On unique continuation  for the modified Euler-Poisson equations

A. Alexandrou Himonas; Gerard Misiołek; Feride Tiǧlay

doi:10.3934/dcds.2007.19.515

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2007, Volume 19, Issue 3: 515-529. Doi: 10.3934/dcds.2007.19.515

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On unique continuation for the modified Euler-Poisson equations

1.
Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556
2.
Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556-5683
3.
Department of Mathematics, University of New Orleans, New Orleans, LA 70148

Received: April 30, 2007

Revised: May 31, 2007

Published: August 2007

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Abstract

It is shown that if a classical solution $(u, n)$ of the modified Euler-Poisson equation (mEP) in one space dimension is such that $u$, $u_x$ and $n$ are initially decaying exponentially and for some later time the first component $u$ is also decaying exponentially, then $n$ must be identically equal to zero and $u$ must be a solution to the Burgers equation. In particular, if $n$ and $u$ are initially compactly supported then $n$ can not be compactly supported at any later time, unless $n$ is identically equal to zero and $u$ is a solution to the Burgers equation. It is also shown that the mEP equations are locally well-posed in $H^s \times H^{s-1}$ for $s>5/2$.

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Mathematics Subject Classification: Primary: 35Q53.

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