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Abstract:

We study a system of two porous medium type equations in a bounded interval, coupled at the boundary in a nonlinear way. Under certain conditions, one of its components becomes unbounded in finite time while the other remains bounded, a situation that is known in the literature as nonsimultaneous blow-up. We characterize completely, in the case of nondecreasing in time solutions, the set of parameters appearing in the system for which non-simultaneous blow-up indeed occurs. Moreover, we obtain the blow-up rate and the blow-up set for the component which blows up. We also prove that in the range of exponents where each of the components may blow up on its own there are special initial data such that blow-up is simultaneous. Finally, we give conditions on the exponents which lead to non-simultaneous blow-up for every initial data.

Registro:

Documento: Artículo
Título:Non-simultaneous blow-up for a quasilinear parabolic system with reaction at the boundary
Autor:Brändle, C.; Quirós, F.; Rossi, J.D.
Filiación:Departamento de Matemáticas, U. Autónoma de Madrid, 28049 Madrid, Spain
Departamento de Matemática, F.C.E y N. UBA (1428), Buenos Aires, Argentina
Palabras clave:Blow-up; Nonlinear boundary conditions; Nonlinear diffusion; Parabolic systems
Año:2005
Volumen:4
Número:3
Página de inicio:523
Página de fin:536
DOI: http://dx.doi.org/10.3934/cpaa.2005.4.523
Título revista:Communications on Pure and Applied Analysis
Título revista abreviado:Commun. Pure Appl. Anal.
ISSN:15340392
Registro:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_15340392_v4_n3_p523_Brandle

Referencias:

  • Chasseigne, E., Vazquez, J.L., Theory of extended solutions for fast-diffusion equations in optimal classes of data. Radiation from singularities (2002) Arch. Ration. Mech. Anal., 164 (2), pp. 133-187
  • Chlebík, M., Fila, M., Some recent results on blow-up on the boundary for the heat equation (2000) Banach Center Publ., 52, pp. 61-71. , Evolution equations: existence, regularity and singularities (Warsaw, 1998), Polish Acad. Sci., Warsaw
  • Cortázar, C., Elgueta, M., Localization and boundedness of the solutions of the Neumann problem for a filtration equation (1989) Nonlinear Anal., 13 (1), pp. 33-41
  • Cortazar, C., Elgueta, M., Rossi, J.D., Uniqueness and nonuniqueness for the porous medium equation with non linear boundary conditions (2003) Differential Integral Equations, 16 (10), pp. 1215-1222
  • DiBenedetto, E., Continuity of weak solutions to a general porous medium equation (1983) Indiana Univ. Math. J., 32 (1), pp. 83-118
  • Ferreira, R., Quirós, F., Rossi, J.D., The balance between nonlinear inwards and outwards boundary flux for a nonlinear heat equation (2002) J. Differential Equations, 184 (1), pp. 259-282
  • Fila, M., Filo, J., Blow-up on the boundary: A survey (1996) Banach Center Publ., 33, pp. 67-78. , Singularities and differential equations (Warsaw, 1993), Polish Acad. Sci., Warsaw
  • Filo, J., Diffusivity versus absorption through the boundary (1992) J. Differential Equations, 99 (2), pp. 281-305
  • Gilding, B.H., Herrero, M.A., Localization and blow-up of thermal waves in nonlinear heat conduction with peaking (1988) Math. Ann., 282 (2), pp. 223-242
  • Hu, B., Yin, H.M., The profile near blowup time for solution of the heat equation with a nonlinear boundary condition (1994) Trans. Amer. Math. Soc., 346 (1), pp. 117-135
  • Lieberman, G.M., (1996) Second Order Parabolic Differential Equations, , World Scientific Publishing Co. Inc., River Edge, NJ
  • Pinasco, J.P., Rossi, J.D., Simultaneous versus non-simultaneous blow-up (2000) New Zealand J. Math., 29 (1), pp. 55-59
  • Quirós, F., Rossi, J.D., Blow-up sets and Fujita type curves for a degenerate parabolic system with nonlinear boundary conditions (2001) Indiana Univ. Math. J., 50 (1), pp. 629-654
  • Non-simultaneous blow-up in a semilinear parabolic system (2001) Z. Angew. Math. Phys., 52 (2), pp. 342-346
  • Non-simultaneous blow-up in a nonlinear parabolic system (2003) Adv. Nonlinear Stud., 3 (3), pp. 397-1118
  • Samarskii, A.A., Galaktionov, V.A., Kurdyumov, S.P., Mikhailov, A.P., Blow-up in quasilinear parabolic equations (1995) De Gruyter Expositions in Mathematics, 19. , Walter de Gruyter & Co., Berlin, Translated from the 1987 Russian original by Michael Grinfeld and revised by the authors
  • Souplet, P., Tayachi, S., Optimal condition for non-simultaneous blow-up in a reaction-diffusion system (2004) J. Math. Soc. Japan, 56 (2), pp. 571-584
  • Wang, M., Fast-slow diffusion systems with nonlinear boundary conditions (2001) Nonlinear Anal., 46 (6), pp. 893-908. , Ser. A: Theory Methods
  • Wang, M., Wang, S., Quasilinear reaction-diffusion systems with nonlinear boundary conditions (1999) J. Math. Anal. Appl., 231 (1), pp. 21-33
  • Ziemer, W.P., Interior and boundary continuity of weak solutions of degenerate parabolic equations (1982) Trans. Amer. Math. Soc., 271 (2), pp. 733-748

Citas:

---------- APA ----------
Brändle, C., Quirós, F. & Rossi, J.D. (2005) . Non-simultaneous blow-up for a quasilinear parabolic system with reaction at the boundary. Communications on Pure and Applied Analysis, 4(3), 523-536.
http://dx.doi.org/10.3934/cpaa.2005.4.523
---------- CHICAGO ----------
Brändle, C., Quirós, F., Rossi, J.D. "Non-simultaneous blow-up for a quasilinear parabolic system with reaction at the boundary" . Communications on Pure and Applied Analysis 4, no. 3 (2005) : 523-536.
http://dx.doi.org/10.3934/cpaa.2005.4.523
---------- MLA ----------
Brändle, C., Quirós, F., Rossi, J.D. "Non-simultaneous blow-up for a quasilinear parabolic system with reaction at the boundary" . Communications on Pure and Applied Analysis, vol. 4, no. 3, 2005, pp. 523-536.
http://dx.doi.org/10.3934/cpaa.2005.4.523
---------- VANCOUVER ----------
Brändle, C., Quirós, F., Rossi, J.D. Non-simultaneous blow-up for a quasilinear parabolic system with reaction at the boundary. Commun. Pure Appl. Anal. 2005;4(3):523-536.
http://dx.doi.org/10.3934/cpaa.2005.4.523