A priori estimates of global solutions of superlinear parabolic problems without variational structure

Discrete and Continuous Dynamical Systems

Volumn 9, Issue 5, 2003, Pages 1277-1292

  Quittner, Pavol ^a   Souplet, Philippe ^b,c  

^a Faculty of Natural Sciences   (Slovakia)

^b UNIVERSITÉ DE PICARDIE JULES VERNE   (France)

^c CENTRE DE MATHÉMATIQUES APPLIQUÉES   (France)

Author keywords

A priori estimate; Semilinear parabolic problem; Universal bound

Indexed keywords

EID: 0038784323     PISSN: 10780947     EISSN: None     Source Type: Journal    
DOI: 10.3934/dcds.2003.9.1277     Document Type: Article

References (49)

1. J. Aguirre and M. Escobedo, On the blow up of solutions for a convective reaction diffusion equation, Proc. Roy. Soc. Edinburgh 123A (1993), 433-460.
2. H. Amann, Existence and regularity for semilinear parabolic evolution equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 11 (1984), 593-676.
3. S. Angenent and R.C.A.M. van der Vorst, A priori bounds and renormalized Morse indices of solutions of an elliptic system, Ann. Inst. Henri Poincaré, Anal, non linéaire 17 (2000), 277-306.
4. H. Brezis and R.E.L. Turner, On a class of superlinear elliptic problems, Commun. Partial Differ. Equations 2 (1977), 601-614.
5. F.E. Browder, On the spectral theory of elliptic differential operators. I, Math. Ann. 142 (1961), 22-130.
6. T. Cazenave and P.-L. Lions, Solutions globales d'équations de la chaleur semi linéaires, Commun. Partial Differ. Equations 9 (1984), 955-978.
7. M. Chipot and F.B. Weissler, Some blowup results for a nonlinear parabolic equation with a gradient term, SIAM J. Math. Anal. 20 (1989), 886-907.
8. M. Chlebík and M. Fila, From critical exponents to blow-up rates for parabolic problems, Rend. Mat. Appl., Ser. VII 19 (1999), 449-470.
9. M. Chlebík, M. Fila and P. Quittner, Blow-up of positive solutions of a semilinear parabolic equation with a gradient term, Dyn. Contin. Discrete Impulsive Syst. (to appear).
10. Ph. Clément, D.G. de Figueiredo and E. Mitidieri, A priori estimates for positive solutions of semilinear elliptic systems via Hardy-Sobolev inequalities, Nonlinear partial differential equations (A. Benkirane et al, eds.), Pitman Research Notes in Math. 343, Harlow, Longman, 1996, pp. 73-91.
11. R. Dautray and J.-L. Lions, Mathematical analysis and numerical methods for science and technology. Vol. 1: Physical origins and classical methods, Springer, Berlin, 1990.
12. K. Deng, Stabilization of solutions of a nonlinear parabolic equation with a gradient term, Math. Z. 216 (1994), 147-155.
13. M. Escobedo and M.A. Herrero, Boundedness and blow up for a semilinear reaction-diffusion system, J. Differ. Equations 89 (1991), 176-202.
14. M. Escobedo and M.A. Herrero, A semilinear parabolic system in a bounded domain, Annali di Matematica pura ed applicata CLXV (1993), 315-336.
15. M.J. Esteban, On periodic solutions of superlinear parabolic problems, Trans. Amer. Math. Soc. 293 (1986), 171-189.
16. M.J. Esteban, A remark on the existence of positive periodic solutions of superlinear parabolic problems, Proc. Amer. Math. Soc. 102 (1988), 131-136.
17. M. Fila and P. Quittner, The blow-up rate for a semilinear parabolic system, J. Math. Anal. Appl. 238 (1999), 468-476.
18. M. Fila and Ph. Souplet, The blow-up rate for semilinear parabolic problems on general domains, Nonlinear Differ. Equ. Appl. 8 (2001), 473-480.
19. p spaces and universal bounds for global solutions, Math. Ann. 320 (2001), 87-113.
20. S. Filippas and F. Merle, Modulation theory for the blowup of vector-valued nonlinear heat equations, J. Differ. Equations 116 (1995), 119-148.
21. A. Friedman and Y. Giga, A single point blow-up for solutions of semilinear parabolic systems, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 34 (1987), 65-79.
22. 1+α, J. Fac. Sci. Univ. Tokyo Sec. IA Math. 13 (1966), 109-124.
23. Y. Giga, A bound for global solutions of semilinear heat equations, Commun. Math. Phys. 103 (1986), 415-421.
24. Y. Giga and R.V. Kohn, Characterizing blowup using similarity variables, Indiana Univ. Math. J. 36 (1987), 1-40.
25. K. Hayakawa, On nonexistence of global solutions of some semilinear parabolic differential equations, Proc. Japan Acad. 49 (1973), 503-505.
26. N. Hirano and N. Mizoguchi, Positive unstable periodic solutions for superlinear parabolic equations, Proc. Amer. Math. Soc. 123 (1995), 1487-1495.
27. B. Hu, Remarks on the blowup estimate for solutions of the heat equation with a nonlinear boundary condition, Differ. Integral Equations 9 (1996), 891-901.
28. O. Kavian, Remarks on the large time behaviour of a nonlinear diffusion equation, Ann. Inst. H. Poincaré, Anal. non linéaire 4 (1987), 423-452.
29. K. Kobayashi, T. Sirao and H. Tanaka, On the blowing up problem for semilinear heat equations, J. Math. Soc. Japan 29 (1977), 407-424.
30. H.A. Levine, A Fujita type global existence - global nonexistence theorem for a weakly coupled system of reaction-diffusion equations, Z. Angew. Math. Phys. 42 (1991), 408-430.
31. H.A. Levine and P. Meier, A blowup result for the critical exponent in cones, Israel J. Math. 67 (1989), 129-136.
32. H.A. Levine, L.N. Payne, P.E. Sacks and B. Straughan, Analysis of convective reaction-diffusion equation (II), SIAM J. Math. Anal. 20 (1989), 133-147.
33. G.M. Lieberman, Second Order Parabolic Differential Equations, World Scientific, Singapore, 1996.
34. A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhäuser, Basel - Boston - Berlin, 1995.
35. 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.
36. S.I. Pohozaev, Eigenfunctions of the equation Δu + λf(u) = 0, Soviet Math. Dokl. 5 (1965), 1408-1411.
37. 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.
38. P. Quittner, A priori bounds for global solutions of a semilinear parabolic problem, Acta Math. Univ. Comenianae 68 (1999), 195-203.
39. P. Quittner, Universal bound for global positive solutions of a superlinear parabolic problem, Math. Ann. 320 (2001), 299-305.
40. P. Quittner, Signed solutions for a semilinear elliptic problem, Differ. Integral Equations 11 (1998), 551-559.
41. 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.
42. P. Quittner, Global solutions in parabolic blow-up problems with perturbations, Progress in partial differ, equations (H. Amann et al, eds.), Pitman Research Notes in Math. 384, Vol. 2, Harlow, Longman, 1998, pp. 77-91.
43. p norms for local existence and for continuation in semilinear parabolic systems are not the same, Proc. Royal Soc. Edinburgh Sect. A 131 (2001), 1435-1456.
44. Ph. Souplet, Recent results and open problems on parabolic equations with gradient nonlinearities, Electronic J. Diff. Equations 2001, 20 (2001), 1-19.
45. Ph. Souplet and F.B. Weissler, Self-similar subsolutions and blow-up for nonlinear parabolic equations, J. Math. Anal. Appl. 212 (1997), 60-74.
46. Ph. Souplet and F.B. Weissler, Poincaré inequality and global solutions of a nonlinear parabolic equation, Ann. Inst. H. Poincaré, Analyse non linéaire 16 (1999), 337-373.
47. P, Indiana Univ. Math. J. 29 (1980), 79-102.
48. H. Zaag, A Liouville theorem and blowup behavior for a vector-valued nonlinear heat equation with no gradient structure, Comm. Pure Appl. Math. 54 (2001), 107-133.
49. H. Zou, A priori estimates for a semilinear elliptic system without variational structure and their applications, Math. Ann. 323 (2002), 713-735.