Continuity of the blow-up time and a priori bounds for solutions in superlinear parabolic problems

Houston Journal of Mathematics

Volumn 29, Issue 3, 2003, Pages 757-800

  Quittner, Pavol ^a  

^a Faculty of Natural Sciences   (Slovakia)

Author keywords

A priori estimate; Blow up; Semilinear parabolic equation

Indexed keywords

EID: 0041737982     PISSN: 03621588     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (57)

1. H. Amann, Linear and Quasilinear Parabolic Problems, Volume I: Abstract Linear Theory, Birkhäuser, Basel, 1995.
2. H. Amann, Linear and Quasilinear Parabolic Problems, Volume II, In preparation.
3. H. Amann, Parabolic evolution equations and nonlinear boundary conditions, J. Differ. Equations 72 (1988), 201-269.
4. H. Amann, Compact embeddings of vector-valued Sobolev and Besov spaces, Glasnik Matematički 35 (2000), 161-177.
5. H. Amann, Linear parabolic problems involving measures, Revista Real Acad. Ciencias, Ser. A Matemáticas 95 (2001), 85-119.
6. ∞-calculus for elliptic operators, Differ. Integral Equations 7 (1994), 613-653.
7. max for the solution of a semilinear heat equation, J. Funct. Anal. 71 (1987), 142-174.
8. T. Bartsch, A. Pankov and Z.-Q. Wang, Nonlinear Schrödinger equations with steep potential well, Commun. Contemp. Math. 3 (2001), 549-569.
9. J.W. Bebernes and A.A. Lacey, Global existence and finite-time blow-up for a class of nonlocal parabolic problems, Adv. Differ. Equations 2 (1997), 927-953.
10. J. Bebernes, C. Li and P. Talaga, Single-point blowup for nonlocal parabolic problems, Physica D 134 (1999), 48-60.
11. J.W. Bebernes and P. Talaga, Nonlocal problems modelling shear banding, Commun. Appl. Nonlinear Anal. 3 (1996), 79-103.
12. C. Budd, W. Dold and A. Stuart, Blowup in a partial differential equation with conserved first integral, SIAM J. Appl. Math. 53 (1993), 718-742.
13. n, Commun. Partial Differ. Equations 27 (2002), 1793-1814.
14. T. Cazenave and P.-L. Lions, Solutions globales d'équations de la chaleur semi linéaires, Commun. Partial Differ. Equations 9 (1984), 955-978.
15. M. Chipot, M. Fila and P. Quittner, Stationary solutions, blow up and convergence to stationary solutions for semilinear parabolic equations with nonlinear boundary conditions, Acta Math. Univ. Comenianae 60 (1991), 35-103.
16. N. Part I: A global variational approach, Arch. Rational Mech. Anal. 153 (2000), 291-316.
17. N. Part II: The forced case, Arch. Rational Mech. Anal. 153 (2000), 317-339.
18. C. Cortazár, M. del Pino and M. Elgueta, The problem of uniqueness of the limit in a semilinear heat equation, Commun. Partial Differ. Equations 24 (1999), 2147-2172.
19. R. Denk, M. Hieber and J. Prüss, R-boundedness, Fourier multipliers and problems of elliptic and parabolic type, Mem. Amer. Math. Soc., to appear.
20. ∞ functional calculus for an elliptic operator on a half-space with general boundary conditions, Preprint.
21. ∞-calculus for elliptic operators with nonsmooth coefficients, Differ. Integral Equations 10 (1997), 201-217.
22. E. Feireisl and H. Petzeltová, Convergence to a ground state as a threshold phenomenon in nonlinear parabolic equations, Differ. Integral Equations 10 (1997), 181-196.
23. C. Fermanian Kammerer, F. Merle, H. Zaag, Stability of the blow-up profile of non-linear heat equations from the dynamical system point of view, Math. Ann. 317 (2000), 347-387.
24. M. Fila, Boundedness of global solutions of nonlinear diffusion equations, J. Differ. Equations 98 (1992), 226-240.
25. M. Fila, Boundedness of global solutions of nonlocal parabolic equations, Nonl. Anal., TMA 30 (1997), 877-885.
26. M. Fila and H.A. Levine, On the boundedness of global solutions of abstract semilinear parabolic equations, J. Math. Anal. Appl. 216 (1997), 654-666.
27. M. Fila and P. Poláčik, Global solutions of a semilinear parabolic equation, Adv. Differ. Equations 4 (1999), 163-196.
28. q spaces and universal bounds for global solutions. Math. Ann. 320 (2001), 87-113.
29. V. Galaktionov and J.L. Vázquez, Continuation of blow-up solutions of nonlinear heat equations in several space dimensions, Commun. Pure Applied Math. 50 (1997), 1-67.
30. Y. Giga, A bound for global solutions of semilinear heat equations. Commun. Math. Phys. 103 (1986), 415-421.
31. Y. Giga and R.V. Kohn, Characterizing blowup using similarity variables, Indiana Univ. Math. J. 36 (1987), 1-40.
32. B. Hu and H. Yin, Semilinear parabolic equations with prescribed energy, Rend. Cir. Mat. Palermo (2) 44 (1995), 479-505.
33. O. Kavian, Remarks on the large time behaviour of a nonlinear diffusion equation, Ann. Inst. H. Poincaré, Analyse Non Linéaire 4 (1987), 423-452.
34. t = -Au + F(u), Arch. Rational Mech. Anal. 51 (1973), 371-386.
35. F. Merle, Contributions a l'étude de certaines équations aux dérivées partielles non linéaires de la physique mathématique, Thesis (Univ. Pierre at Marie Curie, Paris), 1987.
36. F. Merle, Solution of a nonlinear heat equation with arbitrarily given blow-up points, Comm. Pure Appl. Math. 45 (1992), 263-300.
37. N. Mizoguchi, On the behavior of solutions for a semilinear parabolic equation with supercritical nonlinearity, Math. Z. 239 (2002), 215-229.
38. J. Nečas, Les méthodes directes en théorie des équations elliptiques, Masson, Paris, 1967.
39. W.-M. Ni, P.E. Sacks and J. Tavantzis, On the asymptotic behavior of solutions of certain quasilinear parabolic equations, J. Differ. Equations 54 (1984), 97-120.
40. S.I. Pohozaev, Eigenfunctions of the equation Δu+λf(u) = 0, Soviet Math. Dokl. 5 (1965), 1408-1411.
41. P. Poláčik, Domains of attraction of equilibria and monotonicity properties of convergent trajectories in parabolic systems admitting strong comparison principle, J. reine angew. Math. 400 (1989), 32-56.
42. P-spaces, Hiroshima Math. J. 23 (1993), 161-192.
43. P. Quittner, A priori bounds for global solutions of a semilinear parabolic problem, Acta Math. Univ. Comenianae 68 (1999), 195-203.
44. P. Quittner, Universal bound for global positive solutions of a superlinear parabolic problem, Math. Ann. 320 (2001), 299-305.
45. P. Quittner, A priori estimates of global solutions and multiple equilibria of a parabolic problem involving measure, Electronic J. Differ. Equations 2001, 29 (2001), 1-17.
46. P. Quittner, Global existence of solutions of parabolic problems with nonlinear boundary conditions, Banach Center Publ. 33 (1996), 309-314.
47. P. Quittner, Global existence for semilinear parabolic problems. Adv. Math. Sci. Appl. 10 (2000), 643-660.
48. P. Quittner, Multiple equilibria, periodic solutions and a priori bounds for solutions in superlinear parabolic problems, NoDEA, Nonlinear Differ. Equations Appl., to appear.
49. P. Quittner and Ph. Souplet, Bounds of global solutions of parabolic problems with nonlinear boundary conditions, Indiana Univ. Math. J., to appear.
50. P. Quittner, Ph. Souplet and M. Winkler, Initial blow-up rates and universal bounds for nonlinear heat equations, Preprint.
51. ∞-Funktionalkalkül für elliptische Randwertsysteme, Dissertation, Kassel, 1999.
52. Ph. Souplet, Sur l'asymptotique des solutions globales pour une équation de la chaleur semi-linéaire dans des domains non bornés, C. R. Acad. Sci. Paris (I) 323 (1996), 877-882.
53. Ph. Souplet, Blow-up in nonlocal reaction-diffusion equations, SIAM J. Math. Anal. 29 (1998), 1301-1334.
54. Ph. Souplet and Q.S. Zhang, Existence of ground states for semilinear elliptic equations with decaying mass: a parabolic approach, C. R. Acad. Sci. Paris (I) 332 (2001), 515-520.
55. n and dynamical approach to the existence of ground states, Ann. Inst. H. Poincaré, Analyse Non Linéaire 19 (2002), 683-703.
56. T. Weth, On nodal solutions to generalized Emden-Fowler equations, Preprint.
57. H. Zaag, A remark on the energy blow-up behavior for nonlinear heat equations, Duke Math. J. 103 (2000), 545-556.