Quarterly of Applied Mathematics

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A blow-up criterion for a degenerate parabolic problem due to a concentrated nonlinear source

Author(s): C. Y. Chan; R. Boonklurb
Journal: Quart. Appl. Math. 65 (2007), 781-787.
MSC (2000): Primary 35K60, 35K65, 35K57
Posted: October 9, 2007
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Abstract: Let , , , and be real numbers with $q\geq0$ , , , and . This article studies the following degenerate semilinear parabolic first initial-boundary value problem,

$\displaystyle x^{q}u_{t}(x,t)-u_{xx}(x,t)=a\delta(x-b)f\left( u(x,t)\right)$ for $\displaystyle 0<x<1,$ $\displaystyle 0<t\leq T,$
$\displaystyle u(x,0)=\psi(x)$ for $\displaystyle 0\leq x\leq1$ , $\displaystyle u(0,t)=u(1,t)=0$ for $\displaystyle 0<t\leq T,$

where $\delta\left( x\right)$ is the Dirac delta function, and

and $\psi$ are given functions. It is shown that for

sufficiently large, there exists a unique number $b^{\ast}\in\left( 0,1/2\right)$ such that

never blows up for $b\in\left( 0,b^{\ast}\right] \cup\left[ 1-b^{\ast},1\right)$ , and

always blows up in a finite time for $b\in(b^{\ast},1-b^{\ast})$ . To illustrate our main results, two examples are given.

References:

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