An example of blowup for a degenerate parabolic equation with a nonlinear boundary condition - Zurich Open Repository and Archive

Chipot, M; Filo, J (1998). An example of blowup for a degenerate parabolic equation with a nonlinear boundary condition. Zeitschrift für Analysis und ihre Anwendungen, 17(1):89-102.

Abstract

In this paper, a nonlinear parabolic equation of the form $u_t=(a(u_x))_x$ for $x\in(0,1),\ t>0,\ a(u_x)=|u_x|^{p-2}u_x$ if $u_x\geq\eta>0,\ 1<p<2$, with nonlinear boundary condition $a(u_x(1,t))=|u|^{q-2}u(1,t)$, is considered. It is proved that if $qp-3p+2>0$, then the solutions blow up in finite time. Moreover, estimates on the blow-up profile (in $x$) and the blow-up rate (in $t$) for $x=1$ are derived.

Abstract

In this paper, a nonlinear parabolic equation of the form $u_t=(a(u_x))_x$ for $x\in(0,1),\ t>0,\ a(u_x)=|u_x|^{p-2}u_x$ if $u_x\geq\eta>0,\ 1<p<2$, with nonlinear boundary condition $a(u_x(1,t))=|u|^{q-2}u(1,t)$, is considered. It is proved that if $qp-3p+2>0$, then the solutions blow up in finite time. Moreover, estimates on the blow-up profile (in $x$) and the blow-up rate (in $t$) for $x=1$ are derived.

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1 citation in Web of Science®
2 citations in Scopus®

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Item Type: Journal Article, refereed, original work 07 Faculty of Science > Institute of Mathematics 510 Mathematics regularity results; nonlocal variational inequalities English 1998 26 Jul 2010 11:35 05 Apr 2016 13:26 European Mathematical Society 0232-2064 https://doi.org/10.4171/ZAA http://www.math.ethz.ch/EMIS/journals/ZAA/ http://www.zentralblatt-math.org/zbmath/search/