Blow-up of radially symmetric solutions of a non-local problem modelling Ohmic heating

Electronic Journal of Differential Equations

Volumn 2002, Issue , 2002, Pages

  Tzanetis, Dimitrios E ^a  

^a NATIONAL TECHNICAL UNIVERSITY OF ATHENS   (Greece)

Author keywords

Blow up; Global existence; Nonlocal parabolic equations; Steady states

Indexed keywords

EID: 52449120724     PISSN: 10726691     EISSN: 10726691     Source Type: Journal    
DOI: None     Document Type: Article

References (23)

1. S.N. ANTONTSEV &M. CHIPOT, The thermistor problem: Existence, smoothness, uniqueness, blow-up, SIAM J. Math. Anal., Vol. 25, No. 4, (1994), pp. 1128-1156.
2. S.N. ANTONTSEV &M. CHIPOT, Analysis of blowup for the thermistor problem, Siberian Mathematical J. Vol. 38, No. 5, (1997), pp. 827-841.
3. W. ALLEGRETTO AND H. XIE, A non-local thermistor problem, Euro. J. Applied Math., (1995), pp. 83-94.
4. A. BARABANOVA, The blow-up of solutions of a nonlocal thermistor problem, Appl. Math. Lett. 9, No. 1, (1996), pp. 59-63.
5. J.W. BEBERNES AND A.A. LACEY, Global existence and finite-time blowup for a class of nonlocal problems, Adv. Diff. Equns, 2, (1997), pp. 927-953.
6. J.A. CARRILLO, On a non-local elliptic equation with decreasing nonlinearity arising in plasma physics and heat conduction, Nonlinear Analysis TMA., 32, (1998), pp. 97-115.
7. N. CHAFEE, The electric ballast resistor: homogeneous and nonhomogeneous equilibria, In Nonlinear differential equations: invariance stability and bifurcations, P. de Mottoni &L. Salvatori Eds., Academic Press, New York., (1981), pp. 97-127.
8. G. CIMATTI, Remarks on existence and uniqueness for the thermistor problem under mixed boundary conditions, Quart. J. Appl. Math., 47, (1989), pp. 117-121.
9. G. CIMATTI, The stationary thermistor problem with a current limiting device, Proc. Roy. Soc. Edinburgh, 116A, (1990) pp. 79-84.
10. G. CIMATTI, On the stability of the solution of the thermistor problem, Appl. Anal. 73 (1999), no. 3-4, pp. 407-423.
11. M. FILA, Boundedness of global solutions of nonlocal parabolic equations. Proceedings of the Second World Congress of Nonlinear Analysts, Part 2 (Athens, 1996). Nonlinear Anal. 30 (1997), no. 2, pp. 877-885.
12. A.C. FOWLER, I. FRIGAARD, AND S.D. HOWISON, Temperature surges in current-limiting circuit devices, SIAM J. Appl. Math., 52, (1992), pp. 9981011.
13. B. GIDAS, W.-M. Ni AND L. NlRENBERG, Summetry and related properties via the maximum principle, Comm. Math. Phys., 68, (1979), pp. 209243.
14. S.D. HOWISON, A note on the thermistor problem in two space dimensions, Quart. Appl. Math., 47, (1989), pp. 509-512.
15. N.I. KAVALLARIS AND D.E. TZANETIS, An Ohmic heating non-local diffusion-convection problem for the Heaviside function, preprint.
16. N.I. KAVALLARIS AND D.E. TZANETIS, Global existence and divergence of critical solutions of some non-local parabolic problems in the Ohmic heating process, preprint.
17. A.A. LACEY, Mathematical analysis of thermal runaway for spatially inhomogeneous reactions, SIAM J. Appl. Math., 43, (1983), pp. 1350-1366.
18. A.A. LACEY, Thermal runaway in a non-local problem modelling Ohmic heating: Part I: Model derivation and some special cases., Euro. J. Applied Math., 6, (1995), pp. 127-144.
19. A.A. LACEY, Thermal runaway in a non-local problem modelling Ohmic heating: Part II: General proof of blow-up and asymptotics of runaway, Euro. J. Applied Math. 6, (1995), pp. 201-224.
20. J. LOPEZ-GOMEZ, On the structure and stability of the set of solutions of a non-local problem modelling Ohmic heating, J. Dynam. Diff. Eqns 10, No. 4, (1998), pp. 537-559.
21. D.H. SATTINGER, Monotone methods in nonlinear elliptic and parabolic boundary value problems, Indiana Univ. Math. J., 21, (1972), pp. 979-1000.
22. P. SOUPLET, Blow-up in nonlocal reaction-diffusion equations, SIAM J. Math. Anal. 29, No. 6, (1998), pp. 1301-1334.
23. P. SOUPLET, Uniform blow-up profiles and boundary behavior for diffusion equations with nonlocal source, J. Diff. Eqns, 153, (1999), pp. 374-406.