Decay of Heat Semigroups in L∞ and Applications to Nonlinear Parabolic Problems in Unbounded Domains

Journal of Functional Analysis

Volumn 173, Issue 2, 2000, Pages 343-360

  Souplet, Philippe ^a,b  

^a UNIVERSITÉ DE PICARDIE JULES VERNE   (France)

^b CENTRE DE MATHÉMATIQUES APPLIQUÉES   (France)

Author keywords

Heat semigroup; decay; unbounded domains; nonlinear parabolic equations; quenching; stability

Indexed keywords

EID: 0347053866     PISSN: 00221236     EISSN: None     Source Type: Journal    
DOI: 10.1006/jfan.2000.3554     Document Type: Article

References (25)

1. Agmon S. Lectures on Elliptic Boundary Value Problems. 1965;Van Nostrand, Princeton.
2. Ben-Artzi M., Koch H. Decay of mass for a semilinear parabolic equation. Comm. Partial Differential Equations. 24:1999;869-881.
3. van den Berg M. Gaussian bounds for Dirichlet heat kernels. J. Funct. Anal. 88:1990;267-278.
4. Berestycki H., Nirenberg L., Varadhan S. R. S. The principal eigenvalue and maximum principle for second-order elliptic operators in general domains. Comm. Pure Appl. Math. 47:1994;47-92.
5. Brézis H., Cazenave T. A nonlinear heat equation with singular initial data. J. Anal. Math. 68:1996;877-894.
6. Cabre X. On the Alexandroff-Bakelman-Pucci estimate and the reversed Hoelder inequality for solutions of elliptic and parabolic equations. Comm. Pure Appl. Math. 54:1995;539-570.
7. Cazenave T., Haraux A. An Introduction to Semi-Linear Evolution Equations. Oxford Lecture Series in Mathematics and Its Applications. 13:1998;Clarendon, Oxford.
8. Chan C. Y., Kaper H. G. Quenching for semilinear singular parabolic problems. SIAM J. Math. Anal. 20:1989;558-566.
9. Dai Q., Gu Y. A short note on quenching phenomena for semilinear parabolic equations. J. Differential Equations. 137:1997;240-250.
10. Davies E. B. Heat Kernels and Spectral Theory. 1989;Cambridge Univ. Press, Cambridge.
11. Escher J., Scarpellini B. On the asymptotics of solutions of parabolic equations on unbounded domains. Nonlinear Anal. 33:1998;483-507.
12. Escher J., Scarpellini B. Stability properties of parabolic equations in unbounded domains. Arch. Math. 71:1998;31-45.
13. Fila M., Kawohl B., Levine H. A. Quenching for quasilinear equations. Comm. Partial Differential Equations. 17:1992;593-614.
14. H. Ghidouche, Ph. Souplet, and, D. Tarzia, Decay of global solutions, stability and blow-up for a reaction-diffusion problem with free boundary, Proc. Amer. Math. Soc, in press.
15. Gilbarg D., Trudinger N. Elliptic Partial Differential Equations of Second Order. Grundlehren der Mathematischen Wissenschaften. 224:1983;Springer-Verlag, New York/Berlin.
16. Hayman W. Some bounds for principal frequency. Appl. Anal. 7:1978;247-254.
17. Kaplan S. On the growth of solutions of quasilinear parabolic equations. Comm. Pure Appl. Math. 16:1963;305-330.
18. xx+1/(1-u). Publ. Res. Inst. Math. Sci. 10:1975;729-736.
19. Levine H. A. Quenching and beyond: A survey of recent results. Kawarada H. Proc. of the International Conf. on Nonlinear Mathematical Problems in Industry. Gakuto Int. Ser. Math. Sci. Appl. 2:1993;501-512.
20. Lieb E. A. On the lowest eigenvalue of the Laplacian for the intersection of two domains. Invent. Math. 74:1983;441-448.
21. Osserman R. A note on Hayman's theorem on the bass note of a drum. Comment. Math. Helv. 52:1977;545-555.
22. Souplet Ph. Geometry of unbounded domains, Poincaré inequalities, and stability in semilinear parabolic equations. Comm. Partial Differential Equations. 24:1999;951-973.
23. Souplet Ph., Weissler F. B. Poincaré's inequality and global solutions of a non-linear parabolic equation. Ann. Inst. H. Poincaré Anal. Non Linéaire. 16:1999;337-373.
24. p. Indiana Univ. Math. J. 29:1980;79-102.
25. Wiegner M. On the asymptotic behaviour of solutions of nonlinear parabolic equations. Math. Z. 188:1984;3-22.