Quenching Phenomena for a Quasilinear Parabolic Equation with Nonlinear Source

Science & Technology Review ›› 2010, Vol. 28 ›› Issue (06) : 29-31.

PDF(285 KB)
PDF(285 KB)
Science & Technology Review ›› 2010, Vol. 28 ›› Issue (06) : 29-31.
Articles

Quenching Phenomena for a Quasilinear Parabolic Equation with Nonlinear Source

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Abstract

For many quasilinear differential equations, their origin may be traced in biology and astrophysics, as related with generalized reaction diffusion theory, non-Newtonian fluid theory, non-Newtonian filtration theory and the turbulent flow of a gas in porous medium. Quenching phenomena for a class of quasilinear parabolic equations are studied in this paper. In the time domain, nonlinear parabolic equations not necessarily have a continuous solution. For some equations, solutions exist as a whole, for others solutions may not exist. Quenching phenomena are related with cases where solutions do not exist in some time period. This paper considers the extinction of solutions of an initial boundary value problem of the quasilinear parabolic equation ut-div[?滓(|?荦u|2)?荦u]=?姿up in a bounded convex domain of RN with N≥2. This problem is first introduced in the field of geometry. At this time, the principal part of the equation is no longer uniformly elliptic. Using upper and lower solutions and the integral estimate method, two results of the extinction of the solution are obtained. The results obtained can be extended to a more general form of quasilinear parabolic equations.

Key words

extinction / quasilinear parabolic equation / nonlinear source

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Quenching Phenomena for a Quasilinear Parabolic Equation with Nonlinear Source[J]. Science & Technology Review, 2010, 28(06): 29-31
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