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Indiana University Mathematics Journal
Article . 2001 . Peer-reviewed
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Life span of solutions with large initial data in a semilinear parabolic equation

Authors: Mizoguchi, Noriko; Yanagida, Eiji;

Life span of solutions with large initial data in a semilinear parabolic equation

Abstract

The paper deals with the Cauchy problem for the semilinear parabolic equation \[ \begin{cases} u_t=\Delta u +|u|^{p-1}u\quad & \text{in} {\mathbb R}^N\times (0,\infty),\\ u(x,0)=\lambda \varphi(x)\quad & \text{in} {\mathbb R}^N,\end{cases} \] with \(p>1,\) \(\lambda>0\) and \(\varphi\) being a bounded continuous function. The authors show that the blowup time \(T(\lambda)\) of the solution satisfies \[ T(\lambda)={{1}\over {p-1}} |\varphi|_\infty^{1-p} \lambda^{1-p} + o(\lambda^{1-p})\quad \text{ as} \lambda\to\infty. \] Moreover, when the maximum of \(|\varphi(x)|\) is attained at one point, the higher-order term of \(Y(\lambda)\) is determined which reflects the pointedness of the peak of \(|\varphi|.\)

Keywords

Cauchy problem, Nonlinear parabolic equations, semilinear parabolic equation, Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs, blow-up time

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selected citations
These citations are derived from selected sources.
This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Citations provided by BIP!
popularity
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
BIP!Impulse provided by BIP!
15
Average
Top 10%
Average
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