Parabolic problems with dynamical boundary conditions: Eigenvalue expansions and blow up

Atti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, Rendiconti Lincei Matematica e Applicazioni

Volumn 17, Issue 1, 2006, Pages 35-67

  Bandle, Catherine ^a   Von Below, Joachim ^b   Reichel, Wolfgang ^a  

^a UNIVERSITY OF BASEL   (Switzerland)

^b UNIVERSITÉ DU LITTORAL CÔTE D OPALE   (France)

Author keywords

Blow up phenomena; Compact operator; Comparison principles for elliptic and parabolic equations; Existence theory for local solutions; Series representation; Variational characterization of eigenvalues

Indexed keywords

EID: 84887219157     PISSN: 11206330     EISSN: 17200768     Source Type: Journal    
DOI: 10.4171/RLM/453     Document Type: Article

References (21)

1. H. W. ALT, Lineare Funktionalanalysis. 3rd edition, Springer, Berlin, 1999.
2. C. BANDLE - H. A. LEVINE, Fujita type results for convective-like reaction diffusion equations in exterior domains. ZAMP 40 (1989), 665-676.
3. C. BANDLE - W. REICHEL, Solutions of quasilinear second order elliptic boundary value problems via degree theory. In: Handbook of Differential Equations: Stationary Partial Differential Equations, Vol. 1, M. Chipot and P. Quittner (eds.), Elsevier, 2004, 1-70.
4. J. VON BELOW, Parabolic Network Equations. Tübingen, 1994, 3rd ed. to appear.
5. J. VON BELOW - C. DE COSTER, A qualitative theory for parabolic problems under dynamical boundary conditions. J. Inequal. Appl. 5 (2000), 467-486.
6. J. VON BELOW - G. FRANÇOIS, Spectral asymptotics for the Laplacian under an eigenvalue dependent boundary condition. Bull. Belgian Math. Soc. 2005/2006, to appear.
7. J. VON BELOW - G. PINCET MAILLY, Blow up for reaction diffusion equations under dynamical boundary conditions. Comm. Partial Differential Equations 28 (2003), 223-247. (Pubitemid 36618796)
8. D. G. DE FIGUEIREDO, Positive solutions of semilinear elliptic problems. In: Differential Equations (Sao Paulo, 1981), Lecture Notes in Math. 957, Springer, Berlin, 1982, 34-87.
9. J. ERCOLANO - M. SCHECHTER, Spectral theory for operators generated by elliptic boundary problems with eigenvalue parameter in boundary conditions, I. Comm. Pure Appl. Math. 18 (1965), 83-105.
10. J. ESCHER, Quasilinear parabolic systems with dynamical boundary conditions. Comm. Partial Differential Equations 18 (1993), 1309-1364.
11. M. FILA - P. QUITTNER, Large time behavior of solutions of a semilinear parabolic equation with a nonlinear dynamical boundary condition. In: Topics in Nonlinear Analysis, Progr. Nonlinear Differential Equations Appl. 35, Birkhäuser, Basel, 1999, 251-272. (Pubitemid 128158394)
12. G. FRANÇOIS, Spectral asymptotics stemming from parabolic equations under dynamical boundary conditions. Preprint LMPA Nr. 225 (2004), submitted.
13. A. LACEY, Mathematical analysis of thermal runaway for spatially inhomogeneous reactions. SIAM J. Appl. Anal. 43 (1983), 1350-1366. (Pubitemid 14501249)
14. K. MOCHIZUKI - R. SUZUKI, Critical exponent and critical blow up for quasi-linear parabolic equations. Israel J. Math. 98 (1997), 141-156.
15. J. NEČAS, Les Méthodes directes en thÉorie des Équations elliptiques. Masson, Paris, 1967.
16. D. F. RIAL - J. D. ROSSI, Blow-up results and localization of blow-up points in an N-dimensional smooth domain. Duke Math. J. 88 (1997), 391-405. (Pubitemid 127719163)
17. R. SHOWALTER, Degenerate evolution equations. Indiana Univ. Math. J. 23 (1974), 655-677.
18. M. STRUWE, Variational Methods. 2nd ed., Ergeb. Math. Grenzgeb. 34, Springer, Berlin, 1996.
19. J. L. VAZQUEZ - E. VITILLARO, Heat equation with dynamical boundary conditions of reactive type. Preprint, 2004.
20. E. VITILLARO, Global existence and blow-up for Laplace equation with nonlinear dynamical boundary conditions and for abstract evolution equations. To appear.
21. W. WALTER, Gewöhnliche Differentialgleichungen. 7th ed., Springer, Berlin, 2000.