Artículo

Estamos trabajando para incorporar este artículo al repositorio
Consulte el artículo en la página del editor
Consulte la política de Acceso Abierto del editor

Abstract:

We study nonnegative solutions of two porous medium equations with nonlinear coupled boundary conditions and nonnegative nontrivial compactly supported initial data. We describe them in terms of the different parameters appearing in the problem, when solutions blow up in a finite time, and when they exist globally in time. We find three regions, bounded by two curves. One of them is a Fujita type curve. In the first region every solution is global, in the second every nontrivial solution blows up, and in the last one both behaviours coexist. In the blow-up case we find the blow-up rates and the blow-up sets. In particular, we prove that under certain conditions the blow-up sets of the two components of the system are different. This is the first known example of a nontrivially coupled parabolic system showing such a behaviour.

Registro:

Documento: Artículo
Título:Blow-up sets and Fujita type curves for a degenerate parabolic system with nonlinear boundary conditions
Autor:Quirós, F.; Rossi, J.D.
Filiación:Departamento de Matemáticas, U. Autónoma de Madrid, 28049 Madrid, Spain
Departamento de Matemática, F.C.E y N., Universidad Nacional de Buenos Aires, 1428 Buenos Aires, Argentina
Palabras clave:Blow-up; Degenerate parabolic system; Fujita exponents; Nonlinear boundary conditions
Año:2001
Volumen:50
Número:1
Página de inicio:629
Página de fin:654
DOI: http://dx.doi.org/10.1512/iumj.2001.50.1828
Título revista:Indiana University Mathematics Journal
Título revista abreviado:Indiana Univ. Math. J.
ISSN:00222518
Registro:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00222518_v50_n1_p629_Quiros

Referencias:

  • Andreucci, D., Herrero, M.A., Velázquez, J.J.L., Liouville theorems and blow-up behaviour in a semilinear reaction diffusion systems (1997) Ann. Inst. H. Poincaré Anal. Non Linéaire, 14, pp. 1-53
  • Bénilan, P.H., Cortázar, C., Elgueta, M., Uniqueness and non uniqueness of the solutions of a mixed boundary value problem for the porous medium equation (1991) Rev. Un. Mat. Argentina, 37, pp. 10-15
  • Cortázar, C., Del Pino, M., Elgueta, M., On the blow-up set for ut = Δum + um, m > 1 (1998) Indiana Univ. Math. J., 47, pp. 541-561
  • Cortázar, C., Elgueta, M., Vázquez, J.L., Diffusivity determination in non linear diffusion (1991) European J. Appl. Math., 2, pp. 159-169
  • Deng, K., Fila, M., Levine, H., On critical exponents for a system of heat equations coupled in the boundary conditions (1994) Acta Math. Univ. Comenian. (N.S.), 63, pp. 169-192
  • Dibenedetto, E., Continuity of weak solutions to a general porous medium equation (1983) Indiana Univ. Math. J., 32, pp. 83-118
  • Escobedo, M., Herrero, M.A., Boundedness and blow up for a semilinear reaction-diffusion system (1991) J. Differential Equations, 89, pp. 176-202
  • Escobedo, M., Levine, H.A., Critical blow-up and global existence numbers for a weakly coupled system of reaction-diffusion equations (1995) Arch. Rational Mech. Anal., 129, pp. 47-100
  • Fujita, H., On the blowing up of solutions of the Cauchy problem for ut = Δu + u1+α (1966) J. Fac. Sci. Univ. Tokyo Sec. IA Math., 16, pp. 105-113
  • Galaktionov, V.A., Blow-up for quasilinear heat equations with critical Fujita's exponents (1994) Proc. Roy. Soc. Edinburgh Sect. A., 124, pp. 517-525
  • On the blow-up set for the quasilinear equation ut = (uσux)x + uσ+1 (1993) J. Differential Equations, 101, pp. 66-79
  • On asymptotic self-similar behaviour for a quasilinear heat equation. Single point blow-up (1995) SIAM J. Math. Anal., 26, pp. 675-693
  • Galaktionov, V.A., Kurdyumov, S.P., Mikhailov, A.P., Samarskii, A.A., Unbounded solutions of the Cauchy problem for the parabolic equation ut = ∇(uσ∇u) + uβ (1980) Dokl. Akad. Nauk SSSR. Ser. Math. Phys., 252, pp. 1362-1364. , Russian
  • (1980) Soviet Phys. Dokl., 25, pp. 458-459. , English transl
  • Heat localization in nonlinear media (1981) Differentsial'nye Uravneniya, 17, pp. 1826-1841. , Russian
  • (1981) Differential Equations, 17, pp. 1141-1154. , English transl
  • Galaktionov, V.A., Kurdyumov, S.P., Samarskii, A.A., On the method of stationary states for quasilinear parabolic equations (1989) Mat. Sbornik, 180, pp. 995-1016. , Russian
  • (1990) Math. Ussr Sbornik, 67, pp. 449-471. , English transl
  • Galaktionov, V.A., Levine, H.A., On critical Fujita exponents for heat equations with nonlinear flux boundary conditions on the boundary (1996) Israel J. Math., 94, pp. 125-146
  • Gilding, B.H., On a class of similarity solutions of the porous media equation III (1980) J. Math. Anal. Appl., 77, pp. 381-402
  • Gilding, B.H., Herrero, M.A., Localization and blow-up of thermal waves in nonlinear heat conduction with peaking (1988) Math. Ann., 282, pp. 223-242
  • Gilding, B.H., Peletier, L.A., On a class of similarity solutions of the porous media equation (1976) J. Math. Anal. Appl., 55, pp. 351-364
  • On a class of similarity solutions of the porous media equation II (1977) J. Math. Anal. Appl., 57, pp. 522-538
  • 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, pp. 117-135
  • Kalashnikov, A.S., Some problems of the qualitative theory of nonlinear degenerate parabolic equations of second order (1987) Uspekhi Mat. Nauk, 42, pp. 135-176. , Russian
  • (1987) Russian Math. Surveys, 42, pp. 169-222. , English transl
  • Levine, H.A., The role of critical exponents in blow up theorems (1990) SIAM Rev., 32, pp. 262-288
  • Liberman, G.M., Second order parabolic differential equations (1996) World Scientific, , River Edge
  • Pao, C.V., (1992) Nonlinear Parabolic and Elliptic Equations, , Plenum Press, New York
  • Rossi, J.D., Wolanski, N., Global existence and nonexistence for a parabolic system with nonlinear boundary conditions (1998) Differential Integral Equations, 11, pp. 179-190
  • Blow-up vs. Global existence for a semilinear reaction-diffusion system in a bounded domain (1995) Comm. Partial Differential Equations, 20, pp. 1991-2004
  • Samarskii, A.A., Galaktionov, V.A., Kurdyumov, S.P., Mikhailov, A.P., (1987) Blow-up in Problems for Quasilinear Parabolic Equations, Nauka, Moscow, , Russian
  • De Gruyter, W., (1995), Berlin, English transl; Samarskii, A.A., Zmitrenko, N.V., Kurdyumov, S.P., Mihailov, A.P., Thermal structures and fundamental length in a medium with nonlinear heat conduction and volumetric heat sources (1976) Dokl. Akad. Nauk Sssr Ser. Math. Phys., 227, pp. 321-324. , Russian
  • (1976) Soviet Phys. Dokl., 21, pp. 141-143. , English transl
  • Wang, S., Xie, C.H., Wang, M., Note on critical exponents for a system of heat equations coupled in the boundary conditions (1998) J. Math. Anal. Appl., 218, pp. 313-324

Citas:

---------- APA ----------
Quirós, F. & Rossi, J.D. (2001) . Blow-up sets and Fujita type curves for a degenerate parabolic system with nonlinear boundary conditions. Indiana University Mathematics Journal, 50(1), 629-654.
http://dx.doi.org/10.1512/iumj.2001.50.1828
---------- CHICAGO ----------
Quirós, F., Rossi, J.D. "Blow-up sets and Fujita type curves for a degenerate parabolic system with nonlinear boundary conditions" . Indiana University Mathematics Journal 50, no. 1 (2001) : 629-654.
http://dx.doi.org/10.1512/iumj.2001.50.1828
---------- MLA ----------
Quirós, F., Rossi, J.D. "Blow-up sets and Fujita type curves for a degenerate parabolic system with nonlinear boundary conditions" . Indiana University Mathematics Journal, vol. 50, no. 1, 2001, pp. 629-654.
http://dx.doi.org/10.1512/iumj.2001.50.1828
---------- VANCOUVER ----------
Quirós, F., Rossi, J.D. Blow-up sets and Fujita type curves for a degenerate parabolic system with nonlinear boundary conditions. Indiana Univ. Math. J. 2001;50(1):629-654.
http://dx.doi.org/10.1512/iumj.2001.50.1828