[Le comportement désactivant d'une équation parabolique quasi linéaire avec deux sources singulières]
Nous étudions ici le comportement désactivant d'une équation parabolique quasi-linéaire avec un terme de réaction singulier et un flux au bord singulier. Sous certaines conditions sur les données initiales, nous montrons que la désactivation intervient seulement au bord en temps fini. De plus, nous obtenons des bornes inférieure et supérieure du taux de désactivation ainsi que des estimations du temps de désactivation.
In this paper, we study the quenching behavior for a one-dimensional quasilinear parabolic equation with singular reaction term and singular boundary flux. Under certain conditions on the initial data, we show that quenching occurs only on the boundary in finite time. Moreover, we derive some lower and upper bounds of the quenching rate and get some estimates for the quenching time.
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@article{CRMATH_2018__356_7_725_0,
author = {Zhu, Liping},
title = {The quenching behavior of a quasilinear parabolic equation with double singular sources},
journal = {Comptes Rendus. Math\'ematique},
pages = {725--731},
publisher = {Elsevier},
volume = {356},
number = {7},
year = {2018},
doi = {10.1016/j.crma.2018.05.013},
language = {en},
url = {http://archive.numdam.org/articles/10.1016/j.crma.2018.05.013/}
}
TY - JOUR AU - Zhu, Liping TI - The quenching behavior of a quasilinear parabolic equation with double singular sources JO - Comptes Rendus. Mathématique PY - 2018 SP - 725 EP - 731 VL - 356 IS - 7 PB - Elsevier UR - http://archive.numdam.org/articles/10.1016/j.crma.2018.05.013/ DO - 10.1016/j.crma.2018.05.013 LA - en ID - CRMATH_2018__356_7_725_0 ER -
%0 Journal Article %A Zhu, Liping %T The quenching behavior of a quasilinear parabolic equation with double singular sources %J Comptes Rendus. Mathématique %D 2018 %P 725-731 %V 356 %N 7 %I Elsevier %U http://archive.numdam.org/articles/10.1016/j.crma.2018.05.013/ %R 10.1016/j.crma.2018.05.013 %G en %F CRMATH_2018__356_7_725_0
Zhu, Liping. The quenching behavior of a quasilinear parabolic equation with double singular sources. Comptes Rendus. Mathématique, Tome 356 (2018) no. 7, pp. 725-731. doi : 10.1016/j.crma.2018.05.013. http://archive.numdam.org/articles/10.1016/j.crma.2018.05.013/
[1] Some blowup results for a nonlinear parabolic equation with a gradient term, SIAM J. Math. Anal., Volume 20 (1989), pp. 886-907
[2] Quenching for a nonlinear diffusion equation with a singular boundary condition, Z. Angew. Math. Phys., Volume 50 (1999), pp. 564-584
[3] Mathematical analysis of partial differential equations modeling electrostatic MEMS, Courant Lect. Notes Math., Volume 20 (2010)
[4] Admissible solutions for a class of nonlinear parabolic problems with non-negative data, Proc. R. Soc. Edinb., Sect. A, Volume 131 (2001), pp. 857-883
[5] On the quenching behavior of the solution of a semilinear parabolic equation, J. Math. Anal. Appl., Volume 151 (1990), pp. 58-79
[6] Quenching time for a semilinear heat equation with a nonlinear Neumann boundary condition, J. Partial Differ. Equ., Volume 27 (2014), pp. 217-228
[7] Classification of certain qualitative properties of solutions for the quasilinear parabolic equations, Sci. China Ser. A, Volume 61 (2018), pp. 855-868
[8] The quenching behavior of a nonlinear parabolic equation with a singular boundary condition, Hacet. J. Math. Stat., Volume 44 (2015), pp. 615-621
[9] Modeling MEMS and NEMS, Chapman Hall and CRC Press, 2002
[10] A nonlinear heat equation with temperature-dependent parameters, Math. Phys. Electron. J., Volume 12 (2006)
[11] The quenching behavior of a semilinear heat equation with a singular boundary outflux, Q. Appl. Math., Volume 72 (2014), pp. 747-752
[12] Finite time blow-up for a non-linear parabolic equation with a gradient term and applications, Math. Methods Appl. Sci., Volume 19 (1996), pp. 1317-1333
[13] A quenching phenomenon for one-dimensional p-Laplacian with singular boundary flux, Appl. Math. Lett., Volume 23 (2010), pp. 955-959
[14] Quenching phenomenon of positive radial solutions for p-Laplacian with singular boundary flux, J. Dyn. Control Syst., Volume 22 (2016), pp. 653-660
[15] Blowup and existence of global solutions to nonlinear parabolic equations with degenerate diffusion, Electron. J. Differ. Equ., Volume 264 (2013)
[16] Classification of blowup solutions for a parabolic p-Laplacian equation with nonlinear gradient terms, J. Math. Anal. Appl., Volume 436 (2016), pp. 1266-1283
[17] Blow-up and Quenching for Solutions of Some Parabolic Equations, University of Louisiana, Lafayette, LA, USA, 2000 (PhD Thesis)
[18] The boundary quenching behavior of a semilinear parabolic equation, Appl. Math. Comput., Volume 218 (2011), pp. 233-238
[19] The quenching behavior of a nonlinear parabolic equation with nonlinear boundary outflux, Appl. Math. Comput., Volume 184 (2007), pp. 624-630
[20] Blowup time of solutions for a small diffusive parabolic problem with exponential source, Bound. Value Probl., Volume 155 (2016), pp. 1-15
[21] Rate of approach to the steady state for a diffusion-convection equation on annular domains, Electron. J. Qual. Theory Differ. Equ., Volume 39 (2012)
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