Instability of solutions of a semilinear heat equation with a Neumann boundary condition
Authors:
Keng Deng and Cheng-Lin Zhao
Journal:
Quart. Appl. Math. 63 (2005), 13-19
MSC (2000):
Primary 34B18, 35B05, 35B35, 35K60
DOI:
https://doi.org/10.1090/S0033-569X-05-00946-2
Published electronically:
January 19, 2005
MathSciNet review:
2126566
Full-text PDF Free Access
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Abstract: A semilinear heat equation $u_t=u_{xx}+\varepsilon u^p,0<x<1,\ \varepsilon , p>0,$ subject to $u_x(0,t)=0,u_x(1,t)=-u^{-q}(1,t),\ q>0$ is studied. The set of stationary states is characterized, their instability is analyzed, and the large time behavior of positive solutions is discussed.
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DZ K. Deng and C.-L. Zhao, Blow-up versus quenching, Comm. Appl. Anal. 7 (2003), 87-100.
FL M. Fila and H.A. Levine, Quenching on the boundary, Nonlinear Anal. TMA 21 (1993), 795-802.
FM A. Friedman and B. Mcleod, Blow-up of positive solutions of semilinear heat equations, Indiana Univ. Math. J. 34 (1985), 425-447.
L H.A. Levine, Quenching, nonquenching, and beyond quenching for solutions of some parabolic equations, Ann. Mat. Pura Appl. 155 (1989), 243-260. MR1042837 (91m:35028)
PAO C.V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992.
Z T.I. Zelenyak, Stabilisation of solutions of boundary value problems for a second-order equation with one space variable, Differential Equations 4 (1968), 17-22.
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Additional Information
Keng Deng
Affiliation:
Department of Mathematics, University of Louisiana at Lafayette, Lafayette, Louisiana 70504
MR Author ID:
225222
Cheng-Lin Zhao
Affiliation:
Department of Mathematics, University of Louisiana at Lafayette, Lafayette, Louisiana 70504
Received by editor(s):
July 24, 2003
Published electronically:
January 19, 2005
Additional Notes:
The work of the first author was supported in part by the National Science Foundation under grant DMS-0211412
Article copyright:
© Copyright 2005
Brown University
