Existence and asymptotic behavior for a singular parabolic equation
HTML articles powered by AMS MathViewer
- by Juan Dávila and Marcelo Montenegro PDF
- Trans. Amer. Math. Soc. 357 (2005), 1801-1828 Request permission
Abstract:
We prove global existence of nonnegative solutions to the singular parabolic equation $u_t -\Delta u + \chi _{ \{ u>0 \} } ( -u^{-\beta } + \lambda f(u) )=0$ in a smooth bounded domain $\Omega \subset \mathbb {R}^N$ with zero Dirichlet boundary condition and initial condition $u_0 \in C(\Omega )$, $u_0 \geq 0$. In some cases we are also able to treat $u_0 \in L^\infty (\Omega )$. Then we show that if the stationary problem admits no solution which is positive a.e., then the solutions of the parabolic problem must vanish in finite time, a phenomenon called “quenching”. We also establish a converse of this fact and study the solutions with a positive initial condition that leads to uniqueness on an appropriate class of functions.References
- Andrew F. Acker and Bernhard Kawohl, Remarks on quenching, Nonlinear Anal. 13 (1989), no. 1, 53–61. MR 973368, DOI 10.1016/0362-546X(89)90034-5
- R. Aris, The Mathematical Theory of Diffusion and Reaction in Permeable Catalysts. Clarendon Press, Oxford, 1975.
- H. T. Banks, Modeling and control in the biomedical sciences, Lecture Notes in Biomathematics, Vol. 6, Springer-Verlag, Berlin-New York, 1975. MR 0401201, DOI 10.1007/978-3-642-66207-2
- Haïm Brezis and Thierry Cazenave, A nonlinear heat equation with singular initial data, J. Anal. Math. 68 (1996), 277–304. MR 1403259, DOI 10.1007/BF02790212
- Haïm Brezis, Thierry Cazenave, Yvan Martel, and Arthur Ramiandrisoa, Blow up for $u_t-\Delta u=g(u)$ revisited, Adv. Differential Equations 1 (1996), no. 1, 73–90. MR 1357955
- C. Y. Chan and Man Kam Kwong, Quenching phenomena for singular nonlinear parabolic equations, Nonlinear Anal. 12 (1988), no. 12, 1377–1383. MR 972406, DOI 10.1016/0362-546X(88)90085-5
- Juan Dávila, Global regularity for a singular equation and local $H^1$ minimizers of a nondifferentiable functional, Commun. Contemp. Math. 6 (2004), no. 1, 165–193. MR 2048779, DOI 10.1142/S0219199704001240
- Juan Dávila and Marcelo Montenegro, Positive versus free boundary solutions to a singular elliptic equation, J. Anal. Math. 90 (2003), 303–335. MR 2001074, DOI 10.1007/BF02786560
- Juan Dávila and Marcelo Montenegro, A singular equation with positive and free boundary solutions, RACSAM. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 97 (2003), no. 1, 107–112 (English, with English and Spanish summaries). MR 2037228
- Keng Deng and Howard A. Levine, On the blow up of $u_t$ at quenching, Proc. Amer. Math. Soc. 106 (1989), no. 4, 1049–1056. MR 969520, DOI 10.1090/S0002-9939-1989-0969520-0
- J. I. Díaz, Nonlinear partial differential equations and free boundaries. Vol. I, Research Notes in Mathematics, vol. 106, Pitman (Advanced Publishing Program), Boston, MA, 1985. Elliptic equations. MR 853732
- Marek Fila and Josephus Hulshof, A note on the quenching rate, Proc. Amer. Math. Soc. 112 (1991), no. 2, 473–477. MR 1055772, DOI 10.1090/S0002-9939-1991-1055772-7
- Marek Fila, Josephus Hulshof, and Pavol Quittner, The quenching problem on the $N$-dimensional ball, Nonlinear diffusion equations and their equilibrium states, 3 (Gregynog, 1989) Progr. Nonlinear Differential Equations Appl., vol. 7, Birkhäuser Boston, Boston, MA, 1992, pp. 183–196. MR 1167839, DOI 10.1007/978-1-4612-0393-3_{1}4
- Marek Fila and Bernhard Kawohl, Is quenching in infinite time possible?, Quart. Appl. Math. 48 (1990), no. 3, 531–534. MR 1074968, DOI 10.1090/qam/1074968
- Marek Fila and Bernhard Kawohl, Asymptotic analysis of quenching problems, Rocky Mountain J. Math. 22 (1992), no. 2, 563–577. MR 1180720, DOI 10.1216/rmjm/1181072749
- Marek Fila, Bernhard Kawohl, and Howard A. Levine, Quenching for quasilinear equations, Comm. Partial Differential Equations 17 (1992), no. 3-4, 593–614. MR 1163438, DOI 10.1080/03605309208820855
- Marek Fila, Howard A. Levine, and Juan L. Vázquez, Stabilization of solutions of weakly singular quenching problems, Proc. Amer. Math. Soc. 119 (1993), no. 2, 555–559. MR 1174490, DOI 10.1090/S0002-9939-1993-1174490-X
- Stathis Filippas and Jong-Shenq Guo, Quenching profiles for one-dimensional semilinear heat equations, Quart. Appl. Math. 51 (1993), no. 4, 713–729. MR 1247436, DOI 10.1090/qam/1247436
- Changfeng Gui and Fang-Hua Lin, Regularity of an elliptic problem with a singular nonlinearity, Proc. Roy. Soc. Edinburgh Sect. A 123 (1993), no. 6, 1021–1029. MR 1263903, DOI 10.1017/S030821050002970X
- Jong-Shenq Guo, On the quenching rate estimate, Quart. Appl. Math. 49 (1991), no. 4, 747–752. MR 1134750, DOI 10.1090/qam/1134750
- Hideo Kawarada, On solutions of initial-boundary problem for $u_{t}=u_{xx}+1/(1-u)$, Publ. Res. Inst. Math. Sci. 10 (1974/75), no. 3, 729–736. MR 0385328, DOI 10.2977/prims/1195191889
- O. A. Ladyženskaja, V. A. Solonnikov, and N. N. Ural′ceva, Lineĭ nye i kvazilineĭ nye uravneniya parabolicheskogo tipa, Izdat. “Nauka”, Moscow, 1967 (Russian). MR 0241821
- Howard A. Levine, Quenching and beyond: a survey of recent results, Nonlinear mathematical problems in industry, II (Iwaki, 1992) GAKUTO Internat. Ser. Math. Sci. Appl., vol. 2, Gakk\B{o}tosho, Tokyo, 1993, pp. 501–512. MR 1370487
- Daniel Phillips, Existence of solutions of quenching problems, Appl. Anal. 24 (1987), no. 4, 253–264. MR 907341, DOI 10.1080/00036818708839668
Additional Information
- Juan Dávila
- Affiliation: Departamento de Ingeniería Matemática, CMM (UMR CNRS), Universidad de Chile, Casilla 170/3, Correo 3, Santiago, Chile
- Email: jdavila@dim.uchile.cl
- Marcelo Montenegro
- Affiliation: Departamento de Matemática, Universidade Estadual de Campinas, IMECC, Caixa Postal 6065, CEP 13084-970, Campinas, SP, Brasil
- Email: msm@ime.unicamp.br
- Received by editor(s): July 18, 2003
- Published electronically: December 29, 2004
- © Copyright 2004 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 357 (2005), 1801-1828
- MSC (2000): Primary 35B40, 35K55
- DOI: https://doi.org/10.1090/S0002-9947-04-03811-5
- MathSciNet review: 2115077