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Finite time blow-up for the inhomogeneous equation $u_{t}=\Delta u+a(x)u^{p}+\lambda \phi $ in $R^{d}$

Author: Ross G. Pinsky
Journal: Proc. Amer. Math. Soc. 127 (1999), 3319-3327
MSC (1991): Primary 35K15, 35K55
Published electronically: May 17, 1999
MathSciNet review: 1641081
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Abstract | References | Similar Articles | Additional Information

Abstract: We consider the inhomogeneous equation

\begin{equation*}\begin{split} & u_{t}=\Delta u+a(x)u^{p}+\lambda \phi (x) \ \text{in} \ R^{d}, t\in (0,T),\\ &u(x,0)=f(x),\end{split}\end{equation*}

where $a,\phi \gneqq 0$, $\lambda >0$ and $f\ge 0$, and give criteria on $p,d,a$, and $\phi $ which determine whether for all $\lambda $ and all $f$ the solution blows up in finite time or whether for $\lambda $ and $f$ sufficiently small, the solution exists for all time.

References [Enhancements On Off] (What's this?)

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Additional Information

Ross G. Pinsky
Affiliation: Department of Mathematics, Technion-Israel Institute of Technology, Haifa, 32000, Israel

Keywords: Finite time blow-up, semilinear reaction-diffusion equations, critical exponent
Received by editor(s): February 11, 1998
Published electronically: May 17, 1999
Additional Notes: This research was supported by the Fund for the Promotion of Research at the Technion.
Communicated by: Lesley M. Sibner
Article copyright: © Copyright 1999 American Mathematical Society