Földes, Juraj Liouville theorems, a priori estimates, and blow-up rates for solutions of indefinite superlinear parabolic problems. (English) Zbl 1224.35013 Czech. Math. J. 61, No. 1, 169-198 (2011). Summary: In this paper we establish new nonlinear Liouville theorems for parabolic problems on half spaces. Based on the Liouville theorems, we derive estimates for the blow-up of positive solutions of indefinite parabolic problems and investigate the complete blow-up of these solutions. We also discuss a priori estimates for indefinite elliptic problems. Cited in 13 Documents MSC: 35B09 Positive solutions to PDEs 35B44 Blow-up in context of PDEs 35B45 A priori estimates in context of PDEs 35B53 Liouville theorems and Phragmén-Lindelöf theorems in context of PDEs 35J61 Semilinear elliptic equations Keywords:a priori estimate; Liouville theorem; blow-up rate; positive solution; indefinite parabolic problem × Cite Format Result Cite Review PDF Full Text: DOI EuDML Link References: [1] N. Ackermann, T. Bartsch, P. Kaplický and P. Quittner: A priori bounds, nodal equilibria and connecting orbits in indefinite superlinear parabolic problems. Trans. Am. Math. Soc. 360 (2008), 3493–3539. · Zbl 1143.37049 · doi:10.1090/S0002-9947-08-04404-8 [2] H. Amann: Existence and regularity for semilinear parabolic evolution equations. Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 11 (1984), 593–676. · Zbl 0625.35045 [3] D. Andreucci and E. DiBenedetto: On the Cauchy problem and initial traces for a class of evolution equations with strongly nonlinear sources. Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 18 (1991), 363–441. · Zbl 0762.35052 [4] P. Baras and L. Cohen: Complete blow-up after Tmax for the solution of a semilinear heat equation. J. Funct. Anal. 71 (1987), 142–174. · Zbl 0653.35037 · doi:10.1016/0022-1236(87)90020-6 [5] M. F. Bidaut-Véron: Initial blow-up for the solutions of a semilinear parabolic equation with source term. In: Équations aux dérivées partielles et applications. Gauthier-Villars, Éd. Sci. Méd. Elsevier, Paris, 1998, pp. 189–198. · Zbl 0914.35055 [6] X. Cabré: On the Alexandroff-Bakelman-Pucci estimate and the reversed Hölder inequality for solutions of elliptic and parabolic equations. Commun. Pure Appl. Math. 48 (1995), 539–570. · Zbl 0828.35017 · doi:10.1002/cpa.3160480504 [7] Y. Du and S. Li: Nonlinear Liouville theorems and a priori estimates for indefinite superlinear elliptic equations. Adv. Diff. Equ. 10 (2005), 841–860. · Zbl 1161.35388 [8] A. Farina: Liouville-type theorems for elliptic problems. Handbook of differential equations: Stationary partial differential equations, Vol. IV (M. Chipot, ed.). Elsevier/North-Holland, Amsterdam, 2007, pp. 60–116. · Zbl 1191.35128 [9] M. Fila and P. Souplet: The blow-up rate for semilinear parabolic problems on general domains. NoDEA Nonlinear Differ. Equ. Appl. 8 (2001), 473–480. · Zbl 0993.35046 · doi:10.1007/PL00001459 [10] M. Fila, P. Souplet, and F. B. Weissler: Linear and nonlinear heat equations in L {\(\delta\)} q spaces and universal bounds for global solutions. Math. Ann. 320 (2001), 87–113. · Zbl 0993.35023 · doi:10.1007/PL00004471 [11] A. Friedman and B. McLeod: Blow-up of positive solutions of semilinear heat equations. Indiana Univ. Math. J. 34 (1985), 425–447. · Zbl 0576.35068 · doi:10.1512/iumj.1985.34.34025 [12] B. Gidas and J. Spruck: A priori bounds for positive solutions of nonlinear elliptic equations. Commun. Partial Differ. Equations 6 (1981), 883–901. · Zbl 0462.35041 · doi:10.1080/03605308108820196 [13] Y. Giga and R. V. Kohn: Characterizing blowup using similarity variables. Indiana Univ. Math. J. 36 (1987), 1–40. · Zbl 0601.35052 · doi:10.1512/iumj.1987.36.36001 [14] Y. Giga, S. Matsui, and S. Sasayama: Blow up rate for semilinear heat equations with subcritical nonlinearity. Indiana Univ. Math. J. 53 (2004), 483–514. · Zbl 1058.35096 · doi:10.1512/iumj.2004.53.2401 [15] Y. Giga, S. Matsui, and S. Sasayama: On blow-up rate for sign-changing solutions in a convex domain. Math. Methods Appl. Sci. 27 (2004), 1771–1782. · Zbl 1066.35043 · doi:10.1002/mma.562 [16] D. Gilbarg and N. S. Trudinger: Elliptic Partial Differential Equations of Second Order. Classics in Mathematics. Springer, Berlin, 2001. Reprint of the 1998 edition. · Zbl 1042.35002 [17] M.A. Herrero and J. J. L. Velázquez: Blow-up behaviour of one-dimensional semilinear parabolic equations. Ann. Inst. Henri Poincaré Anal. Non Linéaire 10 (1993), 131–189. [18] N. V. Krylov: Nonlinear Elliptic and Parabolic Equations of the Second Order. Mathematics and its Applications (Soviet Series). Vol. 7. D. Reidel Publishing Co., Dordrecht, 1987. · Zbl 0619.35004 [19] G.M. Lieberman: Second Order Parabolic Differential Equations. World Scientific Publishing Co., River Edge, NJ, 1996. · Zbl 0884.35001 [20] J. López-Gómez and P. Quittner: Complete and energy blow-up in indefinite superlinear parabolic problems. Discrete Contin. Dyn. Syst. 14 (2006), 169–186. · Zbl 1114.35093 [21] A. Lunardi: Analytic Semigroups and Optimal Regularity in Parabolic Problems. Progress in Nonlinear Differential Equations and their Applications. Vol. 16. Birkhäuser, Basel, 1995. · Zbl 0816.35001 [22] F. Merle and H. Zaag: Optimal estimates for blowup rate and behavior for nonlinear heat equations. Commun. Pure Appl. Math. 51 (1998), 139–196. · Zbl 0899.35044 · doi:10.1002/(SICI)1097-0312(199802)51:2<139::AID-CPA2>3.0.CO;2-C [23] P. Poláčik and P. Quittner: Liouville type theorems and complete blow-up for indefinite superlinear parabolic equations. In: Nonlinear elliptic and parabolic problems. Progr. Nonlinear Differential Equations Appl., Vol. 64. Birkhäuser, Basel, 2005, pp. 391–402. · Zbl 1093.35037 [24] P. Poláčik and P. Quittner A Liouville-type theorem and the decay of radial solutions of a semilinear heat equation. Nonlinear Anal. 64 (2006), 1679–1689. · Zbl 1092.35045 · doi:10.1016/j.na.2005.07.016 [25] P. Poláčik, P. Quittner, and P. Souplet: Singularity and decay estimates in superlinear problems via Liouville-type theorems. I. Elliptic equations and systems. Duke Math. J. 139 (2007), 555–579. · Zbl 1146.35038 · doi:10.1215/S0012-7094-07-13935-8 [26] P. Poláčik, P. Quittner, and P. Souplet: Singularity and decay estimates in superlinear problems via Liouville-type theorems. II. Parabolic equations. Indiana Univ. Math. J. 56 (2007), 879–908. · Zbl 1122.35051 · doi:10.1512/iumj.2007.56.2911 [27] P. Quittner and F. Simondon: A priori bounds and complete blow-up of positive solutions of indefinite superlinear parabolic problems. J. Math. Anal. Appl. 304 (2005), 614–631. · Zbl 1071.35026 · doi:10.1016/j.jmaa.2004.09.044 [28] P. Quittner and P. Souplet: Superlinear parabolic problems. Blow-up, global existence and steady states. Birkhäuser Advanced Texts: Basel Textbooks. Birkhäuser, Basel, 2007. · Zbl 1128.35003 [29] P. Quittner, P. Souplet, and M. Winkler: Initial blow-up rates and universal bounds for nonlinear heat equations. J. Differ. Equations 196 (2004), 316–339. · Zbl 1044.35027 · doi:10.1016/j.jde.2003.10.007 [30] J. Serrin: Entire solutions of nonlinear Poisson equations. Proc. London. Math. Soc. (3) 24 (1972), 348–366. · Zbl 0229.35035 · doi:10.1112/plms/s3-24.2.348 [31] J. Serrin: Entire solutions of quasilinear elliptic equations. J. Math. Anal. Appl. 352 (2009), 3–14. · Zbl 1180.35243 · doi:10.1016/j.jmaa.2008.10.036 [32] S.D. Taliaferro: Isolated singularities of nonlinear parabolic inequalities. Math. Ann. 338 (2007), 555–586. · Zbl 1120.35003 · doi:10.1007/s00208-007-0088-0 [33] S.D. Taliaferro: Blow-up of solutions of nonlinear parabolic inequalities. Trans. Amer. Math. Soc. 361 (2009), 3289–3302. · Zbl 1175.35072 · doi:10.1090/S0002-9947-09-04770-9 [34] F.B. Weissler: Single point blow-up for a semilinear initial value problem. J. Differ. Equations 55 (1984), 204–224. · Zbl 0555.35061 · doi:10.1016/0022-0396(84)90081-0 [35] F.B. Weissler: An L blow-up estimate for a nonlinear heat equation. Commun. Pure Appl. Math. 38 (1985), 291–295. · Zbl 0592.35071 · doi:10.1002/cpa.3160380303 [36] R. Xing: The blow-up rate for positive solutions of indefinite parabolic problems and related Liouville type theorems. Acta Math. Sin. (Engl. Ser.) 25 (2009), 503–518. · Zbl 1180.35147 · doi:10.1007/s10114-008-5615-8 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.