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| INSTANTANEOUS SHRINKING AND LOCALIZATION OF FUNCTIONS IN Y$_{\lambda}$({m},{p},{q},{N}) AND THEIR APPLICATIONS |
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Citation： |
YUAN Hongjun.INSTANTANEOUS SHRINKING AND LOCALIZATION OF FUNCTIONS IN Y$_{\lambda}$({m},{p},{q},{N}) AND THEIR APPLICATIONS[J].Chinese Annals of Mathematics B,2001,22(3):361~380 |
| Page view： 888
Net amount： 682 |
Authors： |
YUAN Hongjun; |
Foundation： |
Project supported by the Teaching and Research Award Fund for Outstanding Young Teachers in
Higher Education Institutions of MOE, China (No.[2000]26). |
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| Abstract： |
The aim of this paper is to discuss
the instantaneous shrinking and localization of the support
of functions in $\hbox{Y}_{\lambda}(m,p,q,N)$ and their
applications to some nonlinear parabolic equations including
the porous medium equation
$
u_t=\Delta u^m-u^{q},\ m>0,\ q>0
$
and the $p$-Laplace equation
$
u_t=\hbox{div}(|\nabla u|^{p-2}\nabla u)-u^{q},\ p>1,\ q>0.
$
In particular, as an application of the results,
the necessary and sufficient condition for the solutions
of the above $p$-Laplace equation with
nonnegative finite Borel measures
as initial conditions to have the instantaneous shrinking property
of the support is obtained. This is an answer to an open problem
posed by R.~Kersner and A.~Shishkov. |
Keywords： |
Porous medium equation, p-Laplace equation, Instantaneous shrinking, Localization property, Support |
Classification： |
35B05, 35B40, 35K15, 35K55, 35K65 |
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