Blow-up phenomena for some nonlinear parabolic problems under mixed boundary conditions. (English) Zbl 1201.35057
Summary: We determine the lower bound for the blow-up time of solution to equations of the form \(u_t = \text{div} (\rho (|\nabla u|^2)\operatorname{grad} u) + f(u)\) if the solution blows up. Conditions which ensure, that blow-up does not occur, are also presented.
MSC:
| 35B44 | Blow-up in context of PDEs |
| 35K59 | Quasilinear parabolic equations |
| 35K20 | Initial-boundary value problems for second-order parabolic equations |
References:
| [1] | Bandle, C.; Brunner, H., Blow-up in diffusion equations: A survey, J. Comput. Appl. Math., 97, 3-22 (1998) · Zbl 0932.65098 |
| [2] | Galaktionov, V. A.; Vázquez, J. L., The problem of blow up in nonlinear parabolic equations, Discrete Contin. Dyn. Syst., 8, 399-433 (2002) · Zbl 1010.35057 |
| [3] | Levine, H. A., The role of critical exponents in blow-up theorems, SIAM Rev., 32, 262-288 (1990) · Zbl 0706.35008 |
| [4] | Straughan, B., Instability, nonexistence and weighted energy methods in fluid dynamics and related theories, Res. Notes Math., 74 (1982) · Zbl 0492.76001 |
| [5] | Ball, J. M., Remarks on blow up and nonexistence theorems for nonlinear evolution equations, Quart. J. Math. Oxford, 28, 473-486 (1977) · Zbl 0377.35037 |
| [6] | Caffarrelli, L. A.; Friedman, A., Blow-up of solutions of nonlinear heat equations, J. Math. Anal. Appl., 129, 409-419 (1988) · Zbl 0653.35038 |
| [7] | Friedman, A.; Mcleod, B., Blow-up of positive solutions of semilinear heat equations, Indiana Univ. Math. J., 34, 425-447 (1985) · Zbl 0576.35068 |
| [8] | Kielhöfer, H., Existen und Regularität von Lösungen semilinearer parabolisccher Anfangs-Randwertprobleme, Math. Z., 142, 131-160 (1975) · Zbl 0324.35047 |
| [9] | Levine, H. A., Nonexietence of global weak solutions to some properly and improperly posed problems of mathematical physics: The method of unbounded Fourier coefficients, Math. Ann., 214, 205-220 (1975) · Zbl 0286.35006 |
| [10] | Payne, L. E.; Schaefer, P. W., Lower bounds for blow-up time in parabolic problems under Dirichlet conditions, J. Math. Anal. Appl., 328, 1196-1205 (2007) · Zbl 1110.35031 |
| [11] | Straughan, B., Explosive Instabilities in Mechamics (1998), Spring-Verlag: Spring-Verlag Berlin · Zbl 0911.35002 |
| [12] | Payne, L. E.; Philippin, G. A.; Schaefer, P. W., Blow-up phenomena for some nonlinear parabolic problems, Nonlinear Anal., 69, 3495-3502 (2008) · Zbl 1159.35382 |
| [13] | Friedman, A., Remarks on the maximum principle for parabolic equations and its applications, Pacific J. Math., 8, 201-211 (1958) · Zbl 0103.06403 |
| [14] | Nirenberg, L., A strong maximum principle for parabolic equations, Commun. Pure Appl. Math., 6, 167-177 (1953) · Zbl 0050.09601 |
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.
