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Backward uniqueness for solutions of linear parabolic equations

Author: Igor Kukavica
Translated by:
Journal: Proc. Amer. Math. Soc. 132 (2004), 1755-1760
MSC (2000): Primary 35K15
Published electronically: December 22, 2003
MathSciNet review: 2051137
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Abstract: We address the backward uniqueness property for the equation $u_t-\Delta u = w_j\partial_{j}u+v u$ in ${\mathbb R}^n\times(T_0,0]$. We show that under rather general conditions on $v$ and $w$, $u\vert _{t=0}=0$ implies that $u$ vanishes to infinite order for all points $(x,0)$. It follows that the backward uniqueness holds if $w=0$and $v\in L^{\infty}([0,T_0],L^p({\mathbb R}^n))$ when $p>n/2$. The borderline case $p=n/2$ is also covered with an additional continuity and smallness assumption.

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Additional Information

Igor Kukavica
Affiliation: Department of Mathematics, University of Southern California, Los Angeles, California 90089

Keywords: Backward uniqueness, parabolic equation, parabolic inequalities, backward stability
Received by editor(s): February 7, 2003
Published electronically: December 22, 2003
Communicated by: David S. Tartakoff
Article copyright: © Copyright 2003 American Mathematical Society

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