a nonlinear heat equation', Comm. Pure and Appl. Math. 38, 291-295 (1985).
28. Wolanski, N., 'Global behaviour of positive solution to nonlinear diffusion problems with nonlinear absorption through the boundary', SIAM J. Math. Anal. 24, 317-326 (1993).
29. Wu, Y. H. and Wang, M. X. 'Existence and nonexistence of global solution of nonlinear parabolic equation with nonlinear boundary conditions', Chinese Ann Math. Ser. B (1994) to appear.
30. Yin, H. M., 'Blow up versus global solvability for a class of nonlinear parabolic equations', Preprint, Univ. of Toronto, 1992.