September  2007, 19(3): 515-529. doi: 10.3934/dcds.2007.19.515

On unique continuation for the modified Euler-Poisson equations

1. 

Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, United States

2. 

Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556-5683, United States

3. 

Department of Mathematics, University of New Orleans, New Orleans, LA 70148, United States

Received  May 2007 Revised  June 2007 Published  July 2007

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 $s>5/2$.
Citation: A. Alexandrou Himonas, Gerard Misiołek, Feride Tiǧlay. On unique continuation for the modified Euler-Poisson equations. Discrete & Continuous Dynamical Systems - A, 2007, 19 (3) : 515-529. doi: 10.3934/dcds.2007.19.515
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