On the structure and stability of the set of solutions of a nonlocal problem modeling Ohmic heating

Journal of Dynamics and Differential Equations

Volumn 10, Issue 4, 1998, Pages 537-566

  López Gómez, Julián ^a  

^a UNIVERSIDAD COMPLUTENSE DE MADRID   (Spain)

Author keywords

Existence and stability of positive solutions; Global curves of positive solutions; Nonlocal problems

Indexed keywords

EID: 54649085108     PISSN: 10407294     EISSN: 15729222     Source Type: Journal    
DOI: 10.1023/A:1022625024323     Document Type: Article

References (34)

1. Allegretto, V. and Xie, H. (1995). A nonlocal thermistor problem, Eur. J. Appl. Math. 6, 83-94.
2. Amann, H. (1976). Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev. 18, 620-709.
3. Babin, A. V. and Vishik, M. I. (1992). Attractors of Evolution Equations, Studies on Mathematics and Its Applications 25, North-Holland, Amsterdam.
4. Barabanova, A. (1996). The blow-up of solutions of a non-local thermistor problem, Appl. Math. Lett. 9, 59-63.
5. Beresticky, H., Nirenberg, L., and Varadhan, S. R. S. (1994). The principal eigenvalue and maximum principle for second order elliptic operators in general domains, Comm. Pure Appl. Math. XLVII, 47-92.
6. Chen, X. and Friedman, A. (1993). The thermistor problem for conductivity which vanishes at large temperatures, Q. J. Appl. Math. LI, 101-115.
7. Cimatti, G. (1989). Remark on existence and uniqueness for the thermistor problem under mixed boundary conditions, Q. J. Appl. Math. XLVII, 117-121.
8. Cimatti, G. (1990). The stationary thermistor problem with a current limiting device, Proc. Roy. Soc. Edinburgh 116A, 79-84.
9. Crandall, M. G. and Rabinowitz, P. H. (1973). Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch. Rat. Mech. Anal. 52, 161-180.
10. Fiedler, B. and Polácik, P. (1990). Complicated dynamics of scalar reaction diffusion equations with a nonlocal term, Prov. Roy. Soc. Edinburgh 115, 167-192.
11. Fowler, A. C., Frigaard, I., and Howison, S. D. (1992). Temperatures surges in current-limiting circuit devices, SIAM J. Appl. Math. 52, 998-1011.
12. Freitas, P. (1994). A non-local Sturm-Liouville eigenvalue problem, Proc. Roy. Soc. Edinburgh 124A, 169-188.
13. Freitas, P. and Grinfeld, M. (1994). Multiplicity and stability of stationary solutions of an equation modeling ohmic heating, Appl. Math. Lett. 7, 1-6.
14. Gossez, J. P. and Lami-Dozo, E. (1985). On the principal eigenvalue of a second order linear elliptic problem, Arch. Rat. Mech. Anal. 89, 169-175.
15. Graham-Eagle, J. (1995). A non-local equation arising in the theorem of wet combustion, IMA J. Appl. Math. 54, 1-18.
16. Henry, D. (1981). Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics 840, Springer, Berlin.
17. Hirsch, M. W. (1983). Differential equations and convergence almost everywhere in strongly monotone semiflows, In Nonlinear partial Differential Equations (J. Smoller, ed.), AMS, Providence, RI, pp. 267-285.
18. Hirsch, M. W. (1988). Stability and convergence in strongly monotone dynamical systems, J. Reine Ang. Math. 383, 1-58.
19. Howison, S. D. (1989). A note on the thermistor problem in two space dimensions, Q. J. Appl. Math. 47, 509-512.
20. Kato, T. (1975). Perturbation Theory for Linear Operators, Springer, Berlin.
21. Lacey, A. A. (1995). Thermal runaway in a non-local problem modeling Ohmic heating, I. Model derivation and some special cases, Eur. J. Appl. Math. 6, 127-144.
22. Lacey, A. A. (1995). Thermal runaway in a non-local problem modeling Ohmic heating, II. General proof of blow-up and asymptotics of runaway, Eur. J. Appl. Math. 6, 201-224.
23. López-Gómez, J. (1994). On linear weighted boundary value problems, In Partial Differential Equations; Models in Physics and Biology (G. Lumer et al., eds.), Akademie Verlag, Berlin, pp. 188-203.
24. López-Gómez, J. (1996), The maximum principle and the existence of principal eigenvalues for some linear weighted boundary value problems, J. Diff. Eq. 27, 263-294.
25. López-Gómez, J. and Pardo, R. (1998). Invertibility of linear non-cooperative elliptic systems, Nonlinear Anal. TMA 31, 687-699.
26. Nagel, R. et al. (1986). One parameter semigroup of positive operators. Lecture Notes in Mathematics 1184, Springer.
27. Pohozaev, S. I. (1965). Eigenfunctions of the equation Δu + λf(u) = 0, Soviet Math. 5, 1408-1411.
28. Polácik, P. (1989). Convergence in strongly monotone flows defined by semilinear parabolic equations, J. Diff. Eq. 79, 89-110.
29. Polácik, P. (1989). Domains of attraction of equilibria and monotonicity properties of convergent trajectories in parabolic systems admitting a strong comparison principle, J. Reine Ang. Math. 400, 32-56.
30. Protter, M. H. and Weinberger, H. F. (1967). Maximum Principles in Differential Equations, Englewood Cliffs, Prentice-Hall, NJ.
31. Sattinger, D. H. (1973). Topics in stability and bifurcation theory, Lecture Notes in Mathematics 309, Springer.
32. Tzanetis, D. E., Manuscript in preparation.
33. Walter, W. (1989). A theorem on elliptic inequalities with an application to gradient bounds, Math. Z. 200, 293-299.
34. α(Ω̄) solutions for a class of nonlinear degenerate elliptic systems arising in the thermistor problem, SIAM J. Math. Anal. 22, 1491-1499.