Global existence for the heat equation with nonlinear dynamical boundary conditions

Royal Society of Edinburgh - Proceedings A

Volumn 135, Issue 1, 2005, Pages 175-207

  Vitillaro, Enzo ^a  

^a UNIVERSITY OF PERUGIA   (Italy)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 14644417918     PISSN: 03082105     EISSN: None     Source Type: Journal    
DOI: 10.1017/s0308210500003838     Document Type: Article

References (28)

1. R. A. Adams. Sobolev spaces (Academic, 1975).
2. H. Amann. Parabolic evolution equations and nonlinear boundary conditions. J. Diff. Eqns 72 (1988), 201-269.
3. A. Ambrosetti. Nonlinear oscillations with minimal period. In Proc. Symposia in Pure Mathematics, vol. 45, pp. 29-35 (Providence, RI: American Mathematical Society, 1986).
4. H. Brezis. Analyse fonctionelle, théorie et applications (Paris: Masson, 1983).
5. E. A. Coddington and N. Levinson. Theory of ordinary differential equations (McGraw-Hill, 1955).
6. R. E. Edwards. Functional analysis, theory and applications (New York: Holt, Rinehart and Winston, 1965).
7. J. Escher. Global existence and nonexistence for semilinear parabolic systems with nonlinear boundary conditions. Math. Ann. 284 (1989), 285-305.
8. J. Escher. A note on quasilinear parabolic systems with dynamical boundary conditions. In Progress in partial differential equations: the Metz surveys, vol. 2, pp. 138-148 (Harlow: Longman, 1993).
9. J. Escher. Quasilinear parabolic systems with dynamical boundary conditions. Commun. PDEs 18 (1993), 1309-1364.
10. J. Escher. On the qualitative behaviour of some semilinear parabolic problems. Diff. Integ. Eqns 8 (1995), 247-267.
11. L. C. Evans. Partial differential equations. Graduate Studies in Mathematics, vol. 19 (Providence, RI: American Mathematical Society, 1998).
12. V. Georgiev and G. Todorova. Existence of a solution of the wave equation with nonlinear damping and source terms. J. Diff. Eqns 109 (1994), 295-308.
13. M. Grobbelaar Van Dalsen. Semilinear evolution equations and fractional powers of a closed pair of operators. Proc. R. Soc. Edinb. A 105 (1987), 101-115.
14. T. Hintermann. Evolution equations with dynamic boundary conditions. Proc. R. Soc. Edinb. A 113 (1989), 43-60.
15. M. Kirane. Blow-up for some equations with semilinear dynamical boundary conditions of parabolic and hyperbolic type. Hokkaido Math. J. 21 (1992), 221-229.
16. H. A. Levine and L. E. Payne. Nonexistence theorems for the heat equation with nonlinear boundary conditions and for the porous medium equation backward in time. J. Diff. Eqns 16 (1974), 319-334.
17. H. A. Levine and L. E. Payne. Some nonexistence theorems for initial-boundary value problems with nonlinear boundary constraints. Proc. Am. Math. Soc. 46 (1974), 277-284.
18. H. A. Levine and A. Smith. A potential well theory for the heat equation with a nonlinear boundary condition. Math. Meth. Appl. Sci. 9 (1987), 127-136.
19. H. A. Levine and A. Smith. A potential well theory for the wave equation with a nonlinear boundary condition. J. Reine Angew. Math. 374 (1987), 1-23.
20. J. L. Lions. Quelques méthodes de résolutions des problèmes aux limites non linéaires (Paris: Dunod, 1969).
21. J. L. Lions and W. A. Strauss. On some nonlinear evolution equations. Bull. Soc. Math. France 93 (1965), 43-96.
22. R. T. Rockafellar. Convex analysis (Princeton, NJ: Princeton University Press, 1970).
23. I. Segal. Nonlinear semigroups. Ann. Math. 78 (1963), 339-364.
24. J. Serrin, G. Todorova and E. Vitillaro. Existence for a nonlinear wave equation with nonlinear damping and source terms. Diff. Integ. Eqns 16 (2003), 13-50.
25. W. A. Strauss. On continuity of functions with values in various Banach spaces. Pac. J. Math. 19 (1966), 543-551.
26. E. Vitillaro. Some new results on global nonexistence and blow-up for evolution problems with positive initial energy. Rend. Istit. Mat. Univ. Trieste 31 (2000), 245-275.
27. E. Vitillaro. Global existence for the wave equation with nonlinear boundary damping and source terms. J. Diff. Eqns 186 (2002), 259-298.
28. W. P. Ziemer. Weakly differentiable functions. Graduate Text in Mathematics, vol. 120 (Springer, 1989).